Journal of Mathematical Sciences, Vol. 78, No. 6, 1996
BASIC STRUCTURES
OF F U Z Z Y T O P O L O G Y
A. P. Sostak
UDC 515.12
Introduction
One can hardly find many mathematical articles which would be cited as often and in papers of quite
different natures and directions, as L. A. Zadeh's article [210] published in 1965, in which the concept
of a fuzzy set was introduced. Judging by the number and the diversity of authors who use fuzzy sets
with enthusiasm in their work or who have chosen fuzzy sets themselves or mathematical structures based
on them as the subject of investigation, the concept of a fuzzy set turned out to be a very timely and
perspective one. Indirect evidence of its timeliness is given by the fact that practically simultaneously with
L. A. Zadeh, and certainly independently of him, V. N. Salij published his paper [5] in which the notion of
an/:-set was introduced; this notion contains in itself the concept of a fuzzy set and anticipates the notion
of an L-fuzzy set introduced in 1968 by J. A. Goguen [51]. (Unfortunately, V. N. Salij's paper remained
hardly known to specialists working in this area and all the credits of defining the concept of a fuzzy set
are almost exclusively given to L. A. Zadeh and J. A. Goguen.)
Among the first fields of mathematics to be considered in the context of fuzzy sets was general topology.
Aslearly as 1968 C. L. Chang published his paper [28] in which the first definition of fuzzy topology (1.1.1)
was proposed; t h e paper also presented an attempt to develop the foundations of the theory of fuzzy
topological spaces.
C. L. Chang's paper was followed by other works, whose authors either considered different problems
concerning Chang fuzzy topological spaces or proposed alternative definitions of fuzzy topology and developed their own viewpoints on what, in their opinion, is to be the subject of fuzzy topology as an area of
research.
In subsequent years the number of publications in fuzzy topology grew like an avalanche and at present
there exist about a thousand publications of quite different value and originality in which the subject of
fuzzy topology is being addressed.
The large number of existing publications, the diversity of directions, and the ambiguity of terminology
are among the obstacles causing serious problems for a mathematician who is interested in fuzzy topology
but does not constantly follow the whole mass of publications in this area. Unfortunately, there are quite a
few works in which attempts to include the whole "tree of fuzzy topology" would be undertaken. One such
work is our survey [180] published in 1989. However, in this survey the main emphasis was on a discussion
of concrete topological properties of fuzzy topological spaces, while general structures of fuzzy topology
were considered to a significantly smaller degree. Besides, in the years that have passed after writing [180]
the situation in the rapidly developing area of fuzzy topology has substantially changed. Therefore, we
assume that it is quite timely to publish a new survey article.
The main purposes of the present work are the following:
To discuss different approaches to fuzzy topology and the corresponding categories of fuzzy topology.
To trace interrelations between different approaches to fuzzy topology.
To consider some important constructions in the discussed categories of fuzzy topology.
To trace, as far as possible, each one of the directions in fuzzy topology in the unity of the three
basic structures of topological type : topologies, uniformities and proximities, and also in connection with
corresponding syntopogeneous structures.
To trace the principal analogies and deviations in the discussed categories of fuzzy topology if compared
with the situation in general topology.
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i ee Prilozheniya. Tematicheskiye Obzory.
Vol. 14, Topologiya-2, 1994
662
1072-3374/96/7806-0662515.00
9
Plenum Publishing Corporation
To help the m a t h e m a t i c i a n who is not experienced enough in this area not to losse himself in tile
labyrinth of the various directions, branches, and offshoots of fuzzy topology and to understand the logic
and tendencies of the development both of each separate direction and of fuzzy topology as a whole.
A short-coming of this survey is that we say very little about specific topological properties of fuzzy
topological spaces. Moreover, when mentioning one or another topological property we usually do not
give its precise definition, but just confine ourselves to the corresponding references. The main reason for
doing so is that concrete topological properties usually have many different (and sometimes quite distinctive)
extensions in fuzzy topology and to discuss them, even superficially, we would need to increase the volume of
this work substantially. On the other hand, as was already mentioned, some concrete topological properties
of fuzzy spaceswere to a certain degree considered in the survey [180], which, together with the original
papers cited in the text, can be used by the reader for making inquiries.
The pri.ncipal a m o u n t of the work in preparing this survey was done from September to December, 1993,
during the author's stay at R h o d e s University of Grahamstown, South Africa. The author expresses his
gratitude to the staff of the Department of Mathematics (pure and applied) and especially to Prof. W. Kotz6
for excellent working conditions. The author is thankful also to the Hugh Kelly fund for the financial support
of this visit.
Before passing to the main contents of the work, we shall list here definitions and facts from lattice
theory and the theory of fuzzy sets which will be constantly used in the main text.
0.1. L a t t i c e s
Lattices will constantly appear in the main text. Although we use the standard terminology accepted in
lattice theory (see e.g., [21, 50, 55]), for the convenience of the reader the basic definitions are listed below.
We shall furthermore make some important assumptions.
All lattices are assumed to be bounded with a smallest element 0 and a largest element 1.
0.1.1. C o m p l e t e L a t t i c e s . A lattice L ( = (L, _<)) is called complete, if each Subset D C L has the joi.zt
(the supremum) : V D E L. (By the duality principle this is equivalent to the requirement that each D C L
has the meet (the infinum): VD E L.)
0.1.2. D i s t r i b u t i v e L a t t i c e s . A lattice is called distributive if (a A b) V (a A c ) = a A (b V c), or
equivalently, (a V b) A (a V c) = a V (b A c) for any a, b, c E L. A lattice is said to be infinitely distrib~ztive,
or a frame, if a A (vbx) = V(a A bx) for any a E L, {bx: A E A} C L. A complete lattice is called completely
distributive, if
nK }
AEA
for any family {a,~x: A E A, x E Kx} C L.
0.1.3. A mapping c: L --4 L is called an involution, if (aC) c = a for every a E L. An involution is said
to be order reversing if a _< b implies bc _< a c.
A s s u m p t i o n . All lattices considered in this paper are assumed to be complete. (This assumption is quite
inevitable as soon as we have to deal with properties Of topological type structures such as the arbitrary
union axiom.)
All lattices are assumed to be infinitely distributive.
(This assumption is not as crucial as the assumption of completeness, and some parts of the theory can
be developed in the context of lattices more general than frames. However, we need infinite distributivity if
we wish to have a procedure allowing us to generate a topological type structure from some family of fuzzy
sets as from a subbase, i.e., by taking arbitrary joins (unions) of finite meets (intersections) of members of
this family. This procedure, explicitly or implicitly, is employed in m a n y places of the theory.)
Usually lattices are assumed to be equipped with an order reversing involution c: L -4 L.
(This assumption is needed as soon as we have to consider complements of fuzzy sets. Such anecessity
mainly occurs when we have to deal with the concept of closedness together with the concept of openness:
A reader will notice for himself when such a necessity appears.)
All other restrictions on the lattice L will be stated explicitly when needed.
In particular, a large part of the theories of fuzzy uniformities and fuzzy proximities is completely
distributive.
663
0.1.4. A lattice L is called a chain, if for any a, b E L either a _< b or b < a. Each chain is completely
distributive. A chain is called dense if given a, b E L, a < b, a # b, there exists c E L : a < c < b, a # c, b # c.
0.1.5. W a y - B e l o w R e l a t i o n . On each complete lattice a new transitive relation << is defined as follows :
a << b (in words : a is way below b) whenever for any D C L with b < VD there exists a finite subset E C D
such that a < VE. The relation a << b implies a < b; 0 << a holds always. We also mention the following
two important properties of the way-below relation : if c < a << b <_ d, then c << d; if a << c, b << c, then
aVb<< c.
Note that in the unit interval (0, 1] the way-below relation becomes the strictly less-than relation. Also
note that in a complete lattice a << b may never hold unless a = 0.
0.1.6. C o p r i m e E l e m e n t s . An element q E L is called coprime if q < aV b implies q _< a or q <_ b. Let
C P ( L ) denote the set of all coprime elements of L.
0.1.7. N o t a t i o n . Throughout the paper we use 2 to denote the two-point lattice {0, 1} and I to denote
the interval [0, 1] equipped with the standard order < and with involution c : I --+ I defined by a c = 1 - a .
Given a lattice L, let L + := L \ {0}, L - := L \ {1}. If a E L, then let L ~ := L \ i x : x < a},
L~:={xeL: z > a , z # a } .
0.2. F u z z y S e t s
We use terminology which is standard in the theory of fuzzy sets and its applications. See, e.g., [210,
5, 51]. However, for the reader's convenience we reproduce below some of the most important notions
connected with fuzzy sets. Everywhere in the sequel L stands for an arbitrary lattice (see Sec: 0.1).
0.2.1. L - F u z z y S e t s : B a s i c D e f i n i t i o n . Given a set X , by an L-fuzzy (sub)set of a set X we mean a
mapping M : X --+ L. The value M(x) is interpreted as the degree to which a point x E X "belongs" to the
fuzzy set M. A usual (crisp) subset A C X is identified with its characteristic function A = XA : X --+ 2 C L
and thus can be considered as a particular case of an L-fuzzy set. Moreover, crisp subsets of X can be
viewed as 2-fuzzy Subsets of X.
The family of all L-fuzzy subsets of X is denoted L X (so called L-fuzzy power set of X); in particular
2 z will stand for the family of all crisp subsets of X.
0.2.2. O p e r a t i o n s o n L - F u z z y S e t s . Given a family of L-fuzzy subsets of a set X, A4 = {Mj : j E J}
(C L X) its union (join), and its intersection (meet) are defined pointwise, i.e., V,4A = V{l~/lj: j e J} and
A M = A { M / : j E J} are given by the formulas V.Ad(X) = V Mj(x) and AM(x) = A Mj(x) respectively.
jeJ
j~J
In the case where L is equipped with an involution, the complement of an L-fuzzy set M E L X is M c E L x,
defined as M~(x) = (M(x)) ~, x E X.
Given a family of sets {Xj: j E J} and L-fuzzy sets Mj E L xj , the product M = I1 Mj is an L-fuzzy
jEJ
subset of the set X =
1"I X j defined by the equality M ( z ) =
jeJ
A Mj(xj), where zj stands for the j t h
jeJ
coordinate of the point x E X.
It is easy to note that, in the case L = 2, these definitions reduce, respectively, to the usual definitions
of the union, intersection, complement, and product for sets. Besides, the behavior of the introduced
operations is quite analogous to the behavior of their crisp prototypes. In particular, an analogy of the
de Morgan laws hold for them: (vAi) c = AA~, etc. Note, however, that the "complement" introduced
1
J
above is not a complement in the precise sense of lattice theory since the equalities A V A c -- 1 and A A A c = 0
generally do not hold.
Along with the definitions of operations on fuzzy sets presented above, sometimes in "Fuzzy Mathematics" and especially in its applications, other definitions are used. For example, in the case L -- I, the union
and the intersection of A, B E I X are sometimes defined as rnin{A + B, 1} and A. B, respectively. However,
in this work; as in an overwhelming majority of papers on the subject of fuzzy topology, the operations on
fuzzy sets are always realized as defined above.
0.2.3. P r e - I m a g e s a n d I m a g e s o f L - F u z z y S e t s . Let X, Y be two sets and f X -4 Y be a mapping.
Then the pre-image f-I(N) E L X of an L-fuzzy set N E L r is defined by the equality f-~(N)(x) =
(N o f ) ( x ) ( = i ( f ( z ) ) ) , and the image f ( M ) E L Y of an L-fuzzy set M E i X is defined by f ( M ) ( y ) =
664
s u p { ] / ( x ) : x e f - l ( y ) } (the suprenmm of an empty set is 0). In the case L = 2. these definitions are
equivalent to the usual (i.e., crisp) ones.
The properties of images and pre-images of L-fuzzy sets are quite similar to the properties of images and
pre-images of ordinary sets. For example: f - I ( v N j ) = y f - l ( N j ) ' etc.
1
1
0.2.4. A F u z z y P o i n t a n d t h e " B e l o n g i n g R e l a t i o n " [158, 171]. An L-fuzzy point in a set X is an
L-fuzzy set p := pt~o : X --+ L, where x0 E X and t E L + , defined by the equality p(zo) = t and p(x) = 0 if
X ~ X
0 .
In this case, x0 is called the support of p, and t is called its value. A usual point x0 is interpreted as the Lfuzzy point Pzo"
1 An L-fuzzy point p belongs to an L-fuzzy set M (in symbols pEM) if M(xo) > t. Together
with the relation E it is sometimes convenient to consider the relation of so-called quasi-coincidence or
q-coincidence: A fuzzy point ptzo is said to q-coincide with a fuzzy set M, if M(xo) ~ t c. (In the case L = I,
this just means that M(xo) + t > 1; in the case L = 2, both relations obviously reduce to the ordinary
belonging relation.)
The union and intersection of a family of L-fuzzy sets {Mj: j E J} behave with respect to E and q
in some instances like that of the ordinary belonging relation in ordinary (crisp) set theory, but in others
not. In particular, pEA. Mj iff pEMj for all j; ifpqAMj, then pqMj for every j, but the converse does not
J
generally hold. On the other hand, if there exists j for which pEMj, then pE V ~/Ij, but the converse does
not hold; hile pq(VMj) iff pqM for some j .
J
A s s u m p t i o n . In the sequel, when speaking about L-fuzzy sets and L-fuzzy points, we shall sometimes
omit the prefix L (in the cases when this will not lead to ambiguity).
I. F U Z Z Y T O P O L O G I C A L
STRUCTURES
1.1. C h a n g - G o g u e n L - F u z z y Topologies
1.1.1. B a s i c I d e a s a n d D e f i n i t i o n s : S o m e E l e m e n t a r y F a c t s . Historically, the first attempt to
develop the fuzzy counterpart of general topol9gy was undertaken by C. L. Chang in 1968128]. According
to C. L. Chang a fuzzy topology on a set X is a family r of fuzzy subsets (i.e., r C IX) ` which satisfies the
following three axioms :
(IT) 0, 1 E r;
(2T) ifU, V E r , t h e n U A V E r ;
(3T) i f { U j : j E J } C r ,
thenvUjEr.
J
The pair (X, r) is called a fuzzy topological space and the fuzzy sets belonging to r are called open in this
space.
Soon J. AI Goguen [52] proposed a natural generalization of Chang's definition by substituting L-fuzzy
sets for fuzzy sets. Namely, according to 3. A. Goguen an L-fuzzy topology on X is a family r of L-fuzzy
subsets (i.e., r C L X) which satisfies the axioms (1T)-(3T) above; the pair (X, r) is called an L-fuzzy
topological space and L-fuzzy sets U E r are called open in this space. Here L can be any bounded complete
lattice (see 0.1). (In fact, J. A. Goguen puts some additional requirements on the lattice L but we omit
them here since they are nonessential for us at the moment.)
If (X, rx) and (Y, ry) are L-fuzzy topological spaces, and f : X --+ Y is a mapping, then f is called
continuous, if V E ~ implies f - l ( V ) E rx. The category of L-fuzzy topological spaces (for a fixed lattice
L) and continuous mapping will be denoted C F T ( L ) and called the category of Chang-Goguen L-fuzzy
topological spaces.
(In this and the next section we shall usually omit the words "Chang-Goguen" when speaking about
such spaces. On the other hand, in the forthcoming sections, where Chang-Goguen L-fuzzy topological
spaces appear only occasionally, the specifying words "Chang-Goguen" are of importance and therefore
will not be omitted.)
In the special case L = I, the category C F T ( I ) is the category of fuzzy topological spaces as they were
defined in [28] and C F T ( 2 ) is, in fact, just the category T o p of ordinary topological spaces.
665
Although the concept of an L-fuzzy topological space is reasonable for any complete b o u n d e d lattice L,
to develop a substantial theory, one often has to put some additional requirements on L. Among the most
important and often used ones are the requirement of (infinite) distributivity of the lattice L and that L
be equipped with an order reversing involution.
The assumption of an order reversing involution enables us, first of all, to give a reasonable definition
for closedness and some related notions. Namely, an L-fuzzy set M in an L-fuzzy space is called closed, if
its complement M c is open. From the general properties of the lattice L it is clear that the family c~ of all
closed L-fuzzy subsets of a given L-fuzzy space has the following properties (see, e.g., [200]) :
(1C) 0,1 6 o ;
(2C) if A, B 6 cr, then A V B 6 cr;
(3C) i f { A j : j 6 J } C o ,
thenAA i6~r.
J
The concept of closedness naturally leads to the definition of the closure operator on an L-fuzzy space.
Namely, given an L-fuzzy set M in an L-fuzzy topological space (X, r), its closure is defined as M =
A{N : N > M, N 6 a}. One can easily see that the closure operator on an L-fuzzy space can be treated
in many aspects in the same way as the closure operator on an ordinary topological space. In particular,
is the smallest (in the sense of the order <) of all L-fuzzy sets containing (_>) the given L-fuzzy set M.
As in general topology, the following list of properties of the closure operator - : L x -+ L x can be also
used to define an L-fuzzy topology on the set X (see, e.g., [200, 52]):
( l C l ) 0 = 0;
(2C1) M V N = M V N ;
( 3 C I ) M = M;
(4C1) M < M
(M and N are arbitrary L-fuzzy sets in X).
The concepts of closedness and closure operator, as well as that of the interior operator Int: L x --+ L x,
defined in the natural way, can be used to characterize the continuity of mappings of L-fuzzy spaces. Namely,
the following four properties are equivalent for a mapping f : X --+ Y (see, e.g., [200]).
(a) f is continuous;
(b) i f Y 6 ~y, then f - l ( g ) 6 ax;
(c) f ( A ) < f(A) for each A 6 Lx;
(d) f - 1 (Int B) _< Int f - 1 (B) for each B e L w.
1.1.2. B a s i c O p e r a t i o n s . To give a reader insight into how operations on L-fuzzy spaces can be defined,
we shall describe here the operations of taking a subspace, the product, and direct sum (the co-product).
To start with, assume that an L-fuzzy space (X, r) is given, and Y is a crisp subset (i.e., II C X). The
induced L-fuzzy topology on Y is defined as Ty = {Uy = U ]y: U 6 r}, where U ]r denotes the restriction
of U to the set Y (see, e.g., [46, 171]). It is easy to verify that the natural inclusion map i: (I/, Ty) --+ (X, r)
is continuous, in this case, and, moreover, Ty Can be characterized as the weakest (in the sense of C) L-fuzzy
topology on Y for which the inclusion i is continuous.
From this it follows that the subspace (Y, vy) of the space (X, v) is indeed a subobject of (X, v) in the
category C F T ( L ) . (Some authors also consider the operation of taking a subspace in case of an L-fuzzy
subset Y 6 L z (see, e.g., [41, 45, 160, 165], etc.). We however shall not touch on this problem here. Let us
note only, that to define such an operation one has to go out from the category C F T ( L ) into more general
categories (see, e.g., Sec. 1.6).)
Passing now to the operation of product, consider a family of L-fuzzy topological spaces { (X~, v~) : A 6 A }
and let X = 1-IX~ be the product of the corresponding sets and p~ : X --+ X~ d e n o t e the corresponding
A
projection. L e t U P = { V = p ; I ( u ~ ) :
U~6rx, A6A} andB={V~,A...AV~,:
n 6 N , V~, 6 ~ } , i . e . , S
is the family of all finite meets of elements from :P. The product L-fuzzy topology v on X can be defined
now as the family of all joins of elements from B, i.e., r = {W = vvJ: {Yd: j 6 J} C B}. (Extending
J
the standard terminology from general topology to the fuzzy case, one can say that 7~ is a subbase and 6
is a base for the product L-fuzzy topology r.)
666
From the definition of the product L-fuzzy topology it is clear that it can be characterized also as the
weakest fuzzy topology on X, for which all projections px are continuous, or, in other words, 7" is the initial
L-fuzzy topology for the family of all projections. It can be easily seen now that (X, 7") is indeed the product
of the family { ( X x , r x ) : i 6 A} in the category e F T ( L ) , (See, e.g., [52, 205, 200].)
It is important to emphasize that in some aspects the product in Categories e F T ( L ) may essentially
differ from the product in T o p . To begin with, notice, that the projections in the product are not necessarily
open, i.e., U E r does not generally imply pa(U) E rx [159]. (This can be illustrated by the following simple
example: Let X be a set and let 7"1 = {0, 1} and r2 = {0, c, 1}, where c is a constant, different from 0 and 1.
Then, obviously, the constant c is open also in the product (X, rl) x (X, r2), but its image c = pl(c) is not
open in the space (X, rl).) As a consequence of this, the product of two L-fuzzy spaces generally does not
contain a subspace homeomorphic to a factor. (For more about this problem see Sec. 1.2.)
Another deviation from general topology : The formula I'IAx = i-lAx generally does not hold for L-fuzzy
X
X
subsets Ax of L-fuzzy t0pological spaces (Xx, rx), A 6 A. This and some related questions were thoroughly
studied in [136, 137], although in the special case L = I and when the number of factors is finite.
In particular, it is shown in [136] that, given two I-fuzzy spaces ( X I , r l ) and (X2,7"2), the equality
A1 x As = A1 x A2 holds for any fuzzy subsets Ai E I Xi, i = 1,2, iff the following three conditions are
fulfilled :
(1) the fuzzy topologies rl and 7-2 have the same constants, i.e., rt A I = 7-2 f~ I;
(2) the closure of each constant in (Xi, ri), i = 1, 2, is a constant;
(3) M A a = M A o~ for each M 6 IX~,i = 1,2, and each constant a E I.
To define the direct sum of a family of L-fuzzy spaces { (Xx, rx) : i 6 A} consider first the disjoint union
of the corresponding sets X = @Xx, and let r be the L-fuzzy topology on X formed by arbitrary meets of
L-fuzzy sets belonging to B =U{§
A 6 A}, where ~a = {~r E LX: U [x 6 rx and U(z) = 0 if z ~ Xa}.
Now the direct sum can be defined as the pair (X, r) (cf [48]). Obviously, 7- can be characterized also as the
strongest L-fuzzy topology on X, for which all natural embeddings ix: Xx -+ X are continuous (i.e., 7- is
the final L-fuzzy topology for the family of all embeddings) and, therefore, (X, 7-)is indeed the coproduct
of the given family of L-fuzzy spaces in e F T ( L ) .
The Operation of taking a quotient of an L-fuzzy space was studied and used, in particular, in [205, 29,
159, z851.
The general m e t h o d of construction of initial and final L-fuzzy topologies for families of mappings was
probably first worked out by R. Lowen [103]. In fact, R. Lowen considers only laminated (See Sec. 1.2)
I-fuzzy topologies, but his method can easily be extended to the general case of L-fuzzy topologies.
1.1.3. C o n n e c t i o n B e t w e e n C F T ( L ) a n d T o p : S o m e I m p o r t a n t F u n c t o r s . Since fuzzy topology
is a certain extension of general topology, there arises the natural problem of establishing and investigating
the principal interrelations between the category T o p of topological spaces and categories C F T ( L ) (as well
as other categories of fuzzy topology).
To begin with, notice that T o p can be identified with the category C F T ( 2 ) . Furthermore, for any lattice
L, a topological space (X, T) can be viewed as an L-fuzzy topological space just by interpreting sets U C X
as characteristic functions U: X 2+ 2 C L.
Thus, for each lattice L the category Top, in a natural way, can be considered as a (full) subcategory of
the category e F T ( L ) .
Of the functors going in the opposite direction, i.e., from C F T ( L ) into T o p , we first mention here the
iota-functors ~, where ~ e L - (0.1.7) and~ (see [107, 108, 109] (in the case L = I), [166] (in the case where
L is a chain) [102] (in the general case)). The so-called
-te,etf nctor
CFT(L) -+ Top assigns to an L-
fuzzy topological space (X, v) the topological space (X, ~a(~-)), where ~c,(v) is the topology (the s.c. c~-levet
topology) generated by the subbase ~ra(v) = { U - I ( L a ) : U E r} (0.1.7) (in the case where L is a chain,
~ ( r ) = rra(v)); and the functor L: C F T ( L ) -+ T o p assigns to an L-fuzzy space ( X , T ) t h e topological
space (X,~(r)), where ~(r) is the topology generated by the subbase ~(r) = {U-~(L~): U S ~-, c~ e L - } .
In the case where L is a chain, ~(r) = rr(7") and it can be characterized as the weakest topology on X
for which all U E 7- become lower semicontinuous when L is endowed with the order topology. Since ~he
667
continuity of a mapping of L-fuzzy spaces f : (X, 7-x) --+ (Y, wY) guarantees the continuity of mapping
the corresponding topological spaces: f : (X, co(Tx)) ~ (Y,~(7-y)) and f : (X,*(Tx)) ~ (Y,~(rr)), the
functors
and are naturally defined as identities on morphisms, i.e., c~(f) = ,(f) = f.
The functors ca are useful when studying the so called a-level properties of L-fuzzy spaces (see, e.g., [90,
160, 163], etc.) P. Wuyts [208] finds a criterion for a given family of topologies 2 = {T~; a E L} on a set
X to have a fuzzy topology 7- on X with 2 as the family of its a-level topologies, i.e., T~ = L~(r) for each
a ~ L. It is shown [208] that the set F(TI) of all such fuzzy topologies (in case where it is not empty) has a
maximum, but generally does not have minimum. In F(2) at most one fuzzy topology, namely the maximal
one, can be generated by a fuzzy neighborhood system (in the R. Lowen sense [112]); P. Wuyts also gives
the conditions on the family 2 when it is really the case.
In [91] a related problem of generating L-fuzzy topologies by means of families of semi-closure operators
is studied.
The functors ca and t allow us to solve some problems of fuzzy topology by reducing t h e m to the standard
situation in general topology.
Note that all these functors are obviously not embeddings (except for the trivial situation L = 2, where
all of them become identities). In case where L is a chain, the iota functors commute with products [102].
Quite a different type of functor is the so-called hypergraph functor which was introduced by E.S. Santos
(in preprints) and independently by R. Lowen [107, 108] (in the case L = I). Later these functors were
used for investigating different properties of fuzzy topological spaces see, e.g., [160, 163].
Let (X, 7") be an L-fuzzy topological space and let G(7") denote the topology on the product X x L - defined
by the subbase {{(x,a): U(x) > a: x E X,a E L-):
U E 7"} and let G(X, 7")= (X x L-,G(7")). For
every continuous mapping f: (X, 7"x)-+ (Y,'ny)the mapping G(f): G(X, 7-x) -+ G(Y, zy), defined by the
equality G(f)(x, a) = (f(x),a), is continuous and hence G can be considered as a functor G: C F T ( L ) --+
Top. In case where L is a chain, this functor is an embedding.
Interesting connections between the hypergraph functor and the iota functor were established by
A.J. Klein [92] in the case where L is a chain. In particular, he has shown that G(7-) C c(7-) x Tl, where
Tz is the topology on L- consisting of all half open intervals [0,a), where a E L. Besides, the equality
G(7-) = ~(7-) x Tz is valid iff the space (X, 7-) is topologically generated (1.2.2).
In a recent paper by W. Kotze and T. Kubiak [98] a modified version of the hypergraph functor is
investigated; in its definition the relation ~ (instead of the relation >) is involved.
1.1.4. L o c a l S t r u c t u r e o f C h a n g - G o g u e n L - F u z z y S p a c e s . In the literature, one comes across
different viewpoints of the concept of the local structure of a fuzzy space. Some of these differ in a nonessential way. Of the works where this problem is considered in the context L = I, one can mention papers
by R.H. Warren [200, 201] (his approach is based on the use of fuzzy neighborhoods of usual points) and
papers [206, 158, 32, 48, 88, 89, 49] (all these authors work with fuzzy type structures of fuzzy points and
their approaches differ only in the interpretation of the corresponding notions). A comparative analysis of
some of the approaches of the last group was worked out by E.E. Kerre and P.L. Ottoy [88, 89]. To give
the reader an idea about a possible local description of a fuzzy space we present here a brief outline of the
approach developed by P u Paoming and Liu Yingming [158, 159]. A fuzzy set M is called a neighborhood
(a q-neighborhood) of a fuzzy point p in a fuzzy space (X, 7-) if pE Int M (resp., if pq Int M). Let Alp denote
the family of all neighborhoods (resp. q-neighborhoods) of a fuzzy point p in (X, 7"). T h e n :
(1) U E A/'p implies p~U (resp. U E A/'p implies pqU);
(2) ifU, V E.hfp, then U A r E .h/'p;
(3) if U E Alp and U < V, then V E Alp;
(4) for each U E .h/'p there exists Y E A/'p such that Y _< U and V E A/r, for each fuzzy point r'EV (resp.
raY).
Conversely, let to each fuzzy point p in a set X a family 2r of fuzzy sets be assigned in such a way that
conditions (1)-(4) are fulfilled. Then the family 7r o f fuzzy sets U E I x such that U E A/'p whenever pEU
(resp. whenever pqU) is a base of a fuzzy topology 7" (resp. is a fuzzy topology 7" = ~-) on X. Besides, A/'p is
exactly the neighborhood system (resp. the q-neighborhood system) of a fuzzy point p in the space (X. r).
To characterize laminated fuzzy spaces (see Sec. 1.2) one has to supplement conditions (1)-(4) with the
668
following additional axiom:
(5) If the value of a fuzzy point p is t, then the constant t (resp. all constants s > t c) belong to A~.
The local structure of L-fuzzy spaces for lattices L more general than I is studied to a significantly
lesser extent. Of the authors working in this area, one should mention, first of all, the Chinese authors
Wang Guojun [194 :- 196], Zhao Dongsheng [212], Zhao Xiaodong [213], Meng Guangwu [142] and others
(see also Sec. 1.5). An important peculiarity of these researches is the use of so called remote neighborhoo&.
or R-neighborhoods (instead of usual neighborhoods and q-neighborhoods) in the definition of which the
relation "~" is involved. Besides, an appreciable contribution to the understanding of the local structure
of L-fuzzy spaces was made by S.E. Rodabaugh (see [167, 168] et al.) and M.W. Warner [198, 199].
The local structure of L-fuzzy spaces was substantially used (mostly in case L = I) when studying lower
separation properties and some related problems of fuzzy topology (see the works by Pu Paoming and
Liu Yingming [158], A. Srivastava, S.N. Lal, R.K. Srivastava [190, 191,192], M. Sarkar [170, 171], W. Kotz6
[94, 95], D. Adnadjevid [12], A.P. Sostak [9] and others.)
1.1.5. C o n v e r g e n c e S t r u c t u r e of C h a n g - G o g u e n L-Fuzzy Spaces. The convergence theory in
fuzzy topology (mostly, in the context L = I) is being developed in two parallel and essentially equivalent
forms.
The first one of the convergence theories is based on the concept of a fuzzy net introduced by Pu Paoming
and Liu Yingming [158]; in the same article the foundations of the convergence theory on the basis off~zzy
nets were worked out. Recently M.A. de Prada Vicente and M. Macho Stadler [129] presented another
version of fuzzy net convergence theory which appeals to the neighborhoods of fuzzy points (instead of
their q-neighborhoods, as is done in [158]).
Zhao Dongsheng [212] has extended certain parts of the fuzzy net convergence theory to the case of
L-fuzzy topological spaces where L is a completely distributive lattice, and used it to study compactness
type properties of L-fuzzy topologies.
The second theory is based on the concept of a prefilter [110] or a fuzzy filter [73]. This theory was
basically developed by R. Lowen in [110], etc., for laminated I-fuzzy topological spaces (see, 1.2) and later
was extended by R.H. Warren [202] to the case of arbitrary I-fuzzy topological spaces. A certain contribution
to the convergence theory based on prefilters was done by A.K. Katsaras [73] and also by M. Macho Stadler
and M.A. de Prada Vicente, see, e.g., [130, 131, 156, 157]. Prefilters were successfully used in the study of
such properties as compactness and separation axioms (see, e.g., [111, 124, 125, 209, 27] et aI.). Besides,
they play a crucial role in the theory of (Lowen) fuzzy uniformities (Sec. 2.2).
A comparative analysis of the net convergence theory and the prefilter convergence theory is given
in [119].
The a-level approach (cf. 1.1.3) to the study of the convergence structure of fuzzy spaces, both in the
form of fuzzy nets, and in the form of fuzzy filters, was recently developed by M.A. de Prada Vicente and
M. Macho Stadler [128, 130].
Fuzzy convergence spaces, in the spirit of H.R. Fisher's "limesriiume" [44], were introduced by Kyung
Chan Min [144]. The category F l i m of fuzzy convergence spaces is shown to contain, in a natural way, the
category CFT(L) as a bireflective and Fischer's category Lim as a bicorefiective subcategory. One important advantage of the category Fllm, if compared with the categories CFT(L), is its cartesian closedness
(in H. Herrlich's sense [58, 59]). A similar category FCS of fuzzy convergence spaces was considered also
by R. Lowen at aI. [!06].
A very genera/convergence theory based on the concept of a fuzzy filter functor, is developed by P. Eklund
and W. G~ihler [40].
We shall not go into any details in connection with the convergence theory or give precise definitions
or formulations : the interested reader is referred either to the original papers mentioned above, or to our
survey [180] where the theories of R. Lowen [110] and of Pu Paoming and Liu Yingming [158] are discussed
more explicitly.
1.1.6. An E x a m p l e : T h e L-Fuzzy Real Line. In 1974 B. Hutton [67] constructed the L-fuzzy
unit interval I(L), and 4 years later, T.E. Gafltner, R.C. Steinlage, and R.H. Warren [46], by developing
B. Hutton's ideas, defined the L-fuzzy real line ~(L). The fundamental significance of ~(L) for fuzzy
669
topology, and for the whole "fuzzy set mathematics" stems from its topological and algebraic properties,
which enable it to be considered as the fuzzy analogue of the ordinary real line (see, e.g., [67, 46, 161-165,
169, 114, 116-118] etc.). The universal and categorical nature of its construction is also very important
(see, e.g., [163, 165]).
To define the L-fuzzy real line consider the set Z(R, L) of all nonincreasing maps z : ]R --+ L such that
supz(x) = 1, infz(x) = 0, and let the equivalence relation -.~ on Z(R, L) be introduced by setting zl "~ z2,
iff
=
and
=
for all 9 e R, where z ( , + ) = s u p z ( t ) , z ( x - ) = inf z(t). N o w the
t>x
t<x
L-fuzzy real line R(L) can be defined as the quotient of Z(R, L) by the relation ,-,.
The fuzzy topology r on IR(L) is defined by the subbase {lb,ra: b,a E IR}, where fuzzy sets lb,ra:
R(L) ~ L are defined by the equalities lb[z] = z(b-) e, r~[z] = z(a +), respectively. ([z] naturally denotes
the equivalence class of the element z E Z(]R, L).) Sometimes together with the standard L-fuzzy real line
(R(L), ~-), the so called stratified or laminated L-fuzzy real line (R(L), r x) is used; its L-fuzzy topology 7-x is
obtained by saturation of the fuzzy topology 7" with all constants c: R(L) --+ L (in this connection, see also
Sec. 1.2). Note that in the case L = 2, both (R(L), v) and (R(L), v x) in an obvious way can be identified
with the ordinary real line R.
The subspaces of JR(L) defined by (a, b)(L) = {[z]: z E Z(R, L), z(a +) = 1, z(b-) = 0} and [a, hi(L) =
{[z]: z E Z(R,L),z(a-) = 1, z(b +) = 0}, where a,b E IR, a < b, are called, respectively, open and closed
L-fuzzy intervals. In particular, [0, 1](L) is the Hutton L-fuzzy unit interval [67].
By identifying a number a E R with the characteristic function of the set ( - c o , a), we can consider IR as
a subset of I~(L). Moreover, the fuzzy topology r induces on R the ordinary (order) topology 7"< (and r x
induces the L-fuzzy topology wT<; see 1.2.1).
The problem of extending continuous functions from closed subspaces of L-fuzzy spaces with values in
I(L) and in IR(L) (fuzzy versions of the Urysohn lemma and the Tietze-Urysohn Theorem) were thoroughly
studied by T. Kubiak [99-101]; see also [97].
(Note that a construction similar to R(L) was independently proposed by U. H6hle [61].)
1.1.7. O n S o m e G e n e r a l i z a t i o n s of t h e C o n s t r u c t i o n R --+ R(L). Noting the fundamental role
which the. L-fuzzy real line plays in fuzzy topology, S.E. Rodabaugh set the problem to extend its construction to the construction X --+ X(L) for the case of spaces X more general than the real line lt~ (see,
e.g., [162]). Certainly, it is natural to demand that X(L) would contain, in some canonical way, the original
space X, and that X(L) would play a role in fuzzy topology, which is, in a certain sense, analogous to the
role of the space X in general topology. This problem was solved, by different methods, for three special
classes of topological spaces.
A.J. Klein [93] worked out a construction X --+ X ( L ) in the case where X is connected. T h e elements of
X(L) are certain equivalence classes of the mappings u : X -+ L, all preimages of singletons under which
are connected.
R. Lowen [121, 122] proposed a construction X --+ M(X) which assigns to a separable metric space X
an I-fuzzy topological space M(X). The elements of M(X) are probability measures on the cr-algebra of
Borel sets of the space X.
Finally, in [188], a construction X --+ F(X) was defined. This construction assigns to a linearly ordered
topological space X an I-fuzzy topological space F ( X ) ; its' elements are certain equivalence classes of
nonincreasing mappings from X into I.
The constructions X --~ M(X) and X -+ F(X) are of a categorical nature [121,189].
Although the constructions considered are based on essentially different ideas, nevertheless, when restricted to the appropriate classes of spaces, they become close to one another, or even equivalent. In
particular, in the case where X is linearly ordered and connected, the constructions X ( I ) and F(X) are
equivalent [189] and in case a linearly ordered space X has a countable base, the construction F(X) becomes
analogous with (and if besides X does not have isolated points, equivalent to) the construction M(X) [10].
1.2. L a m i n a t e d C h a n g : G o g u e n L - F u z z y T o p o l o g i e s : C a t e g o r i e s L C F T ( L ) .
1.2.1. B a s i c I d e a s a n d D e f i n i t i o n s : S o m e Specific C h a r a c t e r s . When discussing Chang-Goguen
L-fuzzy topological spaces, in the previous section, we have already noticed some significant deviations from
670
the situation with ordinary topological spaces (see 1.1.2). But, probably the most striking deviation, at
least at first glance, is that constant mappings of L-fuzzy spaces are not necessarily continuous. One can
see this by the following simple example: Let X be a set and let 7"1 = {0, 1}, r2 = {0, c, 1}, where c is any
constant different from 0 and 1. Then there are no continuous mappings from (X, 7"1) into (X, v2) at all.
Hence, the set of morphisms between two L-fuzzy spaces can even be empty.
Viewing these and some other (see below) deviations in C F T ( L ) when compared with T o p as serious
drawbacks, t~. Lowen [107, 108, 109, 123] et al. proposes to strengthen the definition of an L-fuzzy topological space (1.1.1) by asking, additionally, that a fuzzy topology must contain all constants c: X -+ L.
(In fact, R. Lowen works in the context L = I, but this restriction is quite insignificant for us at the
moment.)
In the sequel, Chang-Goguen L-fuzzy spaces which satisfy the following strengthening of the axiom (IT) :
(1T x) 7- contains all constants c: X -+ L will be called laminated Chang-Goguen L-fuzzy topological spaces
or Lowen L-fuzzy topological spaces. (Some authors also call them fully stratified L-fuzzy spaces, see,
e.g., [159, 165].)
Laminated L-fuzzy spaces and continuous mappings between t h e m (and continuity is understood here
in the same way as in C F T ( L ) ) form a full subcategory of C F T ( L ) which will be denoted L C F T ( L ) .
The set of morphisms between any two laminated spaces i s n e v e r empty: at least, constant mappings
are, obviously, continuous. Among other advantages of laminated spaces t~. Lowen and P. Wuyts [1_93]
mention the following ones :
(1) The projection in the product of laminated L-fuzzy spaces is open.
(2) In product spaces "slices" are homeomorphic to the corresponding factors.
(3) As a consequence of 1 and 2, many results relating properties of the product space to som e properties
of the quotient space and which hold in Top, carry over to LCFT(L) (but generally not to CFT(L)).
(4) Fuzzy topological structures, compatible with.a vector space structure [72] or with a group structure [45] are translation invariant in L C F T ( L ) but generally not in C F T ( L ) .
(5) Convergence in C F T ( L ) may have some features which P~. Lowen and P. Wuyts consider as patho[ 1/n\
logical. (For example, a "vanishing sequence" of fuzzy points like kx0 )~eN may converge to a fuzzy
point in C F T but it never happens in L C F T ( L ) . )
Pointing out these and some other "advantages" of L C F T ( L ) in comparison with C F T ( L ) R. Lowen and
P. Wuyts insist on restricting the subject of fuzzy topology entirely to the case of laminated spaces [123],
see also [107-109] et al. However, at present, the attitude of most workers in fuzzy topology towards the
property of laminatedness can be compared with the attitude of general topologists towards the axiom of,
say, Hausdorfness, or complete regularity:
One may, certainly, accept it when it is really needed, but it is not reasonable to include axiom T2
or T3,5 as part of the general definition of a topological space. Thus the standard framework for most
specialists working in fuzzy topology is the category C F T ( L ) , but when it is indeed necessary, they impose
the condition of laminatedness on the considered spaces.
It should be noted that a similar to (1TX), but yet a stronger axiom was (independently) introduced
by U. Hhhle in the definition of a probabifistic topological space [60]. Probabilistic topological spaces or,
as they are sometimes called, transition closed fuzzy topological spaces, can be characterized as (I)-fuzzy
spaces, whose fuzzy topologies r satisfy the following axiom:
(1T P) if U E r, then (U + a) A 1 E v and (U - a) V 0 E v for each constant a E I.
1.2.2. C o n n e c t i o n B e t w e e n L C F T ( L ) a n d T o p : S o m e I m p o r t a n t F u n c t o r s . Unlike the category
C F T ( L ) , the category L C F T ( L ) does not contain the category T o p (except for the trivial case L = 2
when L C F T ( L ) = Top). Fortunately, by means of so called omega functors, defined first by R. Lowen
[107, 108, 109], in the case L = I, and then extended by different authors to the case of a general lattice, the
category T o p can be identified with a certain subcategory of the category L C F T ( L ) formed by so called
topologically generated [107] or induced [204] fuzzy topological spaces. Here we shall briefly discuss these
important functors and some related topics.
Given a topological space (X, T), let wT denote the family of all lower semi-continuous mappings from
(X, T) into the usual unit interval I. It can be easily seen that wT satisfies the axioms of the laminated fuzzy
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topology. Thus, assigning to a topological space (X, T) the laminated fuzzy space (X, wT), and assigning
to a continuous mapping f : (X, T x) -+ (Y, T Y) the mapping w(f) = f : (X, wT x) --+ ( Y, osT Y) (which is
also continuous), R. Lowen defines the functor w: T o p --+ L C F T ( I ) [107, 108] et al. This functor is the
right inverse of the functor t(1.1.3): ~ o w = id: T o p --+ Top. R. Lowen interprets the fuzzy space (X,~:T)
as the fuzzy copy of the topological space (X, T).
A fuzzy space (X, 7-) is called topologically generated if 7" = wT for some topology T on X [107-109].
A Chang-Goguen fuzzy topological Space is topologically generated iff it is laminated and weakly induced [135]. (A fuzzy topological space is called weakly induced if all u E 7" are lower semicontinuous when
considered as mappings u : (X, ~- N 2 x ) -> I, [135].) The category w(Top) of all topologically generated
spaces is coreflexive in L e F T ( I ) , the coreflexiones given by Lw: (X,T) --->(X,~wT) [126].
The functor ~; is quite important for the study of fuzzy topological spaces. In particular, by means of w
R. Lowen proposes the concept of "good" extension of a topological property from T o p to L e F T ( I ) as
follows :
Given a topological property P, its extension .P to the fuzzy case is called "good" if a topological space
(X, T) has this property P iff the fuzzy topological spat6 (X, wT) has the property P. (Of course, one
property can have m a n y "good" extensions.)
Several authors tried to extend this functor to the case of lattices different from I. Unfortunately, this
extension was done in different, inequivalent ways depending, in fact, on the choice of one of the possible
order topologies on the lattice L. S.E. Rodabaugh [160, 166] considers a functor wn: T o p -+ L e F T ( L )
assigning to a topological space (X, T) the L-fuzzy space (X, wR(T)), where wR(T) is the L-fuzzy topology
generated by the subbase lrn(T) = {U E L x : U - I ( L ~ ) E T, Va E L} (0.1.7). (The functor wn, as well as
all other functors discussed below, is defined identically on morphisms.)
T. Kubiak's [102] version of omega functor w r : T o p --4 L e F T ( L ) assigns to a topological space (X, T)
the L-fuzzy space (X, wK(T)) where wK(T ) is generated by the subbase ~rn- = {U E LX: U - I ( L a) E T,a' E
L} (0.1.7); i n the case where L is completely distributive, u;K (T) can be characterized also as the family of
all continuous mappings from (X, T) into L endowed with the topology generated by {L ~ : a E L} [102].
Obviously, in the case where L is a chain, wn - w r , and, if L = I, then wn = w K = w. (Notations ~-'R
and ~K are ours. In the original papers the corresponding lattice L is always indicated.)
Now let A: T o p ~ L e F T ( L ) denotes the lamification, or the stratification functor assigning to a
topological space (X', T) the space (X, A(T)), where )~(T) denotes the L-fuzzy topology, obtained from T
by saturating it with all constants c: X -~ L. Then, obviously, A(T) C wK(T) and A(T) C wR(T). If L
is a chain, then oan(T) = ),(T); generally the equality does not hold - for example, if L is a d i a m o n d (see
[166]). The equality ";K (T) = A(T) is guaranteed if L is completely distributive [102].
Clearly, for each L-fuzzy topology 7", 7" C wK ~(7"). If L is completely distributive, then T = (w K (T)) for
each topology T, and the functor "JK, in this case, is left adjoint right inverse of ~ (T. Kubiak [102]).
Under the assumption of complete distributivity of the lattice L, the functor w K preserves products
and images, i.e., w K ( f - l ( T ) ) = f - l ( w r ( T ) ) for an arbitrary mapping f : X ~ (Y,T), where ( Y , T ) i s a
topological space and X is a set [102].
Yet another extension w m of the functor w was introduced by M.W. Warner [197]. Its definition uses
the so-called Scott topology [50] on the lattice L. In the case where L is completely distributive, w w = ~.'z,.,
[102].
1.2.3. A n A l t e r n a t i v e V i e w p o i n t on the L o w e n A x i o m (1T~)was developed by U. H6hle in [65].
According to the result obtained in [65], laminated L-fuzzy topologies can be considered just as the external
version of the topological type objects [193] of the Higgs topos of L-valued sets [53]. From this point of view,
the "constant condition" ( I T A) in Lowen's system of axioms is a consequence of the categorical formulation
of the topological axioms. More precisely, ""the constant condition" is just that condition which permits
"internalizing" L-fuzzy topologies in the category of L-valued sets." [65].
1.3. (L, K ) - F u z z y T o p o l o g i e s : C a t e g o r i e s F T ( L , K)
1.3.1. Basic Ideas a n d D e f i n i t i o n s : S o m e E l e m e n t a r y F a c t s . The "fuzzy topologies" considered
in previous sections were defined as certain subsets 7" of the power set L x of L-fuzzy subsets of X. Thus. to
be consistent, they are preferably to be considered as crisp topological type structures on the families offitzzy
672
sets than fuzzy topologies while the term a fitzzy topology is related to some ftLzzy structure of topological
type on the fuzzy power sets L x. For the first time, the idea of such an approach was probably expressed
in U. Hhhle's paper [63]. However, in that paper, fuzzy topological type structures were considered only
on the power sets 2 x of crisp subsets of X. In more general situations similar ideas in the mid-1980s were
independently discussed in [57, 176, 107, 120, 3, 177, 47]. T. Kubiak [107] was probably the first who
understand the significance of the use of two different lattices L and K : one for creating a fuzzy power set,
and the other as the range for fuzzy topologies.
Let L and K be two fixed lattices (recall that here as always we require that all lattices are at least
complete and infinitely distributive) and let X be a set. An (L, K)-fuzzy topology on a set X is a mapping
T : L x ~ K satisfying the following three requirements :
(1FT) 7-(0) = T(1) = 1;
(2FT) T(U A V) > T(U) A T(V) for any U, V 6 Lx;
(3FT) T(vUj) >_A.T(Uj) for each family {5~: j E J} C L x.
J
J
The pair (X < T) is called an (L, K)-fuzzy topological space (see [176] in the case L = K = I, [180, 103]).
Intuitively, the value T(U) can be interpreted as the degree to which the fuzzy set U is open i n the
corresponding space. Thus, speaking informally, axiom (2FT) just says that the intersection of two fuzzy
sets is not less open than each one of these sets, and the axiom (3FT) affirms that the union of an arbitrary
family of fuzzy sets is at least as much open as is "the least open" of the fuzzy sets belonging to this family.
Given two (L, K)-fuzzy topological spaces (X, Tx) and (Y, Ty), a mapping f : X -+ Y is called continuous
if T x ( f - I ( V ) >__Ty(V)) for each V E L v. Informally, mappings which do not diminish the degree of
openness of fuzzy sets in the pre-image direction are continuous. (L, K)-fuzzy topological spaces and
continuous mappings between them form a category denoted by F T ( L , K).
In the case where the lattice L is endowed with an order reversing involution, one can give a reasonable
definition of the closedness degree for an L-fuzzy set M E L x by setting C(M) = T(MC). The mapping
C: L x --~ K thus obtained is called the closed (L, K)-fuzzy structure of the space (X, 7"). One can easyly
check that the closed (L, K)-fuzzy structure satisfies the following properties [103, 187] :
(1FC) C(0) = C(1) = 1;
(2FC) C(M V N) >_C(M) A C(N) for any M, N E L x ; and
(3FC) C(AM/) _> A.C(Mj) for any family {Mj: j E J} C L x.
l
l
In its turn, the closed (L, K)-fuzzy structure allows us, in a natural way, to define the operator of (L, K)-fuzzy
closure C l: L x x K --+ L x as follows:
Given M E L x and a E K + let C I ( M , a ) = A{N: C(N) _> a , N _> M}. Just from the definitions and
lattice properties it is easy to see that this operator satisfies the following conditions: (M, N E Lx; a E K+):
(1FCl) Cl(M,
> M;
(2FC1) i f M _< g and a < ~, then Cl(M,a) <_C l ( g , ~ ) ;
(3FCl) Cl(Cl(M,a),a) = C I ( M , a ) ;
(4FC1) CI(M V N, a) = CI(M, a) V Cl(Y, a);
(5FC1) C1(0, a) = 0
Conversely, given a mapping C1: L X x K -+ L X satisfying the requirements (1FC1)-(5FC1), one can
construct, in a natural way, an (L, K)-fuzzy topology for which C1 is the corresponding (L, K)-closure
operator. One can atso define, in a natural way, the operator of an (L, K)-fuzzy interior in an (L, K)-fuzzy
space. Now, with the aid of these operators, the property of continuity can be characterized as follows :
The following properties are equivalent for mapping f : (X, Tx) --+ (Y, Ty) of fuzzy (L, K)-fuzzy topological spaces :
(a) f is continuous;
(b) Cx(f-I(N)) >_Cy(N) for each N ELY;
(c) f(Cl(M,a)) <_Cl(f(M,a)) for each M E L X and each a E K+;
(d) f-l(Int(N,
)) <__
for each N e L v and each
e K+.
Sometimes it is reasonable to strengthen, in the spirit of Lowen's approach, the axiom (1FT) in the
definition of an (L, K)-fuzzy topological space. By doing this we come to the following concept :
673
An (L, K)-fuzzy topological space (X, 7") is called laminated, if
(1FT ~) T(c) = 1 for each constant c: X -+ L.
The complete subcategory of FT(L, K) formed by all laminated spaces will be denoted by LFT(L, IK).
In contrast to FT(L, K), the set of morphisms between any two objects of LFT(L, IK) is nonempty (at least,
all constant mappings between laminated spaces are continuous); the category LFT(L, K) is topological (in
H. Herrlich's sense [58, 59]). The category LFT(L, K) is coreflexive in FT(L, K) [7, 8].
1.3.2. R e l a t i o n to C h a n g - G o g u e n L-Fuzzy Topologies. Obviously, a Chang-Goguen L-fuzzy topology is just an (L, 2)-fuzzy topology and, therefore, the category CFT(L) can be identified with the category
FT(L, 2). Further, for each lattice K, Chang-Goguen L-fuzzy topologies can be viewed as (L, IK)-fuzzy
topologies satisfying the additional requirement that T(L X) C {0, 1} C K. Hence LCFT(L), up to an obvious isomorphism, can be considered as a full subcategory of the category FT(L, tK). As is shown in [7, 8]
the subcategory CFT(L) is both reflexive and coreflexive in FT(L, K).
All concepts discussed in 1.3.1 for (L, lK)-fuzzy spaces are in accordance with the corresponding concepts
introduced earlier for Chang-Goguen L-fuzzy spaces. Thus, the present theory of (L, K)-fuzzy topological
spaces can be considered as a natural (in the sense of being "more consistently fuzzy") extension of the
theory of Chang-Goguen L-fuzzy spaces.
On the other hand, by means of the decomposition of an (L, K)-fuzzy topology into a direct system
of Chang-Goguen L-fuzzy topologies one can reduce a certain problem in the category FT(L, ]K) to the
corresponding problem in the category CFT(L) which is usually much easier.
(The method of decomposition of a fuzzy structure into a system of "less fuzzy," or "more crisp" structures
was probably first explicitly usedby C.V. Negoita and D.A. Ralescu [154]. In this particular situation, the
decomposition method was used in [178] (in case L = K = I) and [180], see also [141, 187, 103].)
Namely, given an (L,K)-fuzzy space (X,T) and a E IK, let To = {V E Lx: T(U) >_ a}. Obviously,
To is a Chang-Goguen L-fuzzy topology on X; in the sequel it will be referred to as the a-level L'fuzzy
topology of 7".
Thus an (L, IK)-fuzzy topology 7" can be decomposed into a decreasing (i.e., a < fl implies T~ C To)
family of Chang-Goguen L-fuzzy topologies. Conversely, an (L, K)-fuzzy topology can be restored from its
a-level L-fuzzy topologies To by the formula
= V { T o ( V ) ^ a: a e K } .
The decomposition of (L, K)-fuzzy topologies into systems of their a-level L-fuzzy topologies is functorial
in the following sense:
A mapping f : (X, 7"x) _+ (]i, TY) of (L, K)-fuzzy topological spaces is continuous iff for each a E K the
mapping f : (X, Tax) -+ (Y, TY) of the corresponding a-level Chang-Goguen L-fuzzy topological spaces is
continuous.
1.3.3. Lattice P r o p e r t i e s of Families of (L,K)-Fuzzy Topologies. In order to obtain a deeper
insight into the relations between (L, K)-fuzzy topologies and the corresponding directed systems of ChangGoguen L-fuzzy topologies and to give a satisfactory description on (L, K)-fuzzy spaces (1.3.4), one has
first to analyze the lattice properties of families of (L, K)-fuzzy topologies.
Let ~(X, L, K) denote the family of all (L, K)-fuzzy topologies on a set X. Note first, that .~(X, L, K) is
always a bounded complete lattice [187, 103]:
The discrete fuzzy topology Ta (i.e., Ta(M) = 1 for each M G L x) is the maximal element of .~(X, L, K),
and the antidiscrete fuzzy topology Ta (i.e., Ta(0) = Ta(1) = 1 and Ta(M) = 0 if U # 0, U # 1) is
its minimal element. The infimum of the family % C %(X, L, K) is an (L, K)-fuzzy topology given by the
formula Inf%0(U) = A ~(U); its a-level (a E K) L-fuzzy topologies are characterized by the equalities
TE~o
(Inf.q:0)o =
fl To. The existence of supS0 is ensured by the existence of InfT0 by means of the duality
TE~o
principle. In the case where K is completely distributive, there is also an e~ective description of the
supremum: (sup~0)(U) = V{a 6 K: U e V To}. Its a-level L-fuzzy topologies are characterized in this
TE~/o
case as follows: (sup.~0)o = N{ V Ts: ;3 6 CP(L),~ << a} (see notation in 0.1.5: 0.i.6), [103]; see also
T6T0
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[178] (in the case L = K = I) and [141] (when IK is a chain). (The notation
V "/~,appearing in the above
TE~o
formula, just means the Chang-Goguen L-fuzzy topology generated by the family
U T~ as a subbase, i.e..
T6To
V 7"~ is the family of all joins of finite meets of elements of U Ta.)
TE$o
T6 .'X:o
A problem closely related to the above, is to develop effective methods for constructing (L, K)-fuzzy
topologies from certain families of (L,K)-fuzzy topologies. One quite useful method for the case of a
completely distributive lattice IK Was developed in [103] (see also [178] in the case L = K = I and [141] in
the case where K is a chain):
Let {T~: a 6 K} be a decreasing family of (L, K)-fuzzy topologies on X, where K is completely
distributive. (By decreasing we mean here that a _</3 implies T~(U) >_T#(U) for each U E Lx.) Then the
mapping T : L x -+ K defined by T(U) = V Ta(U) A a for all U 6 L x is a n (L, K)-fuzzy topology on X.
a6IK
Besides, the a-level L-fuzzy topologies of T are characterized as T~ = fq{Tf: /3 E C P ( K ) a n d / 3 << o}
(see 0.1.5, 0.1.6) [1031.
The construction described above is functorial in the following sense: Assume that {Tff: a E K} and
{T{: a 6 K} are decreasing families of (L, K)-fuzzy topologies on sets X and Y respectively, and let
Tx and Ty be the corresponding (L, K)-fuzzy topologies constructed from these families. If the mapping
f : (X, Tff) ~ (Y,T~) is continuous for each a 6 K, then the mapping f : (X, Tx) ~ (Y, Ty) is continuous
also ([103], cf. also [178, 141]).
1.3.4. Basic O p e r a t i o n s in t h e C a t e g o r i e s F T ( L , ]K). To give the reader an idea of how operations
in the categories F T ( L , ]K) are defined, we shall consider the operations of taking a subspace, a product,
and a direct sum.
Let (X, 7") be an (L, K)-fuzzy space and Y C X. Then the corresponding (L, K)-fuzzy subspace is defined
as a pair (Y, Ty), where the induced (L, K)-fuzzy topologyTy : L Y -+ K is defined by the equality
Ty(V) = sup{T(Uv): Uv 6 L X, UvIy = V}.
it is easily seen that the space (ETy) thus obtained is indeed a subobject of (X,T) in the category
F T ( L , K) (see [180, 187] in case L = K).
Passing to the product, consider a family {(Xj,Tj): j E J} of (L,K)-fnzzy spaces and let X = ILYj
be the product of the corresponding sets and, for each j E J, let pj : X -4 Xj denote the corresponding
projection. Then the product of this family can be defined as the pair (X, T) where T is the (L, K)-fuzzy
topology which is initial for the family of all projections, i.e., the weakest one for which all projections
become continuous. Its existence follows from the completeness of the lattice T(.X, L, IK) (1.3.3). To get
an explicitdescription of the product topology T, note that the fuzzy topology Tj = p;l(. j): L x ___>~[,
which is initialfor a projection pj, is defined by ~(V) = Tj(Uj) if V =
for some Uj E L x and
~(V) = 0 otherwise. The product(L,K)-fuzzy topology can be defined now as iF = sup ~ ([180, 187] in
J
the case L = K).
The projections pj: (X, T) --> (XjTj) generally are not open (i.e.,the inequality i/j(pj(V')) _> T(V)
generally does not hold). A sufficientcondition which guarantees the openness of the projections is the
laminatedness of all factors (cf. the situation in L C F T ( L ) ) .
To define the direct sum or coproduct of a family {(Xj,Tj): j E J } of (L, K)-fuzzy spaces let Z = @Xj
be the disjoint union of the corresponding sets and let ej: Xj --+ X denote the corresponding natural
inclusion. Now the direct sum can be defined as the pair (X,T), where T : L x --+ K is the final fuzzy
topology for the family of all inclusions, or, explicitly, T(U) = A{Tj(U/) : Uj = U Izj }, U 6 L x.
p71(Uj)
3
1.3.5. T w o F u n c t o r s in t h e C a t e g o r y F T ( L , K). [178] (in the case L = K = I), [141] (in the case
where K is a chain), [103] (in case K is completely distributive). The functors L and A considered below, are
certain extensions of the iota functor (1.1.3) and the functor of laminification (1.2.2) (which, in its turn,
is an extension of R. Lowen's omega functor (1.2.2)), respectively. Therefore, We use for them the same
675
notation despite the fact that they act in an essentially different situation. Everywhere in the sequel, K is
a completely distributive lattice.
Given an (L, K)-fuzzy space (X, T), let 7.x stand for the weakest laminated (L, K)-fuzzy topology on X
dominating T. Now, assigning to an (L, K)-fuzzy space (X, T) the laminated (L, K)-fuzzy space (X, 7.x)
and leaving the morphisms unchanged, one gets the functor of laminification A: FT(L, K) --+ LFT(L, K).
Explicitly, T x can be expressed as follows: TX(U) = ~V+ {T~(U)A a} for each U E L x.
Further, given an (L, K)-fuzzy space (X, T), let the (L, K)-fuzzy topology 7.' : L z ~ K be defined by the
equality T'(U) = V (tT,(U) A a) (see the notation in 1.1.3 and 1.3.3). If U E L x \ 2 x, then, obviously,
a Eli(+
T'(U) = 0; on the other hand, for U E 2 x T'(U) can be any value in K. Therefore, the space (X,T')
can be interpreted as an object of the subcategory FT(2, K) of the category FT(L, IK). Thus, assigning
to an (L, K)-fuzzy space (X, T) the (2, K)-fuzzy space (X, T') and leaving the morphisms unchanged, the
functor 5: FT(L, K) ~ FT(L, K) with range in the subcategory FT(2, K) is obtained.
Thus, in a certain sense, t can be viewed as the functor of defuzzification.
The functors A and t are connected by the equality A o ~o A = A. Both functors commute with products.
1.3.6. (L,K)-Fuzzy Pre-Topologies. When studying (L,K)-fuzzy topologies, in particular, when
investigating their local structure, one sometimes arrives at the need to consider, along with (L, K)fuzzy topologies, a somewhat more general structure: so-called (L, K)-fuzzy pre-topologies. Namely, an
(L,K)-fuzzy pre-topology on a set X is a mapping T: L x -+ IK satisfying axioms (1FT) and (2FT) of
1.3.1 [181, 187].
Unlike the situation in Top, as well as in CFT(L), where any family of (fuzzy) sets satisfying axioms
like (1FT) and (2FT) can be used as a base to generate a (fuzzy) topology, the situation in FT(L) is much
less definite. In particular, although starting with an (L, K)-fuzzy pre-topology 7~: L x --+ IK (as, by the
way, with an arbitrary mapping ;o: L X _.+ K), one can define "a corresponding" (L, K)-fuzzy topology
T(7 ~) - for example by defining it as the weakest (L, K)-fuzzy topology dominating 7:'; nevertheless, this
(L, K)-fuzzy topology T(T') cannot be considered just an extension of 7:', because the domain of both T'
and T(7 ~) is the whole L X , while they may differ in values.
1.3.7. T h e N e i g h b o r h o o d a n d Q - N e l g h b o r h o o d S t r u c t u r e s of an (L,K)-Fuzzy ( P r e ) T o p o logical Space [181, 187]. To study local properties of (L, K)-fuzzy spaces, one has first to develop the
appropriate tools for the description of the local construction of (L, K)-fuzzy spaces.
Let (X, 7") be an (L, K)-fuzzy (pre)topological space. Its neighborhood structure is a mapping .Af(7.) =
At: :~ • L x -~ K (:~ denotes the totality of all L-fuzzy points in X), defined by the equality Af(x~,U) =
sup{T(V) : V < U, V(xo) >_t}. In the case where K is completely distributive, the neighborhood structure
satisfies the following conditions (p = x~ stands for an arbitrary L-fuzzy point in X) :
(1FN) I f X ( p , U ) > 0, then p~U;
(2FN) sup A/'(p, U) = 1;
UELx
(3FN) N'(p,U, A U2) >_Af(p,UI) AAf(p,U2) for any UI,U~_E Lx;
(4FN) U1 <~ U2 implies N'(p, U1) _~<./~(p, U2);
(bFN) A/(p, U) _< sup N'(p, V) A (rAvA/'(r, V)) for each U e Lx.
v<u
V EL x
rE 3r
Conversely, if X is a set and the mapping Af: ~ x L x --+ K satisfies the axioms (1FN)-(bFN), then the
mapping T(N') = T X -+ K defined by the equality T(U) = inf X'(p, U) is an L-fuzzy pre-topology on X.
p~u
Moreover, N" is exactly the neighborhood structure of the space (X, 7.). For each (L, K)-pre-topology 7" the
inequality T < T(A/'(T)) holds. It becomes the equality in the case where T is an (L, K)-fuzzy topology.
[181] (in the case L = I), [187].
(It is important to emphasize that 7.(3/') may fail to be a fuzzy topology even ia the case when L = IK = I
and 3/(2~ x L x) C 2 C I [181].) By replacing, in the above definition of the neighborhood structure, the relation of belonging E with the relation of quasi-coincidence q, one arrives at the definition of a q-neighborhood
structure Q: ~ x L x --+ K of a space (X,7"). For the q-neighborhood structure we have statements
676
analogous to the ones formulated above for the neighborhood structure, but with the obvious substitution
of the quasi-coincidence relation for the belonging relation. Furthermore. the mapping T ( Q ) : L x --+ K
constructed as above from a q-neighborhood structure, Q: ~ x L x -+ K on a set N, is an (L, K)-fuzzy
topology (and not only an (L, K)-fuzzy pre-topology) on X.
1.3.8. T h e C o n v e r g e n c e S t r u c t u r e o f a n ( L , K ) - F u z z y T o p o l o g i c a l S p a c e [182, 187]. Given an
(L, K)-fuzzy topological space (X, T), its (net) convergence structure is a mapping Con: 9I(X) x K -+ L x
defined b y t h e equality C o n ( s
V{t E L: (VU E L X ( T ( U ) >_ a, ztqU) ~ (/2 is q-final with U ) ) } .
(Here 92(X) stands for the class of all L-fuzzy nets in a set X (1.1.5), s E 91(X),a E N. The notions
concerning fuzzy nets can be found, e.g., in [158] or in our survey [180].)
If s is a subnet of an L-fuzzy net s then Con (/2, a) _< Con (s a) for each a E IK.
Convergence structure can be used to characterize different properties of L-fuzzy topologies. In particular: CI(M,c~) = V{Con(/2,~) : /2 E ~(X)/2 C M } , where C1 is the closure operator (1.3A), and hence
C(M) >_ ~ iff pE Con (/2, cz) for every fuzzy n e t / 2 C M and each fuzzy point p'EM. On the other hand
Con (/2, c~) = V C I ( A m , ~ ) , where s = (s,~),~ez), and Am = V{s,~: n >_ rn}.
mED
A mapping f : (X, Tx) --+ (Y, Ty) is continuous, iff f ( C o n x (s
_< Cony (f(/2),c~) for every c~ E N.
Convergence structure can also be used to describe (L, K)-fuzzy topologies in the spirit of J.L. Kelley's
description of topologies by means of convergence classes, see, e.g., [187].
Let (X, T) be an (L, N)-fuzzy topological space. Then its convergence structure c2(T ) = Con: 9I(X) x
N --+ L x satisfies the following five conditions (c~ E N) :
(1FCS) If s , = p for all n E D, then p~ Con ((s,),,eo, ~).
(2FCS) If/2' is a subnet of s then Con (/2, c~) _< Con (/2', cz).
(3FCS) If p ~ Con (/2, c~), then there exists a subnet/2' in/2, such that p ~ Con (/2", c~) for each s u b n e t / 2 "
of s
(4FCS) Let D and Em for all m E D be directed sets, R(m, ~) = (m, ~(m)) where ~ E n { E m : m E D}, and
let for each pair (m, n) E D x Era an L-fuzzy point s
n) E ~ be fixed. If pm~E Con (/2(m, n)nEEm, &)
for every m E D and p o t Con ((Pro)reED, oe), then p o t Con (s o R, a).
(5FCS) If D C L is a set directed by the order which is induced by the original order of L, then
x ~ C o n ((x~)neD,a) for each x0 and each t _< s u p D .
Conversely, in the case when L is a chain, if a mapping Con: ~ ( X ) x K -+ L x satisfies conditions
(1FCS)-(5FCS), then the mapping C I : L X x K -+ L X defined by the equality C I ( M , a) = V{Con(s c~):
/2 is a L-fuzzy net in M) }, is an (L, N)-fuzzy closure operator.
Besides, if ~b(Con) is the corresponding (L,N)-fuzzy topology, then c~(r
= Con. On the other
hand, r
= 7 for each (L, IK)-fuzzy topology 7", and hence c2 and r are mutually inverse bijections.
Besides, if T1 _> T2, then ~o(T1) _< cp(T2), and if Con1 _> Con2, then r
<_ r
1.4. Hutton Fuzzy Topologies
1.4.1. Basic I d e a s a n d D e f i n i t i o n s . B. Hutton [69] observed that given a lattice L and a set X, the
power set L X = / 2 is again a lattice; moreover L x = / 2 inherits, in a natural way, from L such properties
as completeness, infinite and complete distributivity, and the existence of an order reversing involution
whenever the original lattice L has these properties. Therefore, when defining fuzzy topology, one can start
with a lattice/2 itself (interpreting it as some abstract lattice) and not with a set X and a lattice L and
obtaining the lattice L x of fuzzy sets as the derivative of the first two. In other words, using EklundG/ihler's terminology, one can say that B. Hutton's idea was to take as the ground category [38, 39] for the
fuzzy topology the category L A T of lattices (instead of the category S E T used as the g r o u n d category for
T o p as well as for the categories of fuzzy topology discussed in the previous sections) and to define a fuzzy
topology as a certain subset of a lattice. Here is the precise definition [69, 70] :
By a Hutton topology on a lattice/2 we mean a subset r C s which is closed under arbitrary joins, finite
meets and contains 0 and 1. A pair (/2, 7") is called a Hutton fuzzy topological space. The elements U E 7"
are referred to as open and their complements U c (in the case where s was equipped with an involution
c. s __+ s as clo'sed elements of this space. Given two Hutton spaces (s
and (s
, a morphism
f : (/21,7"1) -+ (/22, r2) in the category H F T of Hutton spaces is defined to be a mapping f - l : /22 --+ s
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preserving arbitrary joins and meets and such that f - 1 (V) E 7"1 for each V E 7"2.
To see the connections between T o p and H I ' T , observe, that a topological space (X, T) can be interpreted
in H F T as the pair (2 x , T) and a continuous mapping f : (X1, T1 ) -4 (X2, T2) corresponds to the morphism
f : (2X1,T1) -4 (2X2,T2) which is defined by the "pre-image mapping" f - l : 2x2 _~ 2xl, i.e., f ( V ) =
f - l ( y ) , V E 2 X2.
The foundations of the theory of Hutton fuzzy spaces are developed in [69, 70]. In particular, there
are considered such properties of Hutton spaces as compactness, separation axioms, metrizability and
connectedness. W h e n developing the theory, B. Hutton emphasized that he tried to work out ""pointless"
definitions for properties and structures which depend purely upon the lattice structures of the collections
of fuzzy sets, and not u p o n the decomposition into the form L X," and "to extract the essence of the usual
topological theorems and generalize their proofs." (An idea of how it was done can be found in 1.4.2.) From
this viewpoint, B. Hutton's approach can be related to "pointless" general topology as was done with the
theory of topological lattices (see, e.g., [33, 71]).
An approach similar to B. Hutton's is discussed in M.A. Erceg's paper [41]. An important contribution
to the investigation of the category H F T was made by S.E. Rodabaugh [163, 167, 168], et al. and by
P. Eklund [36, 37] and P. Eklund and W. Gg.hler [38, 39].
1.4.2. B a s i c O p e r a t i o n s . To illustrate some peculiarities and problems which one encounters when
developing B. Hutton's approach to fuzzy topology, we show here how operations of taking a subspace and
product in H F T are defined [69, 70].
Let (Z;, 7") be a Hutton space and a E/;. For each u E s define an element u , = (u A a) V (a A aC), and let
s = {u, : u E / ; } . Introducing operations V and A on Z;, as restrictions of the corresponding operations
from s t o / ~ , , and assigning to each element u, E Z;, the element ua = u~ A a (= (uC),) E / ; , , one gets a
complete lattice f-, with 1, = a as the maximal element, 0, = a A a c as the minimal element, and the order
reversing involution given by u , -+ u~. The family 7", = {u, : u E 7"} is a H u t t o n topology on the lattice
s and hence (s
is a Hutton fuzzy topological gpace. In fact, it is a subobject of (s
in H F T .
called the subspace of the Hutton space (E, 7.).
To see the connections between this definition and the operation of taking a subspace in other categories
of fuzzy topology, observe the following: Let (X, 7.) be a Chang-Goguen L-fuzzy space and (A, VA), for
some A C X, be its subspace. Then the pair (LA,7.), viewed as a Hutton space in a natural way, can be
identified with the corresponding subspace ((L X)A (7.)A) of the Hutton space (L X, 7").
To define the product in H F T we must first recall the operation | for lattices introduced by B. Hutton [69]. Given a family {s : j E J} of lattices, let a lattice/: = |
be defined as follows : Its elements
are subsets a C n{z:~" : j e J} (H stands for the product in S E T ) such t h a t :
(1) i f t E a and s _< t, then s E a (for s,t E H/: + the inequality s < t just means that sj < tj for each
1
j E J); and
(2) ifbj C f.+ and b = Flbj C a, t h e n ~ = (~?j) E a, where flj = supbj. Now, by letting a _< biff
J
a C b (a, b E/2), the s e t / : becomes a lattice. It is completely distributive whenever a l l / : j are so.
Finally, by letting bc = {x (Vy e b 3j E J such that xj < y~)}, B. H u t t o n defines an order reversing
involution o n / : . (Note : if for every j E J ~:j = 2 z~ for some set Zj, then the lattice |
is in a
natural way isomorphic to the lattice 2nz~). By setting 7r~l(tj) = {s E H/:+: sj < tj} a mapping
rr~-1 : ~:j --+/1 is defined.
Now the product of a family {(/:j, 7"1): j E J} of Hutton spaces in the category H F T can be defined as
the pair (s 7"), where ~: = |
and 7" is the Hutton topology on s which is the initial for the family of all
"projections" rrj : /: - + / : j , j E J. Explicitly, 7" is the family of arbitrary joins of finite meets of elements
of the family {rr~-l(uj): uj E 7"1,J E J} [69]. To see the connections between the product operation in
H F T and in other categories of (fuzzy) topology, take Chang-Goguen L-fuzzy spaces (X1,7"1) and (X2. r2)
and observe that their product (X1 x X2, T) in C F T ( L ) essentially differs from the product (Z:, 7") in H F T
of the corresponding Hutton spaces (L x ' , rl) and (L X2' 7.2) - in particular, the'lattice s is never L xl x.\'_~
except for the crisp case L = 2. On the other hand, in the case where (X1,7"1) and (X1, vl) are o r d i n a l '
678
topological spaces, their product (X1 x X2, T) viewed as a Hutton space (2 xl •
T) in a natural way can
be identified with the product (2 xl x 2 x2, r) of the spaces (2 xl, ri) and (2 x2, 7"2) in H F T .
1.5 " C h i n e s e " A p p r o a c h : T o p o l o g i c a l M o l e c u l a r L a t t i c e s
1.5.1. T o p o l o g i c a l M o l e c u l a r L a t t i c e s V e r s u s H u t t o n ' s A p p r o a c h . In a series of works written
by Wang Guojun ([194-196] et al.) and his successors, mainly Chinese mathematicians (see [212-215] et
aI.), yet a n o t h e r viewpoint on the subject of fuzzy topology was developed.
At first glance it may seem that B. Hutton's approach and the "Chinese" approach are of a similar nature:
Both of them deal with topological type structures on certain lattices. However, originating as if from the
same starting point, the two approaches pursue quite different aims and are being developed, in a certain
sense, in opposite directions. While B. Hutton's principal aim is to develop the "pointless" viewpoint on
the subject of the fuzzy topOlogy and he consequently ignores the local aspects of the topological structure
of his spaces, Wang Guojun extracts some "point-like" elements called molecules in a very general setting
of the so-called molecular lattices and develops an original topological theory for molecular lattices giving
special attention to its local aspects.
Thus, Wang Goujun's approach can be considered as a qualitative generalization of certain parts of
the theory of Chang-Goguen L-fuzzy spaces (Sec. 1.1), especially of those parts which are related to the
investigations by P u Paoming and Liu Yingming [158, 159, 105], et aI.
1.5.2. M o l e c u l a r L a t t i c e s a n d G e n e r a l i z e d O r d e r - H o m o m o r p h i s m s [194].
As an appropriate basis for his theory "Wang Guojun finds completely distributive lattices and co-prime
elements in such lattices.
Given a completely distributive lattice L, let M denote the set of all co-prime elements of L (Sec. 0.1.6).
Wang Guojun shows that, in addition to their V-irreducibillty, the elements of M have the following
important peculiarity which permits them to be used in different situations as "generalized points I' :
If a E L +, then there exists a subset/3(a) C M such that a = V{m: ra E ~3(a)} (i.e., all elements of
L can be "constructed from elements of M") and, if a = Vaj for some a, aj E L, Vj E J, then for each
J
m E/3(a) there exists j such that m E aj (i.e., this "construction" is, in a certain sense, "minimal").
Example: IfL=I x , thenM={zt:
t > 0 , x E X } and/3(:c t ) = { z ~ : 0 < s < l } .
Emphasizing the specific and fundamental role played by co-prime elements in his theory, Wang Goujun
calls them molecules. The molecular lattice is just a completely distributive lattice with special attention
to its molecules. To denote a molecular lattice the entry L ( M ) is used: it indicates both the lattice itself
and the set of its molecules.
A mapping f : L i ( M i ) ~ L2(M2) of molecular lattices is called a generalized order homomorphism, or
G O H , for short, if f(0) = 0 and both f and f - i preserve arbitrary joins, where the mapping f - 1 : L2(M2) -+
L~(M1) is given by f - i ( b ) = V{a E L~: f(a) < b}. Wang Guojun finds a criterion characterizing those
mappings f : M1 --4 M2 which can be extended to a G O H f : Lx(M~) ~ L2(M~); this criterion plays an
important role in the theory.
1.5.3. T o p o l o g i c a l M o l e c u l a r L a t t i c e s [194]. A Topological Molecular Lattice is a pair (L(M); r])
where L(M) is a molecular lattice and 77 is a subset of L(M) containing 0 and 1 and dosed relative to
taking arbitrary meets and finite joins; elements a E r/ are called closed in this lattice.
There are two reasons why the closed topology, and not the open topology is used in the definition of a
topological molecular lattice. Firstl Wang Guojun finds the concept of closedness more appropriate for his
approach (in particular, for the convergence theory), than the concept of openness. Second, Wang Guojun
develops his theory without the assumption that the lattices are endowed with an involution, and, in this
situation, open and closed are no longer dual notions and one has to choose one or the other.
A G O H f : (Ll(M~),r/1) --+ (L2(M~),r/~) is said to be continuous if N E rl2 implies f - l ( N ) E r/1.
Topological molecular lattices and continuous G O H form a category denoted by T M L .
To make use of molecules as pointwise elements, in particular, to develop a local theory in the context
of T M L , one has first to find out what elements are to play the rote of neighborhoods in this situation. In
the absence of the concept of openness, Wang Guojun invokes the notion of a remote neighborhood of a
molecule which is, in a known sense, an antipode of a neighborhood : Given a molecule m in a topological
molecular lattice (L(M), r?), an element P E r/is called a remote neighborhood, or an R-neighborhood (for
679
short) of m, if rn :~ P. We illustrate the opportunities given by the concept of an R-neighborhood by the
following local characterization of continuity :
Let r/(m) denote the set of all R-neighborhoods of a molecule m in (L(M), r/). A G O H I : (Ll(M1),~?l)
(L2(M2),r]2) is called continuous at a molecule rn E M1, if for each N E r]2 ( f ( m ) ) , it holds I - I ( N ) E rh (m).
A G O H is continuous if it is continuous at each molecule m E M1 [194].
The theory of T M L developed in [194-196, 213, 214, 215, 43] and other papers, includes such items
as convergence theory, axioms of separation, locally finite families of elements, axioms of countability,
paracompactness, connectedness, and others.
M. Brown and R. Ertiirk recently obtained an interesting characterization of separation and some other
properties in structures like topological molecular lattices by means of (crisp) bitopologies (see, e.g., [42]).
1.6. T h e P r o b l e m of C h a n g i n g a Base 9 S.E. R o d a b a u g h ' s C a t e g o r i e s
The approaches to the subject of fuzzy topology discussed in Secs. 1.1, 1.2, and 1.3 are being developed
in the context of fized lattices. More precisely, the basis L for the powerset L z in the categories C F T ( L ) ,
L C F T ( L ) , and F T ( L , N) was always assumed to be invariable (as well as the "range" lattice K in the case
of categories F T ( L , N)). On the other hand, in B. Hutton's category H F T the set X, in the definition of a
Hutton space, degenerates as if into a singleton being absorbed by a lattice. Therefore, from the categorical
point of view, this approach incorporates only a change of base. A general theory allowing both a change
of set and a change of lattice was proposed by S.E. Rodabaugh (see, e.g., [163, 165, 167, 168] etc.)
The context for S.E. Rodabaugh's approach is the category F U Z Z [163], the objects of which are triples
(X, L, r) where X is a set, L is a lattice, and ~- C L z is a Chang-Goguen L-fuzzy topology on X; the
morphisms in F U Z Z are pairs (f, c2) ; (X1,LI,vl) --+ (X2,L2,r2) , where f : Xl --+ X2 is a mapping,
~0-1: L2 -+ L1 is a mapping preserving joins, meets, and involution, (i.e., c2-1 is a morphism in the
category of complete lattices with involution), and ~o-1 o V o f E v1, whenever V E r2. Applying P. EklundW. Gs
terminology one can say that the category Set x Lat ~ plays the role of the ground category
for F U Z Z [38, 39].
F U Z Z contains the categories C F T ( L ) and L C F T ( L ) as subcategories, but unfortunately, non-full (and
therefore F U Z Z can hardly be realized just as a generalization of Chang-Goguen's approach). On the other
hand, the category H F T is contained in F U Z Z as a full subcategory.
In [163] and a number of subsequent papers [165-168] et al., S.E. Rodabaugh has thoroughly investigated
the category F U Z Z and some other similarly defined categories; an important contribution to this investigation was also made by P. Eklund [36-37]. Unfortunately we cannot discuss any details of this interesting
theory because such a discussion would inevitably involve a number of new definitions and constructions
which would significantly increase the volume of this work.
In [174, 175], there are given algebraic characterizations up to homeomorphism of objects of the category
F U Z Z and some analogous categories by means of special semigroups constructed on the basis of the endomorphism semigroups (cf., the well-known characterization of topological spaces by means of semigroups
of continuous transformations see, e.g., [132]).
Patterned after S.E. Rodabaugh's approach, T. Kubiak [101] has introduced a category containing categories of F T ( L , K)) type for changing lattices L and K. A category similar to Kubiak's was studied in [179],
see also [180].
1.7. M i n g s h e n g Y i n g ' s Fuzzifying Topologies : S e m a n t i c Analysis o f T o p o l o g y
A specific viewpoint on what can be the subject of fuzzy topology was developed by Mingsheng Ying
[145-148]. Contrary to the approaches discussed in the previous sections, all of which could be united under
the name of point-set lattice-theoretic fuzzy topology, Mingsheng's theory, based on the semantic analysis
of concepts and results of general topology, is to be referred to the so-called model-theoretic fuzzy topology.
(We make use here of S.E. Rodabaugh's terminology, slightly modified, see [167, 168].)
By means of the semantic method of continuous-valued logic, Mingsheng Ying arrives at the concept of a
fuzzifying topology on a set X (which is, in fact, a mapping T : 2 z -~ I satisfying the axioms (1FT)-(3FT)
of Sec. 1.3) and then consistently develops the theory of fuzzifying topologies. The theory developed up until
now [145 - 148] includes such items as local structure of fuzzifying topologies, their convergence structure,
axioms of countability, compactness (including a version of the Tychonoff theorem), connectedness, and
680
others. All these concepts appear to be predicates of multivalued logic and can take vahtes from I.
It is impossible here to go into any details of this very interesting and in our opinion, promising theory."
However, to give the reader at least a superficial perception of the ideas behind this approach, we shall
reproduce the definition of neighborhood structure.
The neighborhood structure of the fuzzifying topological space (X, T) in a point x is a fuzzy predicate
A/',: 2 x ~ I defined by U E Af~ := 3V((V E 7-) A (x E Y C U)), where the belonging and inclusion signs
are interpreted respectively as "z E A" := A(z); "A C B" := Vx(A(x)--+ B(x)), and the logical connectors
and quantifiers are defined in accordance with Lukasiewicz logic :
a(x) --+ b(x) := 1 - a(x) + b(x);
Vx(a(x)) := inf a(x);
zEX
~a(x) =
1
- -
a(x);
3x(a(x)) := "-,(Vx--,a(x))
(see, e.g., [127]).
To the model-theoretic direction of fuzzy topology, together with Mingsheng Ying's theory, belong verb"
interesting approaches developed by U. HShle (to appear in Fuzzy Sets and Syst. in 1994) and U. HShle
and L.N. Stout (unfortunately not yet published). Some ideas, similar to the ones of Mingsheng Ying were
expressed by Z. Diskin [3]. A general theory uniting model-theoretic and point-set lattice-theoretic (with a
fixed basis) approaches to fuzzy topology was recently developed by U. H6hle and A. Sostak [66].
2. F U Z Z Y U N I F O R M S T R U C T U R E S
2.1. H u t t o n L - F u z z y ( Q u a s i - ) U n i f o r m i t i e s
2.1.1. Basic I d e a s a n d Definitions. A concept of a fuzzy (quasi-)uniformity which is in accordance
with Chang-Goguen's approach to fuzzy topology was introduced by B. Hutton [68]. BI Hutton's approach is
based on the observation that if (X, U) is an ordinary (quasi-)uniform space defined in terms of "entourages"
(see, e.g., [203, 11]), then each U E U is a certain subset of the product X x X, and, therefore, can be
identified with the function U: 2 x --+ 2 x defined by U(A) = {y: 3x E A,(x,y) 6 U}, A E 2 x. It is easy
to see that this function preserves unions and A C U(A) for each A E 2 X. Conversely, if U : 2 x --+ 2 x is a
function with these properties, then, by letting U = { (x, y): y E U(z) }, one obtains a subset U C X x X
containing the diagonM, i.e., a typical element of some quasi-uniformity on X. Developing this idea,
B. Hutton arrives at the following concept :
Let L be a completely distributive lattice and X be a set. Let D denote the family of mappings
U: L x -+ L x such that:
(HDI) M <_ U ( M ) for each M E Lx;
(HD2) U(VMj) = vU(Mj) for each family {Mi: j E J} C L x
J
(Note that axiom (HD2) implies that U(0) = 0.)
A n L-fuzzy quasiuniformity on X can now be defined as a subset L/C D satisfying the following axioms :
(HU1) if U ELt, U:< V and V E D, then V E///;
(HU2) if U1, U2 ELt, then there exists V E U such that V _< U1 A U2;
(HU3) for every U E U there exists V E U such that V o V <_ U.
An L-fuzzy quasi-uniformity is called an L-fuzzy uniformity, if
(HU4) U E/.4 implies U -1 ELt
(U-l: L X -+ L x is defined by U-I(N) = A{M: U(M =) < Nz}).
The pair (X, b/) is called a Hutton L-fuzzy (quasi-)uniform space.
A mapping f : (Z, blx) --+ (Y, bly) of L-fuzzy (quasi-)uniform spaces is called uniformly continuous,
if for every V E /4y there exists U E /gz such that U < f-~(V). Explicitly, this means that U(M) <
f - l ( V ( f ( M ) ) ) for every M E L x. The category of Hutton L-fuzzy uniform (resp. quasi-uniform)spaces
\
\
1
and uniformly continuous mappings will be denoted H F U ( L ) (resp. H F Q U ( L ) ) . Obviously, in case L = 2
the categories H F U ( 2 ) and H F Q U ( 2 ) can be identified with the categories U n i f of uniform and Q U n i f
of quasi-uniform spaces respectively [68].
681
Let B C D satisfy axioms (HU2) and (HU3) and let 5 / d e n o t e the family of all U E D such that V < U
for some V E B. T h e n 5/is a quasi-tmiformity on X and B is called a base of 5/. A family "P C D is called
a subbase of an L-fuzzy quasi-uniformity 5 / i f the family B of all finite intersections of elements of 7 ~ is a
base of 5/[68].
An L-fuzzy uniformity 5 / h a s a base B = {Ur: r E R + } such that Ur o U8 _< U~+8 for all r , s E IR+ iffh/
has a countable base [68]. This result plays an important role when studying the problem of metrization
of L-fuzzy topological spaces.
Let 5 / b e an L-fuzzy quasi-uniformity on X. Given U, V E 5/let UaV denote the largest element of D
which is contained (<) in U A V. Then the family { U z x U - 1 : U E 5/}is a base of the smallest L:fuzzy
uniformity ~ containing 5/ (i.e.,/~ D 5/) [68]. The L-fuzzy uniformity 5/is called induced by the L-fuzzy
quasi-uniformity 5/.
2.1.2. C o n n e c t i o n s B e t w e e n L - F u z z y ( Q u a s i - ) U n i f o r m i t i e s a n d L - F u z z y T o p o l o g i e s were studied by B. Hutton [68], B. Hutton and I. Reilly [70] A.K. Katsaras [77], and A.K. Katsaras and C.G. Petalas
[84].
1. Every L-fuzzy quasi-uniformity generates an L-fuzzy topology. Let (X,5/) be an L-fuzzy quasiuniform space. Then the operator Int: L x ~ L X defined by Int(M) = V{N E L x : U(N) <
M for some U E 5/} is the fuzzy interior operator (1.1.1) [68]. The corresponding L-fuzzy topology
on X, r u = {M E L x : Int M = M}, is called generated by 5/. S.E. Rodabaugh [165] gave an alternative description of the generated L-fuzzy topology by means of systems of fuzzy neighborhoods.
2. Every L-fuzzy topology is generated by some L-fuzzy quasi-uniformity. Let (X, 7") be an L-fuzzy
topological space. For every M E 7" let UM: L z ~ L z be defined by the equalities UM(N) = 1~
if N < M, and UM(N) = 1 otherwise. Then /3 = {UM: M E 7"} is a subbase for an L-fuzzy
quasi-uniformity 5/-. The L-fuzzy topology, generated by 5/,-, is just the original L-fuzzy topology
r, i.e., r = 7"u~ [68, 77].
3. If a mapping f': (X,5/x) ~ (Y,5/y) is uniformly continuous, then the mapping f : (X, 7"Ux) --+
(Y, 7"uy) is continuous [68].
4. An L-fuzzy topology is generated by an L-fuzzy uniformity iff it is completely regular [68, 18, 37].
5. A n L-fuzzy topological space (X, v) is normal and Ro [70] iff the fami:y B = {U E D: the range of U
is finite and U(M) is a neighborhood of M for every M e L X ) is a base for an L-fuzzy uniformity,
compatible with 7" (cf., the well-known crisp prototype of this fact : a completely regular space is
normal, iff the family of all its finite open covers is a base for a compatible uniformity),
6. Every compact [107] completely regular [68, 76, 183] L-fuzzy topological space admits a unique compatible L-fuzzy uniformity. Furthermore, this L-fuzzy uniformity is totally bounded (2.1.6) (see [18]).
In this connection, it is interesting to note that if (X,5/) is an L-fuzzy uniform space, (X,~-u) is
corfipact, and M E L X is closed, then for every open N satisfying M < N there exists U E 5 / s u c h
that U(M) < N (see [18]).
7. The family of all I-fuzzy quasi-uniformities compatible with a given I-fuzzy topology is studied
in [86]. In particular, it is shown that the finest fuzzy quasi-uniformity on a given fuzzy topological
space consists exactly of all normal fuzzy neighbouraets. (A mapping A: I X --+ I x is called a
fuzzy neighbournet if A(0) = 0, A preserves joins and M < Int A(M) for each M e I X. A fuzzy
neighbournet is called normal is there exists a family (An)naN of fuzzy neighbournets such that
2
A1 = A and An+l
< An for all n E N [86].)
2.1.3. Basic O p e r a t i o n s . To give the reader an idea of how operations on the objects of the categories
H F U ( L ) and H F Q U ( L ) are defined, we describe here the operations of taking a subspace arid a product.
Given an L-fuzzy quasi-uniform space (X,5/) and a set Y C X, for every M E L y let M* E L X be
defined by M*(x) = M(x) if x E Y and M*(x) = 0 otherwise. Now, for every U E 5/ let Uy: L Y --+ L y
be defined by the equality U y ( M ) = U(M*) ]y. The family {Uy: U E bt} = 5/y is a quasi-uniformity
on Y, and the pair (Y,5/y) is called the subspace of the space (X,5/). One can see that (Y,5/y) is indeed
a subobject of (X,5/) in the category H F Q U ( L ) [165]. On the other hand, if 5/ is a fuzzy uniformity,
then 5/y constructed as above, may fail to be a fuzzy uniformity: as the corresponding subobject of the
fuzzy uniform space (X,5/) in the category H F U ( L ) one has to take the space (Y,5/y) (see notation in
682
2.i.1) [165].
Passing to the product operation, consider a family {(Xj,Uj) : j E J} and let X = 1-IXj be the product
of the corresponding sets. For each j E d let PTt(b/j) = { P 7 1 o I,) o Pj: Vj E b/j}, where Pj: X --+ Xj
stands for the corresponding projection, and let U be the fuzzy quasi-uniformity, generated by the family
OPj-1 (b/j) as a subbase. Then b/is exactly the weakest fuzzy quasi-uniformity on X for which all projections
1
are uniformly continuous, and hence (X, b/) is the product of the considered spaces in the category H F Q U ( L )
(see [165]). Again, if all the spaces were fuzzy uniform, the product nevertheless might fail to be fuzzy
uniform.
The product in the category H F U ( L ) is to be defined as the pair (X,/X) (see notation in 2.1.1) [165].
2.1.4. C o n n e c t i o n s B e t w e e n ( Q u a s i - ) U n i f o r m i t i e s a n d Fuzzy ( Q u a s i - ) U n i f o r m i t i e s (in the case
L = I) were thoroughly investigated by A.K. Katsaras [77]. Here is a brief outline of the main results in
this direction.
Given a set X, let, for each subset W C X x X containing the diagonal &, a mapping Uw: I x -+ I x
be defined by the equality Uw(M)(x) = sup{M(y): (Y, x) E W} for each M E I x. Then, as it is easily
seen, Uw satisfies the axioms (HD1) and (HD2) from 2.1.1. Moreover, if W is an ordinary quasi-uniformity
(resp. uniformity) on X, then {Uw: W E )IV} is a base for some fuzzy quasi-uniformity (resp., a base
for some fuzzy uniformity) T(W) Oil X. Besides, if a mapping f: (X, Wx) --+ (Y, Wy) of (quasi-)uniform
spaces is uniformly continuous, then the mapping f : (X, cy(Wx)) --+ (II, c2(Wy)) of the corresponding
fuzzy (quasi-)uniform spaces is uniformly continuous too. Therefore, ~ can be considered as a functor from
this category ( Q ) U n i f of (quasi-) uniform spaces into the category H F ( Q ) U ( I ) of Hutton fuzzy (quasi-)
uniform spaces.
Let Tw stand for the topology induced on X by a (quasi-)uniformity IV. Then the equality T~(w) =
a~(Tw) holds, i.e., the fuzzy topology generated by ~(W) is just the image of Tw under Lowen's functor
0J. This implies, in particular, that if a topology T is uniformizable, then the corresponding fuzzy topology
~o(T) is fuzzy uniformizable.
Conversely, given a set X and a function U: I x -+ I x satisfying conditions (HD1) and (HD2) of (2.1.1),
define the set Wu = {(x,9) E X x X : M(z) <_U(M)(y) for all M E IX}. As shown by A.K. Katsaras, ifb/
is a fuzzy quasi-uniformity (resp. uniformity) on X, then {Wu: U E/X} is a base for some quasi-uniformity
(resp. uniformity) 9(/,/) on X. Besides, if a mapping f : (X,b/x) -+ (Y,g{y) of fuzzy (quasi-) uniform
spaces is uniformly continuous, then the mapping I : (X, ~(b/x)) -+ (II, q(/./Y)) of the corresponding
(quasi-)uniform spaces is uniformly continuous too. Thus q can be considered as a functor from H F ( Q ) U
into (Q)Unif.
The functor t9 is the left inverse of the functor ~, i.e., q o ~(W) = )IV for each (quasi)uniformity W.
On the other hand, if b/is a fuzzy (quasi-)uniformity, then ~ o ~(b/) is the smallest of all fuzzy (quasi)uniformities which are finer than W and which are induced by (quasi-)uniformities.
2 . 1 . 5 . L-Fuzzy U n i f o r m i t i e s a n d L-Fuzzy R e a l Line. An important role in the theory of L-fuzzy
(quasi-)uniformities is played by the L-fuzzy real line (1.1.6) endowed with the standard L-fuzzy uniformity
described below.
For each e E IR+ consider the mapping B~: L R(L) -+ L ~(L) defined by B~(M) = rt-e, where t =
max{s: M _< (1,) c} (see notation in 1.1.6). Then the family {B~,B~-I: e E R +} is a subbase for an
L-fuzzy uniformity on IR(L) which is compatible with the standard L-fuzzy topology on it, and hence the
standard L-fuzzy topology on 1R(L) is uniformizable [68, 165].
Quite recently Zhang De-xue [211] proved that the laminated L-fuzzy real line (IR(L), r ~) (1.1.6) is also
fuzzy uniformizable, thus solving a problem posed by S.E. Rodabaugh [169, 165].
Let (X,b/) be an L-fuzzy uniform space and M , N E L x satisfy the inequality U(M) <_ N for some
U E b/. Then there exists a uniformly continuous function f : X --+ I(L) such that M(x) <_ f ( x ) ( l - ) _<
f(z)(0 +) ___ N(x) for every x E X. Conversely, assume that for every open L-fuzzy set N in an L-fuzzy
space (X, r) there exists a collection of L-fuzzy sets (Mj)jEj C L x and a collection of continuous mappings
fj: X ~ I(L); j E J, such that Mi(z ) <_ f j ( z ) ( 1 - ) _< fj(z)(O +) <<_N(z) for every x E X. Then
(X, ~') is uniformizable [68]. From the above results it follows, that an L-fuzzy topological space is fuzzy
683
uniformizable iff it is completely regular [68].
An important contribution to the study of L-fuzzy uniformities on L-fuzzy real lines appear in [169, 165].
This enhances in particular the understanding of the general theory of L-fuzzy uniformities a n d fuzzy
(probabilistic) metrization of L-fuzzy topologies.
2.1.6. S o m e O t h e r T r e n d s in t h e T h e o r y o f H u t t o n F u z z y U n i f o r m i t i e s . In addition to the
study of relations between Hutton fuzzy uniformities and Chang-Goguen fuzzy topologies, and between
Hutton fuzzy uniformities and ordinary (crisp) uniformities discussed above, some important connections
between Hutton fuzzy proximities were established recently. (We shall say more about these connections
in the next chapter).
The property of total boundedness, or precompactness, for Hutton's fuzzy uniformities was studied in
[15, 18]. (An L-fuzzy uniformity is called totally bounded [15] if there exists a base B of L/ such that for
every U" E B the set {U(M); M E L x } is finite.)
This definition is compatible with the classic one, i.e., in the case L = {0, 1}, both definitions are
equivalent [18]. In particular, if an L-fuzzy space is compact [107] and completely regular [68], then its
(unique (2.1.2)) L-fuzzy uniformity is totally bounded. Totally bounded L-fuzzy uniform spaces form a
reflective subcategory in the category H F U ( L ) .
Separation properties of L-fuzzy (quasi-)uniformities were discussed in [70, 169, 168]. T h e problem of
fuzzy metrizability of Hutton fuzzy uniformities was considered in [70, 169, 16].
Attempts to study uniform properties of L-fuzzy subsets in L-fuzzy uniform spaces were undertaken in
[169, 168, 84]. In particular, in [34] it is proved that given an I-fuzzy uniform space (X,L/) and a fuzzy
subset M E I x, then C ( M ) C Cpl(M)nPC(M), where C(M), Cpl(M) and PC(M) C I are so-called
spectra of compactness, completeness, and precompactness of M in (X,/,().
In [207] relations between fuzzy bitopologies and fuzzy quasi-uniformities are investigated. The main
result in this direction states, that a fuzzy bitopological space is fuzzy quasi-uniformizable iff it is palrwise
fuzzy completely regular. Moreover, if a fuzzy topology is pairwise compact, then its fuzzy bitopology is
induced by a unique fuzzy quasi-uniformity.
2.2. L o w e n F u z z y U n i f o r m i t i e s
2.2.1. Basic D e f i n i t i o n s . An alternative approach to the concept of a fuzzy uniformity was developed
by R. Lowen [113] (in the case L = I). According to R. Lowen, a fuzzy uniformity on a set X is a set
/d C I x x x satisfying the following axioms :
(LU1) L/is a prefilter on X x X (1.1.5);
(LU2) U(z, z) = 1 for every U E/A and each z E X;
(LU3) if U E L/, then U -1 E/4 (where U - l ( z , y ) = U(y,z));
(LU4) for every U E/A and every e E (0, 1] there exists U., E H such that U, o U, - e _< U;
(LUS) if U~ E/4 for every e E (0, 1], then sup (U~ - e) E/A
e~(0,1]
(cf., N. Bourbaki's definition of a uniformity [2]). The pair (X,L() is called a (Lowen) fuzzy uniform space.
A mapping f : (X, ltx) ~ (Y, g4y) of fuzzy uniform spaces is called uniformly continuous, if (.f •
f)-x (V) E L/x for every V E My. Equivalently, a mapping f : (X,L/x) -+ (Y,/4y) is uniformly continuous,
if for every V E g/y and every r > 0 there exists U E L/such that U - r < ( f • f ) - I (V). T h e category of
Lowen fuzzy uniform spaces and uniformly continuous mappings will be denoted L F U .
By omitting in the above definition the axiom (LU3), A.K. Katsaras introduces the concept of a (Lowen)
fuzzy q~asi-uniformity and (Lowen) fuzzy quasi-uniform space [79, 80]. The corresponding category will be
denoted L F Q U .
A somewhat more general notion than the Lowen fuzzy uniformity, the so called t-uniformity (where
t E I) was recently considered by M.A. de Prada and J. Gutierrez [56].
2.2.2. F u z z y T o p o l o g i e s G e n e r a t e d b y F u z z y U n i f o r m i t i e s [113]. A fuzzy uniformity L / o n a set
X generates a fuzzy topology t(L() as follows :
Let M E I x and U E g/. The fuzzy set U -~ M ~-E I x defined by U -~ M ~- -(z) = sup M(y)/~ U(9, .r) is
yEX
called the section of U over M. Now, by setting M = inf U -< M ~, R. Lowen obtains a closure operator
UE/,/
684
-" I x -+ I and consequently, a Chang fuzzy topology t(L/) on the set X. This fuzzy topology is laminated.
If a mapping f : (X, L/x ) -+ (Y,/gx) of fuzzy uniform spaces is uniformly continuous, then the mapping
f: (X, t(L/x)) -+ (Y, t(L/v)) of the corresponding topological spaces is continuous. Thus, t can be considered
as a functor t: LFU ~ L C F T ( I )
A.K. Katsaras [80] introduced the notion of an n-completely regular fuzzy topological space by means
of a function defined on a given space and taking values in the so called probabiIistic real line (see also
2.3.3) and characterized uniformizable fuzzy topologies as those ones which are n-completely regular [80].
Recently, similar results were obtained by A.S. Masshour and N.N. Morsi [138, 139].
R. Lowen fuzzy uniformities are closely related to so-called fuzzy neighborhood structures~ (These structures, introduced by R. Lowen [1 12], play an important role in certain areas of fuzzy topology.) In particular.
as shown by A.K. Katsaras [80], a fuzzy topology is quasi-uniformizable if it is generated by a fuzzy neighborhood structure.
2.2.3. I n i t i a l F u z z y U n i f o r m i t i e s . P r o d u c t s o f F u z z y U n i f o r m S p a c e s . S u b s p a c e s o f Fuzzy
U n i f o r m S p a c e s [ l l 3 ] . Let X be a set, let {(Yj,L/j): j E J} be a family of fuzzy uniform spaces.
and let {fj: X ~ Yj;j E J} be a family of m a p p i n g s . The weakest fuzzy uniformity on X for which all
these mappings become uniformly continuous, is called the initial fuzzy uniformity for this family. Following
R. Lowen we denote this fuzzy uniformity by sup(fj x fj)-i (L/j). (R. Lowen also gives an explicit description
jEJ
of this fuzzy uniformity, which we shall not reproduce here.)
The operation of taking initial fuzzy uniformities is compatible with that of taking initial fuzzy topologies:
t(sup(fj x fj)-l(L/))= sup(t(L/j)).
jEJ
jEJ
Given a family {(Xj,L/j): j E J } of fuzzy spaces, its product is defined in the usual way, i.e., as the
pair (X,L/) where X = r!Xj is the product of the corresponding sets, and L/is the initial fuzzy uniformity
3
for the family of all projections pj: X --+ (Xj, L/j).
Given a fuzzy uniform space (X, L/) and subset Y C X, the induced fuzzy uniformity L/v on Y is defined as
the initial fuzzy Uniformity for the natural inclusion e: Y -+ X. It is easy to see that L/v = {UIv•
UE
b/}, i.e., fuzzy entourages in L/v, are just restrictions of the fuzzy entourages from/A to the set Y x Y.
2.2.4. C o n n e c t i o n s B e t w e e n U n i f a n d L F U : T w o I m p o r t a n t F u n e t o r s . Unlike B. Hutton's approach, R. Lowen's definition of a fuzzy uniformity is not a direct generalization of a usual (crisp) uniformity
(of., the situation with laminated Chang:Goguen fuzzy topological spaces). Therefore, to establish natural
connections between the categories U n i f and L F U , R. Lowen introduced two functors wu: Unif--+ L F U
and Lu: L F U --+ Unlf, which are, in fact, uniform counterparts of the functors co (1.2.2) and ~ (1.:1.3)
respectively, [113].
The functor coot assigns to a uniform space (X,D) the fuzzy uniform space (X, cov(D)) , where cou(D) =
{U 6 IXxX: Va E [0,1) U - l ( a , ll E D}, and leaves the morphisms unchanged. Since the uniform
continuity of a mapping f : (X, Dx) -+ (Y, 23y) is equivalent to the uniform continuity of the mapping
wv(f) := f : (X, cov(L/x)) ~ (Y,w~,(L/y)), the functor w U embeds U n i f as a full subcategory into L F U
[113].
Now, let (X,L/) be a fuzzy uniform space and let Cu(L/) = { U - l ( a , lli U E L / , a E [0,1)}. T h e n %: is
a uniformity on X. Furthermore, if a mapping f : (X, L/x) -+ (!/', L/Y) is uniformly continuous, then the
mapping Lu(I ) := f: (X, ~a(L/x)) --+ (Y, tu(L/Y)) is also uniformly continuous. (The converse, naturally,
does not hold.) Thus, by letting Lu(X,L/) = (X, Lu(L/) ) and tg(f) = I, the functor Lu: LFU -+ Unif is
defined. This functor is the left inverse of c~u, i.e., the equality Lu(cov )(23) = D holds for every uniformity
23. On t h e other hand, wv (~u(L/)) is the weakest fuzzy uniformity among all fuzzy uniformities which are
finer than L / a n d which are generated by uniformities.
If 7) is a uniformity on X and T~ is the topology induced by it, then t(cov(23)) = w(T~). Similarly, if L/
is a fuzzy uniformity on X, then T,u(u ) = L(t(L/)). Furthermore, a topological space (X,T) is uniformizable
iff the fuzzy topological space (X, co(T)) is fuzzy uniformizable.
2.2.5. a - L e v e l D e c o m p o s { t i o n o f a Fuzzy U n i f o r m i t y . Given a fuzzy uniform space (X,L/) and
a E (0,1], let L/a = {U-1(/3, 1]: U E M,3 < a}. Then L/a is an ordinary uniformity (so-called a-level
685
uniformity) and, besides, i f 0 < /3 _< o < 1, thenL/~ C /d a C b/i a n d U ~ =
U /gO [22]. The fuzzy
uniformity b/ is uniquely determined by the family of its o-level Uniformities in the following sense: If
{/,/a : a E (0, 1] } is a family of ordinary uniformities on a set X satisfying the two conditions above, then
br = {U E IXxX: Vo E (0, 1) V/3 < o U-1(/3, 1] E H a} is a fuzzy uniformity on X and b/= are exactly its
a-level uniformities [22]. A mapping f : (X, blx) --+ (Y, bly) is uniformly continuous iff for each a E (0, 1)
the mapping f : (Z,b/~) ~ (Y,L/~)is uniformly continuous [22].
The method of decomposition of a fuzzy uniformity into the family of its a-level uniformities is quite
useful for the study of various properties of fuzzy uniformities, see, e.g., [24-26].
2.2.6. C o m p a c t n e s s ~ P r e c o m p a c t n e s s ~ a n d C o m p l e t e n e s s of Fuzzy U n i f o r m S p a c e s were first
studied by R. Lowen and P. Wuyts [124, 125]. Here are some results from these papers.
On a compact [107] fuzzy topological space at most one compatible fuzzy uniformity can exist. (It
consists of all neighborhoods of the diagonal.) If Y is a compact subspace of a fuzzy uniform space (X,/g),
then the family of all neighborhoods U -< 1y ~-, U E b/ (2.2.2) forms the fuzzy neighborhood structure (in
the sense of [112]). If a mapping f : (X, t(lgx)) --+ (Y, t(lgy)) is continuous and t(btx) is compact, then the
mapping f : (X, b/x) --+ (Y, Uu is uniformly continuous. Analogous to the situation in general topology, a
fuzzy uniform space is compact iff it is precompact and complete.
(A fuzzy uniform space (X, b/) is called precompact if for every U E b / a n d every ~ > 0 there exist fuzzy
sets M 1 , . . . , M n such that M j x M j - e < U for a l l j = 1 , . . . , n , and VMj >_ 1 - e . The definition of
J
completeness is based on the notion of Cauchy prefilters which are fuzzy counterparts of ordinary Cauchy
filters. Unfortunately it is too bulky to be reproduced here.)
The introduced notions are "good extensions" in the sense of R. Lowen (1.2.2) : a uniform space (X,L/)
is precompact (complete) iff the fuzzy uniform space (X,w v (V)) is precompact (resp. complete).
For every fuzzy uniform space (X,b/) R. Lowen and P. Wuyts constructed a completion (X,/g). This
completion possesses some properties which one could expect in analogy with the situation in ordinary
topology. For example a space is dense in its completion. If (Y, btv) is an ultracomplete (this is a strengthened form of completeness) Hausdorff uniform space, and a mapping f : ( X , / g x ) ~ (Y, blv) is uniformly
continuous, then there exists a unique continuous extension f : (X,btx) -+ (Y,glg) which is, moreover
uniformly continuous.
On the other hand, there are significant departures from the situation in ordinary topology. Particularly,
there does not generally exist an isomorphism between a complete fuzzy uniform space and its completion.
However, the completion of an ultracomplete fuzzy uniform Space coincides with the original space.
2.2.7 P r e c o m p a c t n e s s , C o m p l e t e n e s s ~ a n d B o u n d e d n e s s o f Fuzzy S u b s e t s of F u z z y U n i f o r m
Spaces. In a series of papers by M. Burton [24-26], and in his Ph.D Thesis, the theory of precompactness
was extended to the case of fuzzy subsets of fuzzy uniform spaces. This extension relies heavily on the
technique of Cauchy prefilters developed .in [23]. Unfortunately, we cannot go into details of this interesting and, we think, promising theory, because this would involve a number of auxiliary definitions and
constructions. Below are listed only some of the easily formulated results obtained in this direction.
A uniformly continuous image of a precompact (bounded) fuzzy set is precompact (resp. bounded).
A fuzzy set in the product of fuzzy uniform spaces is precompact iff all its images under projections are
precompact. The product of complete fuzzy sets is complete.
Let (X, b/) be a Hausdorff fuzzy uniform space, M, N E I x, N _< M and assume that M is complete.
Then N is complete, iff N is dosed. If N is precompact, then N is relatively compact.
2.2.8 Fuzzy U n i f o r m i t i e s on H y p e r s p a c e s o f Fuzzy Sets. The techniques of fuzzy uniformities
was successfully applied for the investigation of the hyperspace I x of fuzzy sets of a given uniform space
X; in particular, for the study of different convergence structures on I x. Several quite interesting results
were obtained in this direction by R. Lowen [115] et al..
2.3. H g h l e s A p p r o a c h : Fuzzy T - U n i f o r m i t i e s
At first glance it may seem that B. Hutton's and R. Lowen's definitions of fuzzy uniformities are so
far apart that there could hardly exist a framework which would include both approaches in a natural
way. However, by means of the so-called t-norms U. Hhhle [64] succeeded in introducing the concept of
N
686
a flLzzy t-uniformity, which, for an appropriate choice of a t-norm T, contains in itself both R. Lowen's
and B. Hutton's approaches - - the latter one, however, subject to some restrictions. Moreover, fuzzy
T-uniformities proved themselves to be an appropriate tool for the study of the problem of probabilistic
metrization.
2.3.1 t - N o r m s (see, e.g., [172, 173]). A continuous mapping T: I x I -4 I is called a triangular norm.
or a t-norm for short, if it satisfies the following four conditions :
(1) T(0, 0) = 0, T(x, 1) = T(1, x) = x (boundary condition);
(2) T(z, y) <_ T(z', y') whenever x _< x', 9 ~ Y' (monotonicity);
(3) T(x, y) = T(y, x) (symmetricity);
(4) T(T(x,y),z) = T ( x , T ( y , z ) ) (associativity).
Important examples of t-norms: TM(X, y) = max(z + y -- 1, 0); Tp(z, y) = x. y; Tin(x, y) = min(x, y).
2.3.2 F u z z y T - U n i f o r m i t i e s [64]. Let T be a t-norm. A nonempty subset 7-/C I x x x is called a fiLzzy
T-uniformity if it Satisfies the following axioms :
(TU1) If H E I HxX and for every e > 0 there exists G~ E ~ / s u c h that G~ - e < H, then H E ~ ;
(TU2) if H1,//2 E ~ , then H1 A H2 E ~;
(TU3) if H E ~ , then H(x,z) = 1 for each z E X;
(TU4) if H E 7-/, then H -1 E ~ ;
(TU5) for every H E ~ and every e > 0 there exists G~ E ~ such that Ge ~ G e - e < H, where
V o a = sup T(C(x,z),G(z,y)).
T
z6X
In the case T =Tm this definition is obviously equivalent to that by R. Lowen. On the other hand, as
U. Hhhle showed [62], TM-uniformities are exactly those Hutton uniformities/d which have a base B such
that H - I E B whenever H E B and H(zO)(y) = TM(H(xl)(y),a) for any H E B , z , y E X, and a E (0, 1].
(Fuzzy TM uniformities were also considered in [62] under, the name probabilistic uniformities.)
2.3.3 P r o b a b i l i s t i c M e t r i z a t i o n o f F u z z y T - U n i f o r m i t i e s . The concept of a fuzzy T-uniformky
appeared to be useful also to discover deep relations between structures of fuzzy uniform';~y type and the
problem of so-called probabilistic metrization [64]. Here are some details of these relations :
Let D + be the set of all probabilistic distribution functions p: lR -4 I such that p(0) = 0 (for a related
concept of the fuzzy real line, see Sec. 1.1.6). Every t-norm T induces on D -t- a binary operation CrT defined
by aT(p,q)(x) = sup{T(p(s),q(t)): s + t = x}. Now a probabilistic metric space can be defined as a triple
(X, 5r, T), where X is a set, T is a t-norm, and a so-called probabilic metric jz: X x X -4 D + satisfies the
following conditions :
(PM1) ~'(z, y) = e0 if x = y (where eo(s) = 1 for z > 0 and eo(s) = 0 otherwise);
(PM2) J:(x,y) = 7-(y,x);
(PM3) r
z), .~(y, z))(t) ~_ .~(x, y)(t) for each t E ]R.
A probabilistic metric .T" in a natural way generates a fuzzy T-uniformity 7-/(9v) on the same set : 7-/(.%-) =
{H E IX•
Ve > 0 3n E N such that 5r(x, y)(2 - " ) - e _< H(x,y) Yx, y E X } . Now the main result in this
direction can be given as follows:
A fuzzy T-uniformity 7"t is probabilistic metrizable iff 7-I is 1-Hausdorff (i.e., inf{H(x,y): H E ~ } < 1
for every x, y E X, x # y) and 7-I has a countable base of fuzzy vicinities [64].
Recently the investigation of tb_e problem of probabilistic, or as it is usually called now fuzzy, metrizability
of fuzzy uniformities and some related problems were further developed in a series of papers by N.N. Morsi
et aI. (see, e.g., [13, 14, 139, 1521).
2.4. (L, K ) - F u z z y ( Q u a s i - ) U n i f o r m i t i e s
Fuzzy uniformity-type structures which are in accordance with (L, K)-fuzzy topologies (1.3) were considered in [35]. Here is a brief outline of the approach which was started in [35].
Let L be a completely distributive lattice and let K be a dense chain. (In fact, the approach in [35]
was originally restricted to the case L = K = I; however, without much change it can be extended to the
above-mentioned situation.) Let X be a set and D have the same meaning as in 2.1.1. An (L,K)-fuzzy
quasi-uniformity on a set X is a mapping/X : D -4 K satisfying the following conditions :
(FU0) s u p U ( D ) = 1;
687
(FU1) if U, V E D and U _< V, then 5/(U) < U(V);
(FU2) if U, V E D, then 5/(U A V) _<5/(U) A 5/(Y);
(FU3) U ( U ) = sup(U(V): V ED, V o V <U}.
If, besides,/4(Y) =/,/(V -1) for each Y E D, then U is called an (L, K)-fuzzy uniformity. A mapping
f: (X,5/x) -+ (Y,5/y) of (L, K)-fuzzy (quasi-)uniform spaces is called uniformly continuous ifh/(f -1 (V)) >__
5/(V) for each V E Dy.
Given an (L, K)-fuzzy quasi-uniformity/4 on a set X, tile corresponding (L, K)-fuzzy topology Tu : L x -+
K can be defined by the equality Tu = sup{a: V { N E L x 3U E L x such that 5/(U) > a and U(N) <_
M} = M}. The uniform continuity of a mapping f : (X,5/x) -+ (Y,5/y) implies continuity of the mapping
f: (X, Tux ) -+ (Y, Tuv ). Thus the correspondence/d -+ Tu defines a functor from the category FQU(L, K)
of (L, K)-fuzzy quasi-uniform spaces into FT(L, K).
Among other questions considered in [35] are the following : the decomposition of an (L, K)-fuzzy (quasi)uniformity into decreasing system of its a-level Hutton L-fuzzy uniformities; relations between (L, K)-fuzzy
uniformities and the induced (L, K)-fuzzy topologies; operations on (L, K)-fuzzy uniform spaces and others.
2.5. Fuzzy U n i f o r m i t i e s in A c c o r d a n c e w i t h H u t t o n Fuzzy Topologies. The definition of Hutton
(quasi-)uniformities (2.1.1) is actually a point-free one, in the sense that the extra structure of the lattice
L x is not used; therefore, by a direct replacement of L x by an arbitrary completely distributive lattice s in
Definition 2.1.1, one arrives at the concept which is in accordance with Hutton fuzzy topologies (Sec. 1.4).
To be more precise, in this context a fuzzy uniform space is defined as a pair (s
where s is a completely
distributive lattice and/4 is its subset satisfying axioms (HD1), (HD2) and (HU1)-(HU4) of 2.1.1 (see [70]).
A large number of the concepts and results given in Sec. 2.1 can, in a natural way, be reformulated for this
situation. Moreover, some of the results presented in Sec.2.1, in the context of the category H F U ( L ) , were
originally obtained for fuzzy uniform spaces in the above-mentioned sense. "(This concerns, in particular,
the results cited from [181 and [165].)
However, when defining the category appropriate.for this context (we shall denote this category by
H F U ) one must take into account that in accordance with Hutton fuzzy topologies (Sec. 1.3) a morphism
f from (s
into (s
is to be introduced as a mapping f - i : s --+ s (i.e., going in the opposite
direction) such that f - l ( Y ) E 5/1 whenever Y E L/2 (where f-I(V)(M) := f-1 (V(I(M))) for each M E s
and f(M) = inf{N E L2: M < f - l ( Y ) } ) ; see, e.g., [181.
2.6. On S o m e O t h e r A p p r o a c h e s to t h e C o n c e p t of a Fuzzy U n i f o r m i t y .
2.6.1 Fuzzy U n i f o r m i t i e s on M o l e c u l a r Lattices. Wang Guojun considers fuzzy uniformities on
molecular lattices satisfying some additional requirements [194]. This approach is consistent with the
theory of topological molecular lattices (see 1.5) and, to a certain extent, can be considered as dual to the
approaches discussed in Secs. 2.1 and 2.5.
2.6.2 Fuzzy U n i f o r m i t i e s with a C h a n g i n g B a s i s : S.E. R o d a b a u g h ' s A p p r o a c h . A category
containing categories H F U ( L ) (as nonfull subcategories) for different lattices L was introduced by S.E. Rodabaugh [165]. This Category, which will be denoted here by R F U is in accordance with the categories
of fuzzy topology discussed in Sec. 1.6. The objects of R F U are triples (X, L,/A), where X is a set, L is
a completely distributive lattice, and 5 / a n Hutton L-fuzzy uniformity on X; the morphisms of R F U are
triples (f, qa, ff2): (X1,L1,5/1) ~ (X2,L2,5/2), where f : Z l --+ X2, ha: L1 --+ L2 and r
--+ L1 are
mappings, satisfying certain conditions [165].
Among the problems studied by S.E. Rodabaugh in this context are operations in R F U , relations to
other categories of fuzzy topology, the problem of metrization of fuzzy uniformities, and the role of L-fuzzy
real lines in the theory of fuzzy uniformities.
2.6.3 Fuzzifying Uniformities. Carrying out by means of continuous-valued logic a semantic analysis
of the concepts a n d results of the theory of ordinary uniform spaces, Mingsheng Ying introduced the
concept of a fuzzifying uniformity and developed foundations of the corresponding theory [149]. Fuzzifying
uniformities are the uniform counterparts of fuzzifying topologies which were discussed in Sec. 1.7.
3. F U Z Z Y P R O X I M I T I E S
3.1. Katsaras L-Fuzzy Proximities.
688
3.1.1 Basic C o n c e p t s [76, 15]. Let L be a completely distributive lattice and let X be a set. Then
the binary relation ~: L x x L :~" --+ 2 on L X is called a (Katsaras) L-fuzzy proximity on X if it satisfies the
following axioms (M, N, P E L X) :
(KP1) 5(1,0) = 0;
(KP2) 5(M, N) = 5(Y, M);
(KP3) 5(M V N, P ) = 5(M, P) V 5(N, P);
(IiP4) if 5(M, N) = 0, then there exists P E L X such that 5(M, P) = 5(P c, N) = 0;
(KPh) 5(M, N) = 0 implies M _ N c (i.e., M and N are not quasicoincident).
A pair (X, 5) is called a (gatsaras) L-fuzzy proximity space. A mapping f : (X, 5x) -+ (Y, 5y) is called
proximal, or proximally continuous, if 8x(]l,N) = 1 (M,N E L X) implies 5y(f(M),f(N)) = 1, or
equivalently, if 8y(P,Q) = 0 (P,Q E L y) implies d x ( f - l ( P ) , f - i ( Q ) ) = 0. Let K F P ( L ) stand for the
category of Katsaras L-fuzzy proximity spaces and proximally continuous mappings. Obviously, in the case
L = 2, the category K F P ( 2 ) can be identified with the category P r o x of ordinary proximity spaces (see,
e.g., [4, 6, 153, ii]).
(In fact, A.K. Katsaras ([76] et al.) restricts himself to the case L -= I. In case of an arbitrary completely distributive lattice L, the concept of an L-fuzzy proximity was first introduced by G. Artico and
R. Moresco [15].)
Note also, that, in his first papers on fuzzy proximities [74, 75], A.K. Katsaras used another definition
of a fuzzy proximity, which differs from the one presented here in axiom (KP5) instead of which a stronger
axiom (KPh') "~(M, N) = 0 implies M A N = 0" was accepted. However, this definition turned out
to be unsuccessful, in particular, because of the fact that L-fuzzy proximities, satisfying (KPh') are in
a canonical one-to-one correspondence with the ordinary proximities on t h e same set, and moreover, an
ordinary proximity and the corresponding fuzzy proximity satisfying (KPh') induce on X the same crisp
topology (see, e.g., [15]).
In [79] A.K. Katsaras introduced the concept of a (Katsaras) L-fuzzy quasi-proximity, in fact, in the
paper, the case L = I was considered). The definition of an L-fuzzy quasi-proximity can be obtained from
the above definition of an L-fuzzy proximity by omitting the axiom (KP2) a n d , on the other hand. by
"symmetrizing" axioms (KP3) and (KP1).
3.1.2. Connection B e t w e e n L - F u z z y P r o x i m i t i e s a n d L - F u z z y T o p o l o g i e s . Given an L-fuzzy
proximity space (X,~) and M E L x , l e t I n t M := V{N: ~(N,M c) = 0}. Then Int: L x -+ L x is a
fuzzy interior operator (1.1.1) and the corresponding Chang-Goguen L-fuzzy topology v~ is called induced
or generated by 5. The proximal continuity of a mapping f : (X, Sx) --+ (Y,c~y) implies the continuity
of the mapping f: (X, T~x ) -+ (Y, rsv) and, therefore, the correspondence 8 -+ rs determines a functor
~: K F P ( L ) -+ C F T ( L ) (see [76] in the case L = I; [15]; see also [791 in the context of L-fuzzy quasiproximities).
Every L-fuzzy topology induced by an L-fuzzy proximity is completely regular. Conversely, let r be a
completely regular L-fuzzy topology on X, and define 8: I x x I x -4 2 by setting 8(M, N) = 0 iff there
exists a continuous function f : X --+ I(L) (1.1.6) such that M(x) < f ( x ) ( 1 - ) < f(x)(0 +) < Nr
for all
z E X and 8(M, N) = 1 otherwise. Then 8 is a fuzzy proximity on X and furthermore r = r~ [76] (in the
case L = I); [15].
Given an L-fuzzy topological space (X, ~-) let 8(M, N) = 0 (M, g E L X) i f f M < ( ~ ) c and 8(M, N) = 1
otherwise. Then 8 is an L-fuzzy proximity iff the space (X, r) is normal (in the sense of [67, 70]); if
(X, r) is normal and Ro ([70]), then r 6 = r and ~ is the finest L-fuzzy proximity among those ones which
generate 7" [18].
3.1.3. Connection B e t w e e n L-Fuzzy Proximities and H u t t o n L-Fuzzy Uniformities. Given
an L-fuzzy uniformity/2 on a set X let $u(M, N) = 0 (M, N E L X) iff there exists U E /2 such that
U(M) <_N ~. The binary relation8: L X x L x ~ 2 thus defined is an L-fuzzy proximity on X and besides
L/and ~u induce the same L-fuzzy topology on X [76, 15].
Conversely, an L-fuzzy proximity ~ on X generates a Hutton L-fuzzy uniformity. Let .A = { (M, N) E
L x x L x : $ ( M , N ) = 0}, and for each ( M , N ) E A define a m a p p i n g UMN: L x --+ L z as follows:
UMN(O) = 0; UMN(A) = N c, if A < M (A E L X) and UMN(A) = 1 otherwise. Then S = {UMx:
689
(M, N) E A} is a subbase for an L-fuzzy uniformity/A~ generated by d. An L-fuzzy proximity and the
L-fuzzy uniformity generated by it induce on X the same L-fuzzy topology. Moreover, 5u, = ~ for each
L-fuzzy proximity 5 ([15]; cf. [77] in the case L = I).
Let/A be an L-fuzzy uniformity and p/A :=/46u. Explicitly, p/A can be obtained from the family { U E
U: {U(M): M E L z } is finite} as basis. It follows from here that p/A is exactly the finest totally bounded
(2.1.6) L-fuzzy uniformity coarser t h a n / d (i.e., phr C/d) ([15], cf. [77] in the case L = I).
Let (X, Ux), (Y, Uy) be L-fuzzy proximity spaces. T h e n a mapping f: (X, 8ux ) -'+ (Y, ~uv ) is proximally
continuous iff the m a p p i n g f : (X, pUx) --+ (Y, p/Av) is uniformly continuous. Therefore, in particular, the
spaces (X, ~Ux ) and (Y, Bur ) are isomorphic in K F P ( L ) iff the spaces (X, p/Ax) and (Y, p/Av) are isomorphic
in H F U ( L ) ([15], cf. [77] in case L = I).
The connections between L-fuzzy quasi-proximities and L-fuzzy quasi-uniformities, (which are to a large
extent similar to the ones between L-fuzzy proximities and L-fuzzy uniformities discussed above) were
studied in [79].
3.1.4. Lattice P r o p e r t i e s o f L-Fuzzy P r o x i m i t i e s . I n i t i a l L - F u z z y P r o x i m i t i e s . P r o d u c t s of
L - F u z z y P r o x i m i t y S p a c e s . Let zx(X) denote the family of all L-fuzzy proximities on a set X. Then
A(X) endowed with the relation "<" is a complete lattice [104]. Explicitly, the supremum ~ = sup ~j of a
jEJ
family {~j: j E J} C A(X) can be described as follows:
Given A , B E L x , let ~(A,B) = 1 iff for arbitrary finite decompositions A = A1 V ... V An, B =
B1 V . . . V Bm there exist p E { 1 , . . . , n } and q E { 1 , . . . , m } such that ~i(Ap,Bq) = 1 for all j E J [104].
Let fj : X --+ ( ~ , ~J) be a family of mappings defined on a set X and with values in L-fuzzy proximity
spaces (~,~J)" T h e initial L-fuzzy proximity on X for this family (i.e., the weakest L-fuzzy proximity
making all f/ proximally continuous) is defined by ~ = sup f - l ( ~ / ) , where f~-~($j) is the weakest L-fuzzy
jEJ
proximity on X making fj proximally continuous. (From 3.1.1 it is clear how fj-1 (~j) is to be defined). The
product of L-fuzzy proximity spaces is defined in the usual way, i.e., as the pair (X, ~), where X = H Xj
jEJ
and ~ is the initial L-fuzzy proximity for the family of all projections (see [76, 104] et al.).
All these concepts and constructions are consistent with the corresponding concepts and constructions
for Chang-Goguen L-fuzzy topologies and B. Hutton L-fuzzy uniformities. In particular, given a mapping
f : Z -+ (Y,/A), the equality ~S-~(u) = f-l(~u) holds; if {(Xj,IAj);j E J} is a family of H u t t o n uniform
L-fuzzy spaces and (X,/4) is their product, then ~u = II ~uj, etc.
jeJ
3.2. Artico a n d M o r e s c o ' s Fuzzy Proximities.
As was observed before, the approaches to the concept of a fuzzy uniformity developed by B. Hutton
and by R. Lowen are essentially different. Therefore, as one can guess, the concepts of fuzzy proximities
corresponding to H u t t o n uniformities and to Lowen uniformities, whenever they exist, must also be quite
distinct and rely on the use of different ideas. In the previous section, we discussed fuzzy proximities which
are in accordance with H u t t o n fuzzy uniformities. The subject of this section is the fuzzy proximities
introduced by Artico and Moresco [17]. As it will be shown, these proximities, in a natural way, are related
to Lowen fuzzy uniformities and behave appropriately, from this point of view.
3.2.1 B a s i c C o n c e p t s . A (Artico and Moresco) fuzzy proximity on a set X is a mapping d: I x x I x --+ I
satisfying the following six axioms :
(AMP1) d(0, 1) = 0;
(AMP2) d(M, N) = d(N, M);
(AMP3) d(M, P) V d(g, P) = d(M V N, P);
( A M P 4 ) . i f d ( M , N ) = a, then for c a c h e > 0 there exist P , P ' E I x such that P V P ' = 1, d(M,P) <_
a+~andd(N,P')<a+r
(AMPb) d(U, N) > s u p ( U h W)(z);
~:EX
(AMP6) I U - W] _< r implies [d(M, P) - d(N, P)[ _< r for every P E I x.
(In the original paper [17] in axiom (AMP4) it was additionally required that P A P ' < a. However, as was
shown by N.N. Morsi [151], this condition in the context of the definition is redundant and hence can be
690
omitted.)
The pair (X, d) is called a (Artico and Moreseo) fuzzy proximity space. A mapping f: (X, dx) --+ ( t : dy)
of fuzzy proximity spaces is called proximal, or proximally continuous if d(M, N) < d'(f(M), f ( Y ) ) for all
M, N E I x. (Speaking informally, proximally continuous are those mappings which do not diminish the
degree of nearness between fuzzy sets.) Equivalently, a mapping f is proximally continuous if dy(P, Q) >_
dx ( f - 1 (p), f - 1 (Q)) for any P, Q E I y. The category of G. Artico and R. Moresco fuzzy proximity spaces
and proximally continuous mappings will be denoted A M F P .
N.N. Morsi [151] found a characterization of fuzzy proximities which proved to be quite useful in many
cases. While in the original definition the fuzzy proximity type relation on the fuzzy powerset I x is
involved, N.N. Morsi's characterization relies on the use of the fuzzy proximity type relation defined only
on the crisp powerset. According to N.N. Morsi, a (Artico and Moresco) fuzzy proximity can be defined as
a mapping d: I x • I x -4 I satisfying axioms (AMP'l) - (AMP'6), where axioms ( A M P ' l ) - (AMP'5) are
formulated in the same way as the corresponding axioms (AMP 1) - (AMPb), but referring only to crisp sets
M, P, N E 2 X, i.e., referring to the restriction (t: 2 x • 2 x --+ I of the mapping d. The last axiom (AMP'6)
shows how d is retrieved from its restriction d: d(M, N) = sup{d(M~, No) A a}, where M s -- M -1 [a, 1].
c~EI
Na = N -1 [a, 1]. Observe that, in this case, axioms (AMP4) and (AMPb) can be reformulated, respectively.
as follows :
(AMP"4) : If d(M, N) < a, a E I, then there exists P E 2 z such that d(M, P) < a and d(P c, N) < a.
(AMP"5) : If M U N r 0, then d(M, N) = 1.
This characterization of Artico-Moresco fuzzy proximities shows that these fuzzy proximities are much
closely connected with ordinary proximities than Katsaras fuzzy proximities.
3.2.2. C h a n g - G o g u e n F u z z y T o p o l o g i e s G e n e r a t e d by F u z z y P r o x i m i t i e s . Let (X,d) be a
fuzzy proximity space and M E I x. Then, by letting -~(x) = d(M, x) for each x E X, a fuzzy closure
operator - : I x -+ I is obtained. The corresponding laminated Chang-Goguen fuzzy topology r d is called
generated by d [17].
If a mapping f: (X, dx) --+ (IF,dy) of fuzzy proximity spaces is proximally continuous, then the mapping f : (X, rdx) -+ (Y, vdr) is continuous and hence the correspondence d --+ vd determines a functor
~: A M F P "~ L C F T ( I ) [17].
As shown by N.N. Morsi [151], the fuzzy topology vd is determined by a neighborhood structure in the
sense of R. Lowen [112].
3.2.3. C o n n e c t i o n s w i t h L o w e n F u z z y U n i f o r m i t i e s . Each Lowen fuzzy uniformity/2 on a set X
generates a fuzzy proximity du as follows: du(M,N) = inf sup(U -< M >-- A --< g >--)(x) (see 2.2.1). The
uElt x E X
fuzzy topologies induced on X by/2 and by du coincide. If f : ( X , / / x ) -'+ (Y, g/},) is a uniformly continuous
mapping of fuzzy uniform spaces, then the mapping f: (X, dux ) "+ (II, dur) of the corresponding fuzzy
proximity spaces is proximally continuous [17].
On the other hand, given a fuzzy proximity d, a Lowen fuzzy uniformity//d is defined [17, 19]. (Its definition is very natural but quite bulky, therefore, we do not reproduce it here.) If a mapping f : (X, dx) --+
(Y, dv) of fuzzy proximity spaces is proximally continuous then the mapping f : (X, LQx ) --+ (Y, Udr) is
uniformly continuous, and hence the assignment d --+ Ud determines a functor ~ : A M F P --+ L F U [19]. An
important connection between fuzzy proximities and generated fuzzy uniformities is given by the formuIa
dud = d [19]. On the other hand, given a fuzzy uniformity/2, let pll = Udu. The fuzzy uniformity p / / i s
precompazt (2.2.7); p/~ can be characterized as the coarsest fuzzy uniformity whose induced fuzzy proximity
is du [19].
3.2.4. F u z z y P r o x i m i t i e s G e n e r a t e d by C r i s p P r o x i m i t i e s . Unlike the situation with Katsaras
fuzzy proximities, crisp proximities cannot be directly viewed as a special case of Artico-Moresco proximities. To clarify the natural way in which the category P r o x can be embedded into A M F P one can use the
following procedure described in [17].
Given a crisp proximity zx on a set X, let a mappingTk : I x x I X --+ I be defined as follows :-A(M, N) =
sup{a E I: M-][a, 1]zxN-][a, 1]}, M , N E I x. WhenTk is a fuzzy proximity on X. Besides, a mapping
f : (X, Ax) --+ (IF, Ay) of proximity spaces is proximally continuous iff the mapping f : ( X , A x ) --+ (y~'Ay)
691
of the corresponding fuzzy proximity spaces is proximally continuous. Therefore, the correspondence A --+-/~
determines an embedding functor ,-~: P r o x --4 A M F P .
Topological structures, induced by ~ andNA, are connected in the following "natural" way : r-e. = w(Tzx)
(w is Lowen's functor, see 1.2.2).
3.3. (L, K)-Fuzzy Proximities.
An approach to fuzzy proximities, consistent with (L, IK)-fuzzy topologies and, to a certain extent, with
(L, K)-fuzzy uniformities, was presented in [184, 133]. The context, in which the corresponding theory is
being developed, is essentially restricted to the case where L and • are sublattices of I; this assumption
will be accepted throughout this section.
3.3.1. Basic Definitions. An (L,K)-fuzzy proximity on a set X is a mapping 5: L z • L x --4 K
satisfying the following axioms (A, B, C E LX)"
(FP1) 5(0, 1) = 0;
(FP2) 5(A, B) = 5(B, A);
(FP3) 5(A, B V C) = 5(B, A) Y 5(A, C);
(FP4) 5(A, B) > sup(A + B - 1)(x);
(FP5) 5(A,B) > inf{5(A,E) V 5(B, EC): E 6 Lx}.
The pair (X, 5) is called an (L,K)-fuzzy proximity space. A mapping f : (X, Sx) --~ (Y, Sy) of fuzzy
proximity spaces is called proximally continuous, or proximal, if 5y (f(A), f(B)) >_5x (A, B) for any A, B 9
L x. The category of (L, K)-fuzzy proximity spaces and proximally continuous mappings will be denoted
by FP(L, K).
Sometimes it is convenient to deal with those (L, K)-fuzzy proximity spaces which satisfy the following
stronger versions of axioms (FP4) and (FP5):
(FP4') 5(A, B) >_ sup(A ^ B)(z);
2:
(FP5') 5(A,B) >_ inf{5(A, C) VS(B,D): C V D = I ; C , D 9
3.3.2. R e l a t i o n s w i t h K a t s a r a s Fuzzy P r o x i m i t i e s [133]. Note first that the category K F P ( L )
in a natural way Can be identified with the category FP(L, 2) as well as with the full subcategory of any
category FP(L, K), the objects (X, 5) of which satisfy the additional requirement 5(L X x L X) C 2 (C K).
Now let (X, 5) be an (L, K)-fuzzy proximity space. Then for each a 9 K a relation ~ C L x x L x can
be defined by letting Az~aB if 5(A,B) >_ a. It is easy to see that zx~ satisfies the axioms (KP1), (KP2),
(KP3), (KP4) of (3.1.1) and the following weakened version of the fifth axiom:
(KP5'~) A-~oB implies sup(A + B - 1)(z) < a.
(We shall call the relation satisfying axioms like those of zx~, an a-Katsaras L-fuzzy proximity). The system {zx~ : a 9 K} of a-level a-Katsaras L-fuzzy proximities of a given (L, IK)-fuzzy proximity is decreasing
and lower-semicontinuous: a~ = gl zx~, for each a 9 K +. Conversely, given a decreasing system D =
cr~(a
{zx~: a 9 K* } of a-Katsaras L-fuzzy proximities, the formula 5(A,B) = sup{ a~(A,B) A a: a 9 K + } determines an L-fuzzy proximity 5. Furthermore, if D is lower-semicontinuous, then z~~ = { (A, B) : 5(A, B) >_
a}.
3.3.3. (L,K)-Fuzzy Topologies Induced by (L,K)-Fuzzy P r o x i m i t i e s [133]. Let (X, 5) be an
(L, K)-fuzzy proximity space, a 9 K + and A 9 L x. Then the a-closure of A or the closure of A at the
level a is defined by the equality A-~ = A V (1 - V{B: 5(A, B) < c~}), or, equivalently, in the pointwise
formA a
AV (V{xX: 5(xX~
> a , } ) . In the last formula a fuzzy point x x is defined as usualwith
the exception that for technical reasons the case ,~ = 0 (i.e., degenerate fuzzy point with support x) is
not excluded. The operator A --+ ~ thus defined is a fuzzy closure operator (1.1.1). Let r ~ stand for
the corresponding Chang-Goguen L-fuzzy topology. The system of Chang-Goguen L-fuzzy topologies thus
obtained is decreasing and lower-semicontinuous : r~ = Cl to, for each ~ 9 K +. Therefore, one can apply
to it the machinery described in 1.3.3. In particular, one can, by means of the formula T(U) = V (r~(U)A a),
Ot
U 9 L x, introduce an (L, K)-fuzzy topology 7~ = T: L x -+ N. In the sequel, ~ will be called induced or
692
generated by the fuzzy proxinfity 5.
If a mapping f: (X, 5x) --+ (Y, 5y) of (L, N)-fuzzy proximity spaces is proximally continuous, then the
mapping of the corresponding (L, K)-fuzzy topological spaces f : (X, 7~x ) --+ (}", T~v) is continuous, and
hence, the correspondence 6 ~ 7~ in a natural way determines a functor from FP(L, IN) into FT(L, K).
This functor is anti-isotone; if 5 _< 5', then T~ > 7~,.
3.3.4 Initial (L, IK)-Fuzzy Proximities. P r o d u c t in FP(L, IK). Lattice P r o p e r t i e s of (L, IK)Fuzzy P r o x i m i t i e s [133]. Let X be a set, let (Yj,6j), j E J, be a family of (L,K)-fuzzy proximity spaces, and let fj: X -+ Yj, j 6 J, be a family of mappings. Then' the equality 6(A,B) =
k
inf{supinf Sj(fj(Ap), fj(Bq)) : A = V Ap, B = ~/ Bq,k E N}, where A , B E L X, defines the initial (L,K)p,q
1
p=l
q----I
fuzzy proximity, i.e.,the weakest (L, K)-fuzzy proximity on X among those ones for which all mappings fj
are proximally continuous. By means of initial (L, K)-fuzzy proximities one can define various operations
on (L, K)-fuzzy proximity spaces in a standard way. In particular, the product of a family { (Xj, 6j ),j E J}
of (L, K)-fuzzy proximity spaces is defined as a pair (X, 5), where X = FIXj is the product of the prodJ
uct of the corresponding sets, and 5 is the (L, lK)-fuzzy proximity, initial for the family of all projections
pj : X --+ Xj. Let P(L, IK,X) stand for the set of all (L, [<)-fuzzy proximities on X. Then P(L, E, X)
endowed with the order "<" becomes a complete lattice. Its largest element (i.e., the weakest fuzzy proximity) is the antidiscrete (L, E)-fuzzy proximity ha : L z x L x --+ K given by 6a (A, B) = 0 if A = 0 or
B = 0 and 6~(A, B) = 1 otherwise. Its smallest element (Le., the strongest fuzzy proximity) is the so-called
discrete (L, [<)-fuzzy proximity 5d: L x x L x --+ K defined by' the equality 5d(A, B) = sup(A + B - 1)(X).
(In accordance with the terminology accepted in the theory of proximity spaces (see, e.g., [11, 153] et al.)
it is natural tosay, that 5 is weaker than 6' if 6 >_ 6~.)
3 . 3 . 5 : R e l a t i o n s to A r t i c o - M o r e s c o Fuzzy P r o x i m i t i e s . At first glance there is a remarkable
resemblance between the system of axioms given in 3.2.1 and 3.3.1 (at least in the case L = K = I).
The axioms (AMP1), (AMP2), (AMP3) and (AMP5) coincide with the axioms (FP1), (FP2), (FP3), and
(FP4'), respectively, and the axiom (AMP4) implies the axiom (FP5'). Therefore the category A M F P
formally can be considered as a full subcategory of F P ( I , I). However, the resemblance between the two
approaches is only superficial. Although, starting from a similar systems of axioms, they are developed
in quite different directions. In particular, G. Artico and R. Moresco use their fuzzy proximity to define
the closure operator A(x) = 6(A, x) thus obtaining a laminated Chang-Goguen fuzzy topology, and, as
a result, they get a functor ~: A M F P --+ L C F T ( I ) . Furthermore, Artico-Moresco fuzzy proximities
are used to establish natural connections with Lowen fuzzy uniformities. On the other hand, (L, K)fuzzy proximities are introduced and employed as the proximal counterpart of (L, lK)-fuzzy topologies and
(L, K)-fuzzy uniformities. Unfortunately, until now the relations between (L, K)-fuzzy proximities and their
uniform counterparts are not sufficiently well studied. See some results in this direction in [186].
Notice also that the concept of a (L, K)-fuziy proximity generalizes the notion of a Katsaras L-fuzzy
proximity while no Katsaras fuzzy proximity can be an Artico-Moresco fuzzy proximity at the same time
because the axiom (AMP6) cannot hold for it.
3.3.6. S o m e C o n n e c t i o n s b e t w e e n P r o x i m i t i e s a n d (L,K)-Fuzzy P r o x i m i t i e s . The category
P r o x of proximity spaces can, in an obvious way, be identified with the categoryFP(2, 2). More generally,
for any L, K C I the category P r o x can be, up to a natural isomorphism, viewed as the full subcategory of
FP(L, K) formed by objects (X, 5) satisfying the following two requirements: 6(L X) C 2 and 5(L x \ 2 X) =
{0).
A connection of a different type is established by the functor -,, : P r o x -+ A M F P (3.2.4) where A M F P
is formally considered as a subcategory of FP(I, I) (3.3.5). Clearly, the full subcategory of F P ( I , I) formed
by objects of (X, ~) type, where a is a proximity on X, is isomorphic to Prox.
4. F U Z Z Y S Y N T O P O G E N E O U S
STRUCTURES
The concept of a syntopogeneous structure introduced by A. Csazar [31] enabled him to develop a united
viewpoint of the three main structures of set-theoretic topology : topologies, uniformities, and proximities;
693
and to evolve a theory which includes as special cases, the foundations of the classical theories of topological
spaces, uniform spaces, and proximity spaces.
In fuzzy topology the classical (Csaszar's) theory of syntopogeneous structures is reflected mainly in
the form of three different theories of fuzzy syntopogeneous structures. The first one of these theories,
developed in the papersby A.K. Katsaras and C.G. Petalas [84, 85, 78, 81] etc., presents a united approach
to the theories of Chang-Goguen I-fuzzy topologies (Sec. 1.1), Hutton I-fuzzy uniformities (Sec. 2.1), and
Katsaras I-fuzzy proximities (Sec. 3.1). The second one, developed mainly mostly by A.K. Katsaras [82, 83],
presents a united approach to the theories of laminated Chang I-fuzzy topological spaces (Sec. 1.2), Lowen
fuzzy uniformities (Sec. 2.2), and Artico-Moresco fuzzy proximities (Sec. 3.2). The third one, set forth
in [186] offers a united approach to the theories of (I, I)-fuzzy topologies (Sec. 1.3), (I, I)-uniformity type
structures (which are defined in a different way than the ones introduced in Sec. 2.3), and (I, I)-proximities
(Sec. 3.3).
Although all these approaches axe being developed in the context of the lattice L = I, we assume that
there are no serious obstacles which prevent generalization of the first theory to the case of a completely
distributive lattice L. On the other hand, the dependence of the last two theories on the extra algebraic
and order structure of the unit interval is much heavier.
Let us mention also that recently several papers were published mainly by N.N. Morsi et al. (see,
e.g., [140], etc.), where syntopogeneous structures, in the context of fuzzy neighborhood spaces [112], were
introduced and studied.
Unfortunately, we cannot go deeper into a discussion of the theories of fuzzy syntopogeneous structures,
their achievements and problems. Such a discussion would inevitably demand the introduction of a large
amount of new concepts and arguments. On the other hand, we would like to mention a reason why an
interest in fuzzy syntopogeneous structures seems to be natural and justified.
A specialist in fuzzy topology constantly faces the problem of selecting from numerous possible "natural"
extensions of concepts and constructions from the set-theoretic topology to the fuzzy case, one or Several
(but not too many) extensions which are "optimal" in a certain sense. Besides, this choice must, of course, be
done in a way coherent with the whole context of the work. And here the "syntopogeneous approach" which
permits one to comprehend the whole context of the research, often gives an indication of the appropriate
choice.
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