I nternat. J. Mah. Math. S ci.
Vol. 3 No. 3 (1980) 433-443
433
PROPERTIES OF COMPLEMENTS IN THE LATTICE
OF CONVERGENCE STRUCTURES
C.V. RIECKE
Department of Mathematics
Cameron University
73505 U.S.A.
Lawton, Oklahoma
(Received October Ii, 1979)
ABSTRACT.
Relative complements and differences are investigated for several con-
vergence structure lattices, especially the lattices of Kent convergence structures
and the lattice of pretopologies.
Convergence space properties preserved by
relative complementation are studied. Mappings of some convergence structure
lattices into related lattices of lattice homomorphisms are considered.
KEY WORDS AND PHRASES. Convergence structure, pretopology, limitierung, elative
pseudo-complement, pseudo-difference, continuous lance.
1980 ATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
54A05, 06D15.
INTRODUCTION.
The author classified the relative complements and differences for the
lattice of Kent convergence structures on a nonempty set in [8].
This paper in-
vestigates further the convergence space properties preserved by these complements
and their relationships to some of the standard lattice operations such as products
434
C.V. RIECKE
and quotients.
[3], [4] and [ii]
The definitions used are essentially those of
vergence structure on X considered as a map q
U(X) and
set of filters on X).
X
with a con-
8(F(X)) (the power set of the
+
are the set of ultrafilters on X and the prln-
q
cipal ultrafilter generated by {x}. For a convergence space (X,q), let %q,
regula topological
q be the topological, pretopological, and complety
COS
of q.
cosue
cl(A) of a subset A is {x
The
q-//mlt Set of
F
a filter
F
is ad
q(F)
(x
if z is the greatest element with x^z<y and the
y<xvz.
of x
pseudo-difference
2.
RELATIVE COMPLEMENTS.
is -x
of x
modifi-
q(x)} and the
F}.
rzive to y (x,y)
pseudo-difference of y and x (y-x)
If L is a complete lattice with 0 and i
the least and greatest elements, then the
the
E
q(x) for some filter F with A
An element z of lattice L is the pseudo-complement
if z is the least element with
F
and
pseudo-complement of x
is
x*
x,0 and
l-x (a change of notation from [8]).
The relative pseudo-complement and pseudo-difference of two convergence
structures in C(X) were described in [8] as:
q,r(x)
{FI F
q-r(x)
{F
or
F +G
G n
_c F for some G
r(x) \q(x)}
B(X) or is in q(x) for all G
r(x)}
For q and r limitierungs of Fischer [3] or pseudo-topologies the same descriptions hold for relative pseudo-complements and pseudo-differences in the
lattices of limitierungs or pseudotopologies.
In P(X), the lattices of pretopol-
ogles on X, pseudo-differences do not exist from [9].
Relative pseudo-complements
also fall to exist in P(X) even though, from [9], P(X) is pseudo-complemented.
EXAMPLE 2.1:
complement and x
filter
On an infinite set X, let A be an infinite subset with infinite
X.
F q(r)-converges
If q(r) is the finest pretopology on X such that an ultrato x if and only if
then q,r does not exist in P(X).
F
or
F
is free and contains
A(X-A
435
LATTICE OF CONVERGENCE STRUCTURES
Many properties of convergence structures are preserved by relative pseudoIf r is a limitierung (resp. pseudotopology, pretopology, topol-
complementatlon.
ogy), then from [8], q,r is the same type of structure for any convergence structure q.
PROPOSITION 2.2:
(i)
(ii)
(ill)
is T (Hausdorff).
1
I (Hausdorff) if r
q,r is T if r is T
3.
3
q,r is T
q,r is compact if and only if r is compact and for any ultrafilter
F,
(iv)
(v)
For any convergence structures q and r on X:
ad (G) for some G
ad (G)
r
F.
c
q
q,r is T-regular [5] if r is T-regular.
q,r is first countable
(-countable,
[2]) if r is first countable
-countable).
(vi)
q,r is second countable if r is second countable and q,r has at
most countably many discrete points.
In addition, for any pretopology q on X:
(vii)
q,r is a completely regular topology if r is a completely regular
topology.
(viii)
(ix)
q,r is m-regular [6] if r is m-regular.
q,r is C-embedded [6] if r is C-embedded.
PROOF:
(ii)
If
The proof of (1) is in [8].
F
E
q,r(x) with G
c
F
since r is regular so cl
to x.
(iii)
(iv)
If q,r(x)
{}
Suppose
F
e q,r(x) and
r(x) \ q(x) so
If q,r(x)
G ccl
G ccl
r
G
_
q,r
then q,r is T
is obvious.
G
r(x) \ q(x) then cl rG
for some
F
I
so
for some G
q,r
F
.
and cl
Clq,r
q,r
r(x) \ q(x).
E
r(x) \ q(x)
F q,r-converges
Then
ClrG
e
ClxrG _c Clq,xrG _c clx(q,r) C clx(q,r)F since q,Xr < X(q,r).
{},
X(q,r)(x)
{}
so
q,r
is T-regular.
436
C.V. RIECKE
(v)
F
E
q,r(x) with G E F and G
some
H
with filterbase of cardlnallty less than
If
B
Let
r(x) \q(x), then H
be a countable basis for (X,r).
is a countable basis for
(vii)
E
for any cardinal
B’
Then
r(x) \q(x) for
u {x
{}}
q,r(x)
(X,q,r).
Since q,r is topological from
[8], suppose A
A.
is q,r-closed and x
cl (A), any real valued continuous function on (X,r) which
r
Then if x
separates x and A is also q,r-continuous.
If x
cl (A), then q,r is
r
discrete at x so x and A can be separated by a q,r-continuous, real-
valued function.
(viii)
If
F
r(x) \ q(x) with cl 0rG
q,r(x) and G
E
r(x) \ q(x) then from
(vii), if r is the completely regular modification of r, cl rG
G c cl
= cl(q,r F and if q,r
q,r
(q,r) and the conclusion follows.
(q,r)G
cl
(ix)
=
is discrete at x, so is
From [8], q,r is pseudo-topological if r is pseudo-topological.
If r
is Hausdorff and u-regular, then q,r has the same properties from (i)
and (viii) so by [6], q,r is C-embedded if r is C-embedded.
(i)
COROLLARY 2.3:
is the finest first countable structure coarser
If r
q,r
than r, then (q,r)
(ii)
.
(q,r)
_
If Rr, the finest regular structure coarser than r [7], is T I, then
q,Rr < R(q,r).
PROOF:
<
for every convergence structure q.
Since r
o
<
r, q,r
Conversely, if
countable filterbase.
F
Then G
o <
o
q,r and q*r being first countable implies q,r
q,r(x)
E
then
G
q,r(x) so F
F
for some G
E
r(x) \ q(x) with
(q,r)(x).
As q,Rr is T from (ii) of Proposition 2.2 and q,Rr < q,r, then q,Rr <_ R(q,r).
3
The converses of the statements in Proposition 2.2 fail to be true since if
q <_ r, q*r is discrete.
for T
3.
In (ii) of Proposition 2.2, one cannot substitute regular
LATTICE OF CONVERGENCE STRUCTURES
F/AMPLE
2.4:
{x,y} and q be the finest convergence structure
Let X
(i)
Fxy generated
on X for which the principal filter
x.
437
Then 0 is regular but not T
I
q*
and
by {x,y} converges to
is not regular.
(ii) Let r be a convergence structure on an infinite set X for which Rr # r
and Rr is
TI,
such as a non-regular
finer than some
T2,
T2-convergence
regular topology.
structure which is
R(r,r) #
Then 1
r,Rr.
The following description of the convergent ultrafilters of the pseudo-dlf-
ference q-r of two convergence structures is given in [8]:
LEMMA 2.5:
F
An ultrafilter
q-r converges to x if and only if
F q-converges
to x or does not r-converge to x.
Because q-r can have so many convergent ultrafilters, most convergence space
properties are not preserved.
This can also be observed from the result of [9]
that the image of the map q + l-q is the lattice of pseudotopologles.
ample, one can readily show that q-r is not pretopologlcal if q
For ex-
r, r is T 1 and
q is not discrete and l-q is not regular if q is T and not discrete.
I
A few
properties can be easily seen to be preserved.
PROPOSITION 2.6:
For any convergence structures q and r on X,
(1) q-r is T 1 if and only if q is T 1 and the pretopologlcal modification
r
of r is indiscrete.
(il) q-r is Hausdorff if and only if l-q < r.
(ill) q-r is compact if q is compact.
(iv) q-r is compact if and only if no ultrafilter
F (l-r)^q-converges
to
every point.
For complements of product convergence structures there exist relationships
If
to the complements in the original spaces.
nondegenerate convergence spaces with products
convergence structure defined on
EXe by:
F
{(Xq)
(HXe,q=),
Hwq e-converges
r} is a family of
let
wq e
to x
denote the
(x=)
if and
C. V. RIECKE
438
py(F) qy-COnverges
only if the projection
tions, p or
pe
will be
In the subsequent four proposi-
will denote the appropriate projection or quotient map.
PROPOSITION 2.7:
FI
Hwq
F.
for some y
Xy
p,’odct convergence s5"mcZue.
called the
with
to
q=
If
r= are convergence structures on Xe for e
and
e F
then in C(HX ):
> i
(1)
(Hqe), (Hre)
<
(li)
F converges
to x
H (q,re)-
(xe)
py(F) q,ry-converges
(Hqe),(Hre)
with respect to
if and only if
for some y e F.
to x
(Hqe),(re) Hw(qe,re).
(iv) H(qe,re)
(Hqe),(Hre) if and only
(v) (q)*
Hwq e *
PROOF: (i) If F H(qe,re)-converges to
(iii)
verges to x
so for each
if each
re
is indiscrete.
there exists a filter
G
on X
xe or G _c p(F) and Ge re(x=) \ qe(xe)"
vergent to x and HGe
(Hra)(x) \ (Hqe)(x) or HG
converges to x since HGe -c Hp(F) _c F.
Pc(F)
(ii)
F (Hq),(Hr )-converges
Suppose
some
G
for some
(iii)
(iv)
(Hre)(x)\ (Hqa)(x).
y so py (F)
qy,ry(x
HGe
Then
is
G
Hr,-con-
x and
F
Then
In the latter case,
with
x or
G
F
for
py(G) ry(xT)\ qy(xy)
The converse is similar.
follows immediately from (ii) and the definition of a weak product.
If
IF[
G
> 1 and
G where
c
Hqe
c
(Hwqe )-converges
qy-converges
each
qe,re
(since {X }
(v)
(x).
to x
).
Pc(F) qa,ra-con-
(x e) then each
x
to
Hwqe
<
x
c
to x
then for
for some x
(xe)
xy and qy
F
c
Y
X
so must
any filter on X and x
Y
Y
if
c
y and
Hqe -converge
is indiscrete
Thus if
is indiscrete and it follows that each
r (x)\
q(x
for each x
is a direct consequence of (iv) since
q*
X ).
q*O.
y
Fy
to x and
Hqe*ra
re is
<
X
Y
Then
F
Y
let
G
p (G)
Y
Hwqa*ra’
indiscrete
LATTICE OF CONVERGENCE STRUCTURES
PROPOSITION 2.8:
If q
439
are convergence structures on X
and r
F,
for
then in C(HX ):
(i)
(ii)
<-
(Hq)-(Hr)
and H (-r) agree on convergent ultrafilters.
-(Nr
PROOF:
N(q-r)
(i)
p(G)
B(X).
(Nr)(x).
C
Let
so
F
In either case,
(-r)(x)
e
and
F
let 0
for all
p(F)+p(G) e q(x) or
p(F)+p() _c p(F4) and F+
so
Then
p(F)
IIw(-r)-converges
to x.
be an ultrafilter in [-(r
p(F)
converges to x
all
(x)for
01,
e r
Then
p(F)+p()
[(Hqa)-(Hr )](x).
(ii)
p(F) q-r
(x) and
If x
)](x).
# r
(xa)
for some
The reverse in-
equality is similar.
From Lemma 2.5, one can show that equality does not hold in Proposition
2.8(i) even for q topological.
The product operation can also be viewed as a lattice operation on C(X).
PROPOSITION 2.9:
(NXa,Hq)
The map
"
H[C(Xa)]
C(HX)
an equivalence relation on
{x
be the quotient space with quotient structure q, A
F
H[(Xa,qa)(a
F)]
is a complete join homomorphism.
If (X,q) is a convergence space with
for
defined by
a filter on
{ A
X,
e
F}.
Let f"
x
A} for A
X, let X/~
X and,
C(X/) be the map f(q)
C(X)
The subsequent propositions are readily established.
PROPOSITION 2.10"
f is a complete meet homomorphism.
PROPOSITION 2.11:
For q and r in C(X), in C(X/):
(i)
(ii)
q,r _< q,r
r*
(r)*
exist A e
(iii)
if and only if for each x and
F
with A
F
in
r(x), there does
not
y # # for all y in X.
-q._< (-q) with equality if and only if for each x, F # x in q(x) and
A
F, A
x.
q.
440
C. V. RIECKE
(iv)
N (x).
q
plies N (y)
r
For A a nonempty subset of X and
filter on A where
A e
F
and
F
FA
{A n
q(x)}, i.e.,
BIB
fA(q)
(i)
PROPOSITION 2.12:
(ii)
N (x) imq
q,r if yx and N r (y)
q,r
If q and r are pretopologies
F
a filter on X with A
F} and
e
fA:
C(X)
C(A) be
/
F,
FA be
let
fA(q)(x)
the
{FAI
is the subspace structure on A.
is a complete lattice epimorphism.
fA
fA(q,r)
For any q and r in C(X),
fA(q),fA(r)
and
fA(q-r)= fA(q)-fA(r).
As one would expect, Proposition 2.12 establishes that the restriction of
the relative complements to a subspace are the complements of the restrictions.
3.
LATTICE OPERATORS INDUCED BY RELATIVE COMPLEMENTS.
The relative pseudo-complement and pseudo-difference induce four obvious
self-maps of C(X) for each convergence structure q:
(i)
f*(q)" f*(q)(r)
q,r
r,q
(ii)
f,(q)"
f,(q)(r)
(iii)
f (q)"
f (q)(r)= q-r
(iv)
f (q)"
f (q)(r)= r-q
Of these maps, (i) and (iv) were considered in [8].
Only (i) and (iv) will be
considered here since (ii) and (iii) have similar lattice properties if consid-
ered as maps of C(X) into its dual.
If F is a cardinal, a subset A of a lattice L is prime with
joins in L
if for any subset {x
Y
Y
F} with vx
A, some x
rpect to F-
e A.
A conver-
gence structure q of C(X) is join prime if each q(x)\ {x} is prime with respect
to finite joins in
{r(x)
\
{}
r
C(X)}.
As an extension of a result in [8] one
has
PROPOSITION 3.1:
For any convergence structure q on X:
(i)
f"(q)
is a complete meet homomorphism.
(ii)
f*(q)
is a F-join homomorphism if and only if
q(x) \ {x}
is prime with
441
LATTICE OF CONVERGENCE STRUCTURES
respect to F-joins in F(X) for any cardinal F.
f*(q)
(iii)
PROOF:
is bijective if and only if q is discrete.
(i) is a result of [8] while the proof of (ii) parallels the result
of [8] for finite joins.
PROPOSITION 3.2:
f_(q)
(ii)
(iii) is a property of complete lattices.
(i)
f_(q)
is a complete join homomorphism.
is complete with respect to F-meets for a cardinal F if and only
if each q(x) is complete with respect to F-meets in C(X) for each x.
f (q) is bijective if and only if q is indiscrete.
(iii)
PROOF:
[8].
(i) is from [8] while the proof of (ii) is similar to Theorem 4.2 of
(iii) is dual to Proposition 3.1(iii).
From Proposition 3.2 one can observe that f (q) is a complete lattice homo-
morphism if and only if q is a pretopology.
F <
,
If F and
are infinite cardinals with
by choosing the cardinal of X large enough so that if y e X and q(x) is
discrete for y
# x
and q(y) is closed with respect to F-meets but not -meets,
then f (q) is a F-homomorphism that is not an -homomorphism.
Using the given four lattice operators, one can construct maps of certain
sublattices of
C(X) into the duals of their lattices of homomorphisms (with
ordinatewise order).
For example, if L(X) is the lattice of limitierungs on X
and P(X) the lattice of pretopologies, one can define
q,r and f
P(X)
+
co-
similarly, where
LL(p P)
fL:*
L(X)
L
L
by
fL*(q)(r)
is the dual of the lattice of homo-
morphisms of L(X) and q,r is the relative pseudo-complement in C(X).
The succeed-
ing two propositions follow directly from the definitions and properties of
pseudo-complements and differences.
PROPOSITION 3.3:
(i)
f
*
L
is a lattice embedding and a complete join homo-
morphism.
(ii)
If
fp* is a complete
cC(x,v) denotes the
lattice embedding.
join semilattice of join-homomorphisms of C(X) then
442
C. V. RIECKE
cC(x,v)
C(X)
the map f
PROPOSITION 3.4"
dual of
(ii)
f
(i)
is f
(q)(r)
r-q.
is a complete meet homomorphism of C(X) into the
f
cC(x,v).
is an embedding.
In a partially ordered set (L,<-), let x<<y if and only if for every up-directed set D,
y-<sup D implies
pe ce L
x<-d for some d in D.
c0nuou if x
sup{y
e
L
Then from Scott
y<<x}
[I0],
for all x in L.
.
a
The in-
duced topology is that topology for which U ! L is open if U is a terminal set
and if S ! L is directed, sup S exists and is in U, then S n U
#
Since a
compactly generated lattice is continuous, we have from [8] that C(X), L(X) and
P(X) are continuous with C(X) and L(X) having continuous duals.
[i0],
Since also from
a function between complete lattices is continuous in the induced topologies
if and only if it is join-preserving, one has immediately from Propositions 3.1
and 3.2:
PROPOSITION 3.5"
(i)
f*(q)
For any convergence structure q on X:
is continuous if and only if each
q(x) \ {}
is prime with respect
to joins in F(X).
(ii)
f (q) is continuous.
PROPOSITION 3.6:
fL
and
fp
are continuous in the induced topologies.
Since join-prime elements q determine when
note that if
q(x)
convergent to x.
f*(q)
is a homomorphism, one may
is join-prime, there exists at most one ultrafilter
F
not q-
Also, the join-prime elements of C(X) form ameet-sublattice of
C(X) but not a join-sublattice.
A number of special types of convergence structure lattices are continuous
lattices by virtue of being retracts of
sition 2.10 of [i01.
C(X) in the induced topologies and Propo-
Some examples are the lattices of
T1-structures,
pseudo-
t,pologies, locally bounded structures and locally compact structures which can
443
LATTICE OF CONVERGENCE STRUCTURES
be shown to be continuous by virtue of the standard modification maps.
In [8], Theorem 5.1, the incorrect statement is made that the map
topological modification is a join homomorphism.
of pre-
If q is the cofinite topology
and r is the finest convergence structure for which each prin.cipal ultrafilter
converges to each point, then (qvr) #
qvr.
Therefore
cannot be used to show
P(X) is a continuous lattice.
REFERENCES
i.
Carstens, A. M. The lattice of pretopologies on an arbitrary set S, Pacific
J. Math. 29(1969) 67-71.
2.
Feldman, W. A.
47(1973)
Axioms of countability and the algebra
C(X), Pacific J. Math.
81-89.
Limesrume, Math. Ann. 137(1959)
269-303.
3.
Fischer, H. R.
4.
Hearsey, B. V. and D. C. Kent.
31(1972) 105-118.
5.
Kent, D. C., G. D. Richardson and R. J. Gazik. T-regular closed convergence
spaces, Proc. Amer. Math. Soc. 51(1975) 461-468.
6.
Kent, D. C., K. McKennon, G. Richardson and M. Schroder.
gence in C(X), Pacific J. Math. 52(1974) 457-465.
7.
Richardson, G. D. and D. C. Kent. The regularity series of a convergence
space, Bull. Australian Math. Soc. 13(1975) 21-44.
8.
Riecke, C. Complementation in the lattice of convergence structures, Pacific
J. Math. 69(1977) 517-526.
9.
Riecke, C.
Convergence structures, Portugaliae Math.
Continuous conver-
Ideals in convergence structure lattices, submitted.
i0.
Scott, D. Continuous lattices, Lecture Notes in Math., v. 274, SpringerVerlag, Berlin-New York, 1972.
ii.
Szasz, G.
Introduction to lattice theory, Academic Press, New York, 1963.