209
Internat. J. Math. & Math. Sci.
(!987) 209-216
Vol. i0 Nc. 2
ORDER COMPATIBILITY FOR CAUCHY SPACES
AND CONVERGENCE SPACES
D.C. KENT
Department of Pure and Avvlied Mathematics
Washington State University
Pullman, WA 99164-2930
and
REINO VAINIO
BO AKADEMI
Matematiska Institutionen
Fnriksgaan 3
SF-20500 BO 50
Finland
Received July 28, 1986)
A Cauchy structure and a preorder on the same set are said to be compat-
ABSTRACT.
Howover,
ible if both arise from the same quasi-uniform convergence structure on X.
there are two natural ways to derive the former structures from the latter, leading
to
"strong"
and "weak" notions of order compatibility for Cauchy spaces.
These in
turn lead to characterizations of strong and weak order compatibility for convergence
spaces.
Preordered Cauchy space, preordered convergence space,
weakly preordered space, quasi-uniform convergence space.
KEY WORDS AND PHRASES.
1980 MATHE4ATICS SUBJECT CLASSIFICATION CODE.
54A20, 54E15, 54F05
INTRODUCTION
L. Nachbin,
[3],
introduced order compatibility between a uniform structure
and a preorder @ on the same set X by requiring the existence of a quasi-uniformity
g on X such that
gU
g-I
[2],
An analogous procedure was used by
to define order compatibility between a Cauchy
on X; the principal deviation from Nachbin’s
and a partial order
structure
g.
and @
the authors in an earlier paper,
definition is the replacement of "quasi-uniformity" by "quasi-uniform convergence
structure".
However we required in
[2]
.
that the quasi-uniform convergent structure
employed be coarser than that generated by the order
If the latter restriction
is removed, a weaker form of order compatibility is obtained.
This paper examines both types of order compatibility for Cauchy spaces.
"strong"
version is the subject of Section
assuming that
I; here
we extend the results of
@ is a preorder rather than a partial order.
derived from a "preordered Cauchy
space"
is called a
The
[2]
by
A convergence space
"preordered convergence space",
and these are shown to be precisely the locally convex convergence spaces for "hich
the preorder is closed.
In Section 2, we consider the "weak" form of order compatibility, a topic not
previously studied. The characterization of "weakly preordered Cauchy spaces", is
given in two forms, the first of which provides a convenient comparison with the
D.C. KENT and R. VAINIO
210
preordered Cauchy spaces of Section I.
The second leads to a simple characterization
of weakly preordered convergence spaces.
l’ne terminology and noaio of
reference.
[2]
will be used ill his paper inhou further
is a preorder
However we shall always assume in this present paper that
(i.e., a transitive, reflexive relation).
PREORDERED CAUCHY SPACES (STRONG COMPATIBILITY).
From Definitions 2.1 and 2.2 of [2], we obtain the definitions of preordered
is
uniform convergence space and preordered Cauchy space by simply ussuming that
I.
a preorder instead of a partial order. The latter term will be abbreviated "p.c.s".
is a partial order is called an ordered Cauchy pace
) for which
A p.c.s. (X,
), we have "strong" compatibility
(abbreviated "o.c.s.") In a p.c.s. (X,
because of the requirement in Definition 2.1, [2], that the filter
between @ and
,
<>
,
The relaxation of this requirement leads to
be in o.
on X x X generated by
the "weak" compatibility studied in the next section.
In this section, we consider triples of the form (X, @ ,C ), where (X,
is a
is a
preordered set and C a Cauchy structure on X. In determining when (X, ,C
[2],
486,
on
defined
p.
as
Cauchy
of
<
filters,
C
set
on
the
p.c.s., the preorder
plays a vital role since it is required in formulating two of the following
conditions (see also p. 487,
(OC)I
(OC)
If
2
,
,
<
and
,
C;
then
implies x _< y.
x <
(0c) 3
C whenever
<
C
[2]):
The first condition defines "local convexity" for a Cauchy structure on a preordered
The second condition
is the convex hull of
() D
set; recall that
-I)
^
is antisymmetric relative to Cauchy equivalence
asserts that the preorder < on
classes, and the third turns out to be equivalent to the order
being closed in
XXX.
It was shown in Theorem 2.9,
an o.c.s, iff conditions
we show that when
(OC) I,
[2],
(0C2),
is a preorder,
(X,
is a partial order,
that when
and (OC) 3 are all satisfied.
,
(X,
,
) is
In this section
C) is a p.c.s, iff the same conditions
This task is made easier by the fact that all lemmas and Propositions in
is a
Section 2, [2] prior to Theorem 2.9 remain valid under the assumption that
hold.
preorder rather than a partial or_der. Whereas the statement of Theorem 2.9, [2]
is a preorder, the proof of this theorem must be altered,
remains valid when
since it makes explicit use of the assumption that
particular Proposition 2.8,
[2]
is valid when
is antisymmetric.
Since, in
is a preorder, the principal theorem
(0C)2,
,
(OC)3
and
of this section reduces to showing that (OC) I,
imply that (X,
is a p.c.s. Thus for the remainder of this section we assume that (X
C is a
preordered set with Cauchy structure which satisfies
(OC)I’ (OC)2’
and (OC)
3.
the elements of X is defined as
)= an mquvalne rlan
{y ( X: x y} be the equivalence
Let E
y iff (x,y) (
follows: x
x
>
be the filter of oversets of E
class containing x, and let <E
Given
(Y.
-I.
ORDER COMPATIBILITY FOR CAUCHY SPACES AND CONVERGENCE SPACES
LE4A I.I.
<E >
For each x
<Ex>
X,
(
y in (X,q
C)
for all y (
C.
x
It is obvious that E
PROOF.
(y) N
Y
y, the conclusion follows by (OC)
and
LEMMA 1.2.
such that
Ex
PROOF.
If
Furthermore, (C
and
for all y { X.
x
<>
In particular,
Since E
E
x
y
N
-I.
then there is x
qc
y in (X,
such that F x F
Choose F
Ex.
I.
is a filter on X such that
"
(
-l(y),
211
X
Ex.
for all y
clearly x ( F implies F c E
The second statement follows by Lena 1.
From Lemma 2 and
,
E
x
my
LEMMA 1.3.
<>
c
(OC)3,
the next lemma follows easily.
,
Assume that
and E
x
(i)
E
(2)
n,
(3)
<8> c
(
y
.
<>
are filters on X such that:
c
The following statements are equivalent:
C
(:x) N (x:)
(X,
THEOREM 1.4.
,
C)
(OCI) (OC)2
is a p.c.s, iff conditions
and (OC)
3
are
satisfied.
PROOF.
As we have noted previously, it is enough to prove that the three
conditions are sufficient.
we see that (X,
,C)
C, where
compatible with
and
v
(,C)
,C
to show that
-i
j=l
,C
x
j)
[2]
A
[2],
K1
_c
<
2’
2
K1 hl
x
and E
Since
K
I 2
2
y
K1 K2 and K2 KI. But then
D
from Lemma 3 that E
x
K
<
2’
and
,
,C
’’
n.
j
>
K.
x
both finer than
K2,
If E
X.
for some
x
2hen there must
Otherwise, there must
such that
and, for nonequivalent elements x and y, E x
x
D 6, one can show that
and
K2
(OC)
2
I
requires that
1N
K
K2
{
If
:
I
finer than
K },
a contradiction.
finer than
I
we conclude that
such that
and
i
(
<>
[2]
x
K
leads to
Thus if
C
Assume instead that there is an ultrafilter
for some
and so E 6
Then < > c x
X
the proof is complete.
such that
A
C, and it follows
The argument given in paragraphs 2 and 3 of the proof of Theorem 2.9,
( K
for all
( C such that
the conclusion that there is
I
2.4, [2]
where
for all x
E
be the set of all ultrafilters
(
is
C
implies
x
x
such that <
x
>
5
Suppose
finer than
x
o,C
<@>, and
be (by Lemma ) ultrafilters
Let
and
is characterized in Proposition
we may assume
by Lemma i.
be an ultrafilter
<> _c
C
,C
In view of Proposition 2.7, [2], it is sufficient
C
In other words, we must show, as in the
x
that
j))
C
x ( X, then
,C
(,C)-I.
By Proposition 2.4,
@
U(:
is compatible with
proof of Theorem 2.9,
n
From Proposition 2.6, (generalized to preordered sets),
is a p.c.s, iff
D.C. KENT and R. VAINIO
212
X by Lemma 2.
x
A repetition of a previous argument shows that every such ultrafor the same x
X. If ]L is the set of all such
must contain E
filter
ultrafilters
and
X
N{
’2
:
IL
it follows that E
I
It remains only to show that
K and
are in
C
K
can deduce that
<
statement
C
2
Let
2’
and so
]L.
6 ]( and
Then
K, from which it follows by (OC) 2 that X N
C
C, and the proof is complete.
If (X,
PROPOSITION 1.5.
PROOF.
N
(
As in an earlier paragraph of the proof, we
G
D
<
and
IN 2
Thus
X
and
x
,
C) is a p.c.s., then
X.
is closed in X
It is shown in Proposition 2.7, [2], that (OC) is equivalent to the
3
U{Q
is closed under
Since <>
@,
o@,C and
:
compositions @=U{Do
<> o: o,C
the last equality follows from 4.1.5, p. 301,
to the product convergence structure on X
on X compatible with
where
the closure being taken relative
X derived from the convergence structure
[I],
the latter convergence
qc
structure is precisely
Let (X,q)
[I],
By Proposition 2.6,
,C
o,C
,C
<>
U{
be a convergence space (in the sense of Fischer), let
q
C be the
set
of all q-convergent filters, and let q(x) be the set of all filters which q-converge
to x.
It is well known that the following statements are equivalent:
(a)
(b)
There is a Cauchy structure C on X such that q
q
C is a Cauchy structure on X.
(c)
For x, y
(
q
c
X, q(x) and q(y) are either equal or disjoint.
If (X,) is a preordered set and q a convergence structure on X, we define the
triple (X, @, q) to be a preordered convergence space if there is a Cauchy structure
C on X such that (X, ,C
qc
be a preordered set, q a convergence structure on
Let (X,
PROPOSITION 1.6.
X.
is a p.c.s, and q
C q) is a p.c.s.
q) is a preordered convergence space iff (X,
C be a
Let (X, @ q) be a preordered convergence space, and let (X,
Then (X, @
PROOF.
p.c.s, such that q
q
Note that
c
q
D
C.
From the fact that (X, @, C) satisfies
(X, @, cq) has the same properties.
(OC) and (OC) 3,
Furthermore the latter space also satisfies (OC) 2, since for complete spaces (OC) 2
follows immediately from (OC) 3.
it is easy to deduce that
The converse argument is trivial.
Let (X, ) be a preordered set, q a convergence structure on X.
q) is locally convex
q) is a preordered convergence space iff (X,
THEOREM 1.7.
Then (X,
,
,
and @ is closed.
PROOF.
Using Proposition 1.5, it is a simple matter to verify that the two
properties specified for (X,
satisfies (OC)
and (OC)
3.
,
,
q) are equivalent to the assertion that (X, @,
Since
(X,
q)
q)
is complete, Theorem 1.4 and
Proposition 1.6 imply the desired conclusion.
2.
WEAKLY PREORDERED CAUCHY SPACES (WEAK COMPATIBILITY).
a preorder on X, and
As before, we assume that X is a set,
structure on X.
(X,
,
a Cauchy
C) is defined to be a weakly preordered Cauchy space if
there is a quasi-uniform convergence structure
on X such that:
(I) the uniform
213
ORDER COMPATIBILITY FOR CAUCHY SPACES AND CONVERGENCE SPACES
convergence structure
(2)
U{
}.
v
j
-I
derived from
=
has C as its set of Cauchy filters;
This definition differs from that of preordered Cauchy
space only in the condition <
which is required for the latter but not for
,
>
our present definition.
We abbreviate weakly preordered Cauchy space by "w.p.c.s.".
on X, let
be the quasi-uniform convergence structure on
Given a preorder
Y
X with base consisting of all finite filter intersections of the form
0{x i x Yi
n}. If, in addition, C is a Cauchy structure
i
(xi’ y’)
on X, let
where the lattice meet i taken in the lattice of all
T,C
^
’
quasi-uniform convergence structures on X.
C
in
as follows:
Q
fl
iff
or there is (x,y)
fl
As we shal see,
that
o,C
Finally, we define a preorder on filters
I,C
9
C and
and
such that
Qplay
the same role in the theory of weak compatibility
respectively, play in that of strong compatibility.
and
PROPOSITION 2.1.
Let (X,) be a preordered set with Cauchy structure
has a base of sets consisting of all finite intersections of the form
j
n}, where
j
n.
Y,
{
j xj
j
PROOF
each
I 2 " n fl where
i and i (I) "i i
C
(i.e., 3i
C )" By
i i
fom
and there are two possibilities for
i i
(xi, yi)
where
j,
Consider a composition of the
x
Then
,
or (2)
i’
i
and
examining four possible cases, it is easy to verify that the existence of the
composition
i Q i+l"
implies that
we obtain for the entire chain of compositions
i
i+l
Thus if
Q
I
n’
and
and
61 %0...0 %.
We thus see that filters of the form indicated in the proposlton are in
and a base for
involves taking finite compositions and intersections of
YC
such
Y,C
It can be shown by a straightforward set theoretic argument that any
filters.
finite composition of filters of the indicated form can again be written as a finite
intersection of filters of the same form, which is the desired conclusion
PROPOSITION 2.2.
and
C
is compatible with
PROOF.
,
A triple (X,
,
T,C
C)
v
is a w.p.c.s, iff
(T,)
(X,
,
U{
:
,
},
T,
-I
C) is a w.p.c.s, according to the
definition of this term. Conversely, assume that (X,
C) is a w.p.c.s, derived
from a quasi-uniform convergence structure
v
and let
be the associated
Given the two conditions,
,
uniform convergence structure
and
since
u{ne
:
C
^
are compatible with C
u}
a u{ne:e
so is
,e
,
s-I
Then it is easy to see that
Also
C
since (x,y)
D{0
x x
:
C
6
@
and
o}
T,
We next introduce conditions on a triple (X, ,C which lead to a characterization of a w.p.c.s, similar to that given for a p.c.s, in Theorem 1.4.
It turns
out that only two such conditions are needed
to
(OC) 2 and
(OC)3,
respectively.
(woc)2
and (woc) which are analogous
3
There is no form of "local convexity" involved
in the characterization of weak compatibility, and so we shall later use (woc)
to
describe a single condition (not related to
and
(wo)
3
(OC)p
which can replace both (woc)
2
D.C. KENT and R. VAINIO
214
(woc)2
(woc)
,
If W
@
3
@,
are in C,
Q
and
WA
then
C.
implies (x,y) E @.
If (X, ,)
PROPOSITION 2.3.
satisfied.
is a w.p.c.s., then (woc)
and (woc)
2
are
3
,
Q, and Q. Then, by Proposition 2.1,
C
x
Thus () x
and
T,
T,C
()
and by Proposition 2.2,
x)
x
By Proposition 2.1, x
y
i, and so (x,y)
is a w.p.c.s, iff conditions (woc) and (woc) are
THEOREM 2.4. (X, ,C
2
Assume
PROOF.
x
r,
3
satisfied.
Using Proposition 2.2, we must show:
PROOF.
=P’
(2)
:
U{
}.
T,c
T,c
x
implying
Te,
If (x,y)
N,
Let
-1
j x.j
_
so that
N
which implies
x
x
Q
E
,e
implies
T
,e x
(.j j))
_DG=
then x x
and so (x,y)
,
,
"i
j@ j.
N
x,i c.
@
and
@N
and
ZG
(woe)3"
by
By
where
x,
be distinct ultrafilters finer than
indices i, j such that
This implies
,
G for some f
By Proposition 2.1,
Proposition 2.1, we may assume
l,...,n.
(Te, C
T,
It is sufficient to show
(2)
for j
V
@, then x
If (x,y)
(I).
(x,y)
,
where P
j Q j,
Then there are
and
i@"i Thu
where
by
(woc)2.
Rearranging the indices if necessary, let l,...,m be the indices such that, if
,
j _< m, then there are ultrafilters
where
j @
’3
in
g
Let
j
finer than
and
7
_m H
Thus
such that
jlj.
preceding paragraph leads to the conclusion that
ultrafilter
H
{
which implies
We now introduce the condition (woc
qC(x)
q(y)
iff (x,y) {
D__
for a triple (X,
,
for every
is
).
-i.
It is significant that this condition is formulated entirely in terms of
q
the convergence structure derived from
is a w.p.c.s,
iff condition (woc) is satisfied.
(X,
In both directions of this proof, we use the characterization of a
THEOREM 2.5.
PROOF.
w.p.c.s, given in Theorem 2.4.
.
Assume that
x"
qe
y in
(x,y)
A
@
argument,
established.
(X,. ,
x
satisfies (woc)
n
e,
-iThus If (x,y)
By (woc) 2,
q(y)
implies
and.
2
implies x
which_l and
A
A
(woc)
q(x),
and so
3.
and
then
If
qC(y),
?(x)By (woc)
3,
@ and
then
x.
(q(y).
qC(x) qC(y).
implying
q(x),
j,
j
The reasoning of the
1 and that N
C and the proof
complete.
(woc)
x
m
A
By the symmetric
Thus (woc)
is
and
215
ORDER COMPATIBILITY FOR CAUCHY SPACES AND CONVERGENCE SPACES
Conversely, assume that (X,,) satisfies (woc)
Q
Q
and
Considering only the nonobvious case, we can assume that there
@ such that $
x
C and N
C. We can also assume there is
9
@ such that $
(b, a)
Let
I.
C.
5,
is (x,y)
where
C and O
C
qc(a)
Then
qC(x)
qc[a)
@0-I and (b,y)
-1.
(woc)l
Thus (x,a), (a,b), (b,y) are all in @-i, which implies (x,y)
-I. Thus
@ N -I, and again by
(x,y)
(woc)l we conclude q(x) qC(y). Consequently,
0
implying
0
C and so (woc)2 is established.
qC(x) qC(y),
To prove (woc)3, let x Q 9. If x
9 6 then q (x) q (y), and (x,y)
qC(x).
qc(b)
Likewise,
qC(y ).
By
(a,x)
,
follows by (woc)
x
and
C,
a
9
q(y)
I.
Otherwise there are elements a, b in X such that (a,b)
But this implies q (a)
C.
I]
qc(b),
so that (y,b)
-I
@
q (x), so that (x,a)
and
I,
[i
Thus (x,a), (a,b), (b,y) are all in
@
and so (x,y)
@.
Next, we consider weak compatibility for convergence spaces.
preordered set, and let q be a convergence structure on X.
weakly preordered convergence,
(X,
C) is a w.p.c.s, and q
,
,
If (X,
space if there is a Cauchy structure
,
q) is a
on X such that
qc.
q) is a weakly preordered convergence space, then it is an immediate
consequence of Theorem 2.5 that every Cauchy sructure
C
compatible with q has the
(X, @, C) is a w.p.c.s.; in particular (X, @,
property that
Let (X,) be a
The triple (X,
q)
has this property.
Thus we obtain
For a preordered set (X,
COROLLARY 2.6.
with convergence structure q, the
following statements are equivalent.
(X, @ q) is a weakly preordered convergence space.
(X, @, q) is a w.p.c.s.
(a)
(b)
(c)
is a Cauchy structure on X such that q
If
qc
then (X, ,C) is a w.p.c.s.
One final characterization of weakly preordered convergence space which makes no
reference to the notion of Cauchy structure is given in the next theorem.
The triple (X, @
THEOREM 2.7.
q) is a weakly preordered convergence space iff
the following conditions are satisfied:
-i
(a) (x,y)
q(y)
implies q(x)
(b)
@-i
(x,y)
@
PROOF.
If (X,
implies q(x) q(y)=
,q
.
is a weakly preordered convergence space, then (a) follows
immediately from Corollary 2.6, and (b) follows from Corollary 2.6 and the fact that
q
q
c
for some Cauchy structure
C q)
Conversely, assume the two conditions; it suffices to show that (X,
satisfies condition (woc)
I.
From (a) and (b), we see that for any x, y
q
X, q(x) and
q(y) are either equal or disjoint; thus C is a Cauchy structure. Also (a) gives
"half" of condition (woc)l, and the other "half" comes from (b), since q(x)
q(y)
implies q(x)
In case
q(y) #
,
and so (x,y)
-i.
is a partial order, it is appropriate to speak of a
weakly ordered
Cauchy space or a weakly ordered conve.rg.ence space in the case of weak order
compatibility.
(X,
,C
In this case the compatibility is indeed "weak", since any triple
where (X, ) is any poset and
C any T Cauchy
a weakly ordered Cauchy space, and likewise if q is any T
structure will constitute
2
convergence space then
D.C. EENT and R. AINIO
216
,
q) will be a weakly ordered convergence space. It is well known that, in
(X,
C)
general, every T 1Cauchy space has a variety of T completions. Thus if (X,
is any weakly ordered Cauchy space and (X’, C’) is any T 1 completion of (X,C), then
can be defined on X’ in many ways so as to make (X’, ’, ’) a
a partial order
,
’
weakly ordered Cauchy completion of (X,
,
treatment of this topic may be found in
[2].
C).
In the case of strong compatibility the completion theory is much more complicated. Not all ordered Cauchy spaces have ordered Cauchy completions. A detailed
REFERENCES
i.
GHLER, W. Gr,,.dqtrr,,-- der Ana3ysi. Bsnd
und Stuttgart 1977.
2.
KENT, D. C. and VANIO, R.
Sci. 8 (1985), 483-496.
3.
NACHBIN, L.
1965.
!
Birh,,=r Vrag, Basel
Ordered Cauchy Spaces, Internat. J. Math. and Math.
Topo.logy and Order, Van Nostrand Studies, No. 4,
Princeton, N.