483
Internat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 483-496
ORDERED CAUCHY SPACES
D. C. KENT and R. V AINIO
Department of Pure and Applied Mathematics
Washington State University
Pullman, WA 99164-2930
(Received May I, 1985)
ABSTRACT:
This
paper
is concerned with the notion of
"ordered Cauchy
2.
is given a simple internal characterization in Section
space"
which
It gives a discription
and a
of the category of ordered Cauchy spaces which have ordered completions,
4 through
of the "fine completion functor" on this category. Sections
construction
ordered comple6 deals with certain classes of ordered Cauchy spaces which have
not preserve
does
completion
fine
tions; examples are given which show that the
From these
such properties as uniformizability, regularity, or total boundedness.
is needed.
results, it is evident that a further study of ordered Cauchy completions
uniformizability,
KEY WORDS AND PHRASES: Ordered Cauchy spaces, ordered completions,
regularity, total boundedness.
1980 AMS MATHEMATICS CLASSIFICATIONS CODES: 54A20, 5415.
0. INTRODUCTION.
Cauchy spaces provide a completion theory much more general than that obtained
from metric or uniform spaces, and much simpler than that arising from uniform
convergence spaces.
R. Ball
[i], [2]
has shown the applicability of Cauchy spaces
in the study of lattice completions; the compatibility between lattice and Cauchy
structures is obtained by requiring the lattice operations to be Cauchy-continuous.
While this compatibility criterion is quite natural it is not appropriate for more
general order structures, such as partial orders, and may be too restrictive even
for some lattice applications.
L. Nachbin
[9]
has defined an "ordered uniform
space" by requiring
that the
uniform and order structures arise from a common source, that being a quasi-niform
structure on the same underlying set.
We adapt Nachbin’s method to define an
"ordered uniform convergence space", and from this the definition of "ordered
The resulting notion of "ordered Cauchy space"
Cauchy structure" follows naturally.
is given a simple internal characterization in Section 2.
The development of a completion theory for ordered Cauchy spaces is compli-
cated by the fact that some ordered Cauchy spaces do not have ordered completions.
In Section 3, we describe the category of ordered Cauchy spaces which have ordered
completions, and construct the "fine completion functor" on this category. The
remaining sections examine certain classes of ordered Cauchy spaces which have
ordered completions; examples are given which show that the fine completion does
not preserve such properties as uniformizability, regularity, or total boundedness.
From these results, it is evident that a further study of ordered Cauchy completions
is needed.
D. C. KENT AND R. VAINIO
484
-
PRELIMINARIES.
1.
Let X be a set, and F(X) the set of all (proper) filters on X. If
G
for all F E
then
D
denotes the
,8 E F(X), and F N G #
,
F N G
,
A c X, A
If
{F N G: F
generated by
for some F
X
filter on
<A>
we denote by
,
}.
G E
G
and some
On the other hand, if
U
we say that
fails to exist.
the filter of all oversets of
A; one departure
from this convention are the fixed ultrafilters, which are denoted by
for
xEX.
A.
Compositions are defined as follows:
A
B
{(x, y)
X
z
there is
H c X, then
and
addition, G E F(X x X)
filters on
{A(H): A
6
,
X
X x X are defined in the obvious way. If
{y E H: (x, y) A for some x H}; if, in
E
.Q
,
(H), and ()
F(X) then A(),
and
{A(H): H
denote the
},
}, {A(H): A
and
}, respectively.
H
DEFINITION I.I.
If
A(H)
generated by the sets
A set
of filters on
X
convergence structure on
(u)
(u 2)
(u 3)
(u 4)
x E X} is designated
A, B of X x X,
(x, z) E A and (z, y) B}. Compo-
for subsets
such that
sitions and inverses of filters on
A c X x X
{(x, x):
X x X, the diagonal
Turning to the product set
by
whenever
0
is called a
quasi-uniform
c__
0
5,
whenever
and
X x X
if:
6
implies
and
=
exists
6
satisfies the additional condition
,
(u 5)
o
-I
implies
is called a uniform
then
E o
conver.gence
X.
structure on
Quasi-uniform convergence structures are natural generalizations of quasiUniform convergence structures were introduced by Cook and Fischer
uniformities.
[3]
another term often used for the same concept is "pseudouniforme Struktur"
(see
[6]).
A subset
@
’ ,
of a quasi-uniform convergence structure @ is called a base for
if, for each
there is
’
for any quasi-uniform convergence structure
Qu(X)
The set
c
such that
is
One convenient
: c__ < A >}.
A
of all quasi-uniform convergence structures on
X
complete lattice under the natural ordering (the dual of set inclusion).
0,
Qu(X),
then
O
j,
j
where for each
^E
has for a base all finite compositions
OA
or
j
A"
If
o E
uniform convergence structure on X induced by o.
"^"
(X),
then
I
v -I
formq
If
n
is the
The lattice operations "v"
applied to quasi-uniform convergence structures (including uniform convergence
structures) will always be taken relative to the lattice
DEFINITION 1.2.
(c I)
x
C
A set
for all
of filters on
x 6 X
X
is a
Qu(X).
Cauchy
structure on
X
if:
485
ORDERED CAUCHY SPACES
,
If
(c3)
X,
on
implies
C
U
exists, then
and
is a base for
of a Cauchy structure
A subset
c
such that
in
there is
!
and
C
(c2)
E F(X):
there is associated the induced Cauchy structure
}.
x
E
if, for each
With every uniform convergence structure
On the other hand, one can associate with each Cauchy structure
,
with base consisting of all finite
a finest compatible uniform convergence
x
intersections of filters of the form
(one can show
where
that this set of finite intersections is closed under finite compositions).
associated with a Cauchy structure C
x
defined by:
If
then f
f:
(X,
in
(XI, )
qc
x E
iff
(X2, 2)
Caucy-continuous if
is
Also
q
is the convergence structure
X
on
is a function from one Cauchy space to another,
C2
f()
whenever
We next consider the notation and terminology associated with an order
We shall always assume that (X,
on X.
relation
{(x, y)
order relation with its graph
(X, ).
(X, ) be a poset" let
({x}), (A), and ()
Let
(x)
hull of
A
A
is
^
(A)
Indeed, it will usually
rather than the more conventional
F(X).
X, and
Then
-I (x), -i (5),
A, and
A
and
A
is convex if
(X, )
A Cauchy space
is
The convex
respectively.
A
^
similar
^
convex if
lo.call
E C
whenever
(X I,
i
is increasing if
2.
X x X.
are the increasing hulls of x,
-I(A),
notation applies to filters.
If
X, A
x
The decreasing hulls are
respectively.
x
(X,)
be convenient to designate the poset by
notation
is a poset, and we identify the
y}
(X 2,
and
f(
I
2 are posers,
2 (equivalently,
a function
if
f(xl)
f
(X I,
f(x 2)
i
/
(X2,
2
x
E
whenever
x2.
ORDERED CAUCHY SPACES.
Throughout this section, (X, ) will be
DEFINITION 2.1.
arbitrary poset.
,
be a uniform convergence space. Then (X,
(X,
is an ordered uniform convergence space if there is a quasi-uniform convergence
D{:
@}.
on X such that <>
structure
and
V
is an
DEFINITION 2.2. Let (X,) be a Cauchy space. Then (X,
a
is
If
o.c.s.)
there
uniform convergence
ordered Cauchy space (abbreviated
) s an ordered uniform convergence space,
on X such that (X,
structure
Let
,
=
-I,
,
,
and C
C
We see that
(X, @, )
is an o.c.s, iff
on X
quasi-uniform convergence structure
definitions).
.
and
are both determined by some
(in the sense described in the preceding
We shall obtain a more precise chmracterization of of this concept
by making use of a particular quasi-uniform convergence structure
structed directly from
and
o=
Let
quasi-uniform convergence structure on
X
.
F(X x X):
generated by
<>
If
o
,
conbe the
is a Cauchy
D. C. KENT AND R. VAINIO
486
(X, ), we define
structure on
uniform convergence structure
be a poser.
(X, )
Let
LE4A 2.3.
we also define the associated
,C
^
,C (,C)
v
-I
(,)
,
For arbitrary filters
X, the
on
following statements are equivalent:
(a) O() U
exists.
-i(.)
(b)
; U
(c)
@() U
exists.
-I()
exists.
(d)
(3x) U <@> exists.
(e)
(3
x
(X,
Let
I
in
n
X.
a Cauchy structure on
C
If
as follows:
of filters
exists.
C
be a poset and
C
defined on
<x>
<>
C such that
then
<
() U
I
A relation
is
iff there is a finite set
U
80n)
all exist.
This relation is clearly reflexive and transitive, but not generally anti-symmetric.
PROPOSITION 2.4.
Let
(X,))
be a poset,
,C
where S
PROOF:
$
gn
(
<>
<
j
Assume
I
()
<
<,j,
C
x
j):
n.
<
in
U
all exist.
C.
<>
[x]
,C
o ,C
o
C)
Then there is
[l
Thus
n
-1(,)
C such that
in
)
<>
<>
l]
x
I
[x
c
,C
,C contains all filters of the indicated form.
has a base of filters of the form <>
< >
of the form
where
x
6
Furthermore,
< >
< >,
n
that is, a finite intersection of filters
I
C
is a basis filter for
6j
C
or
Finite intersections of such filters are also in
and so
where each
Then
n})<>
j =I
exists; since each component filter in this composition is in either
-I (5) x
X.
a Cauchy structure on
9-I(.j)
({
has a base of filters of the form
C
A straightforward set-theoretic argument shows
@,C can be expressed in the form specified in
that all such basic filters for
the statement of the proposition.
,
Let
PROPOSITION 2.5.
If
PROOF.
-I () x
iff
Then
C.
()
(X, )
x
x
Proposition 2.4, there are filters
such that
n
j
If
x
(x) U <@ >
U(I()
j, ( x)
proof is complete.
poser, C
x
())
,C
I’’’’ Rn
and
is in
x
Then
G,C
i
n
in
since
@
by
C with
({
exists, then
exists.
is clear.
<
This implies
.j
<
Otherwise, for some
.3 <
and the
|
PROPOSITION 2.6. Let (X, @)
C) is an o.c.s, iff
Then (X,
,
X.
be a poset and C a Cauchy structure on
compatible
is
and
U{ 6:
,C
C
with C-
PROOF.
X, and
a Cauchy structure on
the preceding proof shows that
x
Conversely, suppose
j
!R
be a
If the two conditions are satisfied, then
Definitions 2.1 and 2.2.
Conversely, assume that
(X,
, C)
(X,
,
is an o.c.s., and let
C) is an
o.c.s, by
be as in
487
ORDERED COUCHY SPACES
Definition 2.2.
U{A6:
Then
C
and
(X, )
If
a Cauchy structure on X, then
y.
x
implies
With the help of Proposition 2.5, we have
{(:
(
x x
x
the conclusion follows from these observations.
,C
x
if
T2-ordered
A poser with Cauchy structure is defined to be
y. This terminology coincides with the definition of
x
and the fact
,C
C
is a poset and
x
iff
,C
PROOF.
(x,y)
thus the
D
CC
are both compatible with
PROPOSITION 2.7.
U{N
U{A 6:
D
The second follows because
first condition holds.
that
}
A
implies
"T2-ordered’
for
convergence ordered spaces in [8] if we identify a convergence structure as a
complete Cauchy structure. It foiiows from Proposition 1.2, [8] that if (X, ,)
is closed in X x X.
poser with Cauchy structure, then
C) which is locally convex is said to
(X,
then
and
If
C whenever
satisfy Condition (0C)
T2-ordered
is a
,
.
A poser with Cauchy structure
(X,
,
C)
<
is said to satisfy Condition
(0C)2
then we say that Condition (OC)
T2-ordered
,
<
(X, ), C) is
Finally, if
is satisfied.
3
An ordered Cauchy space satisfies conditions (OC) I, (OC) 2,
PROPOSITION 2.8.
and (OC)
3
PROOF.
(X,
Let
be an ordered Cauchy space.
C
immediate consequence of Propositions 2.6 and 2.7.
that
and (OC)
Also
U
-I
<
<>
),C v(,C )-i
^
<
Finally, assume
satisfied.
are in
x
and
-I
,^ x
(x.)
,C
being symmetric, is also in
, n. c.
(x)
<>-- -I()
() x
-I) ((,C) I.
,,C<
.
..
,and so
,C
Let
(X, @
be a
is an o.c.s, iff Conditions
and (OC)
,
poser
(OC) I,
B x B
o, C
implication first under the assumption that
C
x
n
-i(j)
in
,C
x
with
8(j):
<
>
there is an index j,
j
~<
j
Sj)^
C
j
for
n})
j
K x
with Cauchy structure.
(OC2),
and (OC)
3
implies
C
Then
are satisfied.
_
n
In
We prove this
is a free filter; later, this
j
<>.
such that
n
such that
’3 ~< ’3’
-I (j)
and
I’
x
n
and
If K is an ultrafilter finer than
contrary to our assumption that
latter case we see easily that
j
C
is compatible with
then by Proposition 2.5 there are filters
,,C
then either
j
x
and this filter,
restriction will be removed.
I
is
when the three conditions are satisfied.
O,C
other words, we must show that
([
(o,C
Thus
In view of Propositions 2.6, 2.7, and 2.8, it remains only to show
PROOF.
that C is compatible with
If
(OC)
Then, by Proposition 2.5,
(.A) x (N)
Since
C
()
x
|
THEOREM 2.9.
(X, ,C
and
Thus
is an
3
this fact will be used to establish
--< >
C, then
If
2.
-I
C
is a compatible with
,C
Condition (OC)
Proposition 2.6 also asserts
Sj
x
~< j;
(j)
is free, or else
x
thus by (OC)
K.
In the
and
(OC)2,
D. C. KENT AND R. VAINIO
488
{I,..., m} (m <_ n) be
ultrafilter K finer than
Without loss of generality, let
if
{I
j
(j)
m
x
},
there is an
E x K
c
(j)
the preceding paragraph,
Indeed, suppose
than
j, k
such that
either
Zj
exists for some
conclusion that
to conclusion
when
c_
Also, 6
c_
I
5<
where
(0C)2,
Since
I
clude that
()]
x
C.
and k are
Since j
c
we conclude that
3)
is necessarily of the form
C
,C
is not free, then by (OC)
j
finer
k
it follows that
D
)-I()
K k) U
C
Either alternative leads to the
Zk
Z
j
C
is free and
x
X.
IThus C
By our previous results,
is the filter described earlier in the proof.
-<
U
( x I)
exists or
C,
()-l(j) x 9(j))U( x
<I
Either way, it follows that
n.
leads to the conclusion that
x x
x
K
x
and
as in
Applying the same" reasoning with indices reversed leads
<>
implies
]I
},
and by
I
exists for some j,
c__
n
(Kj
K
.)^
m}
5.
j
Kj
Since
exists, or else
where
x
x
c
{l,...,m},
where
I
6
j
,C
x
Zk
and
is a free filter in
If
such that
C
We next show that ={. j=l
{I,..., m}. Then there are ultrafilters
j < k
Zj < Z k
arbitrary indices in
..
Z.
U < >
{I
)
x
Aj
c
_< j _< m, let
For each j,
the indices for which,
i
<
Using
(0C)2
|
Starting with
again, we con-
and the proof is complete.
The three conditions which characterize ordered Cauchy spaces are all quite
The properties
natural.
"locally convex" and
"T2-ordered"
are commonly assumed in
the study of ordered topological and convergence spaces, and the condition (OC)
2
of Definition 1.2. Cauchy
is a natural extension to ordered spaces of axiom (C
3)
structures intrinsic to distributive lattices and lattice ordered groups studied by
Ball
([I], [2])
are examples of ordered convergence structures.
Since Cauchy structures are primarily used as a means for constructing com-
pletions and compactifications, it is natural to turn our attention now to
completions of ordered Cauchy spaces.
spaces have T
2
While it is well known that all T
2
Cauchy
completions (indeed, a variety of different completions are described
in the literature), it turns out that not all ordered Cauchy spaces have ordered
In the remainder of this paper, we characterize those ordered
Cauchy completions.
Cauchy spaces which do have such completions, and examine several special cases.
3.
A triple
if
,
ORDERED CAUCHY COMPLETIONS.
(X, )
(X,
C)
will be called a
is a poset and
C
a
T2-ordered
poser with
T2-ordered
Cauchy structure on
Cauchy structure
(X, ).
Thus a
poset with T
not
(OC)
ordered Cauchy structure is required to satisfy condition (OC) but
2
3
or (OC)
An
increasing, Cauchy-continuous function from one such space
2.
to another will be called a
morphism.
The category having posets with
T2-ordered
Cauchy structures as objects and morphisms as maps will be designated PCS.
,
Let
OCS be the full subcategory of PCS whose objects are the ordered Cauchy spaces.
PCS, let C be the Cauchy structure compatible with the
-I
o
v
It follows from
o,C
,C
,C
Proposition 2.7 that (X,
@C) is T2-ordered. Furthermore,
If
(X,
C)
,
uniform convergence structure
so by Proposition 2.6,
(X,
,
is an o.c.s.
We shall call
,C
C
,C
The ordered
and
ORDERED CAUCHY SPACES
Cauchy modification of C relative
(X, )).
to
489
f: (X, ), C)
If
I’ C1
(X I,
is
f:(X,
a map in PCS, then it is a simple matter to verify that
is also a morphlsm.
F(X,
OCS defined for objects by
F: PCS
,
OCS is a reflective subcategory of PCS relative to the
PROPOSITION 3.1.
reflector
(X, ),)C)
)
and fixed
on maps.
(X, ,C) in PCS is complete, then we shall refer to C as a
X and (X, C as a convergence, s.pac. For complete
If an object
convergence
structure on
objects in OCS, the
T2-ordered
X x X.
product convergence on
property means simply that
A complete
o.c.s,
is closed in the
will be called an ordered
cpnvergence space; the latter term has the same meaning as
ordered
pace"
"T2-ordered
convergence
Let OCON be the full subcategory of OCS whose objects are
[8].
in
ordered convergence spaces.
An ordered completion
DEFINITION 3.2.
,
)
(X,
(X I, O I,
i
((X I,
(Xl’
I
)
((X I,
I’ CI)
I,CI),
and a morphlsm
I’is CI)’
(X)
)
is dense in
X
I).
X*
,
of an o.c.s.
@:(X,
obviously a Cauchy completion of
,
(X, C),
{[]:
two filters
C}
-I
.
Starting with a T
2
X
X*
and
j.
[].
Reed
The completion
Cauchy
Let
be the set of all Cauchy equivalence classes; let
j(x)
are
,
in C are equivalent if
the natural injection, defined by
standard form if
)
and
C), then
(X,
(X, C), and so it will be con-
is an ordered completion of
venient to review some aspects of Cauchy completions.
space
)
which is an ordered Cauchy embedding (meaning that
one-to-one morphisms and
If
(XI’
is a complete o.c.s.
X
((X I,
CI),
)
X* be
is in
[i0] showed that every Cauchy completion
is equivalent (in the usual sense) to one in standard form.
The same is, of course,
true for ordered completions.
The maln goal of this section is to determine which objects in OCS have ordered
completions; the full subcategory of OCS determined by these will be denoted by
,
COCS.
It is clear that (X, 9, C)
COCS iff (X,
) is isomorphic to a Cauchy
subspace of an object in OCON; we seek, however, an internal characterization of
such objects. For this purpose, it is necessary to introduce a new order relation
on the Cauchy filters of an o.c.s.
Let
U x
(X,
,
, ,v
C) be an o.c.s. For
x}. Note that if
X:
Now, for
or
C
<
transitive subset
define
L>
of
I’’’’’ n
I O 2 and 2
I [] and I
in
exists, or
v
I
i
also, if
]"
L
{x
X: x
-l(y)
LN
and
and
U
(y).
iff any one of the following is true:
<
> exists. Let
be the smallest
Cx C containing
C such that
C let
C then
U
@
other words,
n "
then
Note that
I OI
iff there are
I
2
C implies
for every
Before proceeding further, we give an example to
illustrate the difference between the relations
and
2
EXA}LE 3.3. Let X
R
{(0,0)} be the Euclidean plane with the usual
order and with the origin deleted. Let D(x) be the
neighborhood filter at x with
respect to the usual topology. Let
be the filter on X generated by all sets
of
the form
{(x,0): 0 < x <
for n 6 N
and let
be generated by nil
Gn
Q
in
D. C. KENT AND R. VAINIO
490
X
structure on
i
{(0,y): 0 < y <
H
sets of the form
,
D(x)
which contains
}.
Let C
for each
be the smallest Cauchy
x
X
in
,
along with
that
..
It
and
and B
B
i
is easy to verify that
(X,
However
both consist of the closed first quadrant (excluding, of
and
’$
U
C)
is an o.c.s.; also note
Q
course, the origin), and consequently
and
We now introduce three conditions which are analogous to
(OC) 3" we shall show that an o.c.s. (X, @, )
(OC)2,
(OC) I,
and
has an ordered completion iff it
satisfies these new conditions.
E C implies
(COC)
(cot)
(X,
,
C)
*
(X*, *)
PROOF.
<
,
all exist.
Q
3
}.
(X, ,)
If
*
i
[])
([,], [])
([3], ])
PROPOSITION 3.5.
If
I’
,
Assume that
((x,,
c), J).
(COC) 2
If $,
Proposition 3.4 that
[3]
partial order,
If
3
(COC)
X* x X*: 3
in
<
}.
([], [])6
If
Furthermore,
and consequently
*
,
(X,
)
(X,
,
C ,3
)
and
Uj
U
>) U
)71
Q,
then
(j(3)
(-I(3)
U <
or
U>
is both closed and
)I
(COC)3.
has an ordered Cauchy completion
and
([3], [])
[,].
O
L>
Since
has an ordered Cauchy completion, then
O <
then it follows by
@I
since
[]) ! *(J(3)
[]).
is a
([]’ [])
@1 (j(3)
j-l(,(j(.)
[,])
L$
O3,
([], [3])E
and
By Proposition 3.4,
[])
D <
implies
It was shown that the o.c.s.
(COC)2,
)
n
is in the closure of
A direct argument shows that
<
,
(X,
is a completion of
are all in
(COC)I, (COC)2,
satisfies
PROOF.
((3)
(COC) 2, then
j: (X,@
i"
transitive,
C)
as
is satisfied, then
all have traces on %
x
are all in
([
exists, then
I(j(3)
X*
n
n
I"’’’
J ()
!
j)
{([3], [])
Thus 3 x
(COC)
on
satisfies
(COC)3
The proof is concluded by observing that if
(X,
*
I"
!
and there are
!I
[i])
I’ I )’
(X*,
If
First, let
J (’)’ J
,
C
is an order-embedding.
standard form, then
and so
>) E
L3
x _< y.
X*; if, in addition,
PROPOSITION 3.4.
([.],
implies 3
and
N <
be an o.c.s., and define the preorder
{([3], [])
is a partial order on
then
O
implies
3
Let
follows:
(-I()
C,,Q,
2
(cot)
(() N < U 3 >)
CI,
Q
[,]))
(3)
< U
>
But
and consequently
> 6 C
,
(X,
C) of Example 3.3 fails to satisfy
and thus we see that COCS is a proper subcategory of OCS
491
ORDERED CAUCHY SPACES
(COC)I, (COC)2,
We next show that the conditions
and (COC) are also sufficient
3
First, recall Wyler’s completion
for the existence of an ordered completion.
arbitrary T
), j) of an
[]: e}.
((X*,
{j()
THEOREM 3.6.
(*C)
(XI’ I’ CI) is
(X*, *,
morphism
(X,
Let
)
satisfy
(COC)I, (COC)2,
(X, ), C).
is an ordered completion of
a morphism and
(X*, *,
f*
,
C)
(X, ), defined by
Cauchy space
2
I’ I
I’ CI
(XI’
(X I,
and
OCON, then there
f
such that
,
(COC)3.
f:(X,
If
Then
)
is a unique
f*
j.
We have previously observed that conditions
(COC) 2 and (COC) 3
(X, @)
(X*, @) is an order embedding. It is also clear
that j
(X*, *C) is Cauchy-continuous, and that j(X) is dense in
(X, C)
-I is also
X*. The fact that
Cauchy-continuous follows from (COC) and the
<
result (used also in the preceding proof) that
()
U > j-l(, (j() []))
(along with the dual equality).
PROOF.
guarantee that
Given a function
f*([])
by
y
f as specified, let
where
C
f()
f*
be the natural extension defined
Then
)
f*: (X*,
(X1, C I)
is
Cauchy-continuous by the well-known extension properties of Wyler’s completion
(see [5]), and
To show that
f($)
f*: (X*,
f*
O f()
*C)
(XI, i)
is Cauchy-continuous by Proposition 3.1.
is increasing, we first note that
I"
in
Since
I
$
in
implies
f()
is complete, we have also that
and it follows from Proposition 1.2,
[8] that., f*([$])
yf()
Yf(0
f(0,
f*([])"
|
Henceforth, the ordered completion of (X, @, ) defined in Theorem 3.6 will
be denoted by (X*, >*, C*) rather than (X*
*)
Following [4], we define an ordered completion ((X*, i,
j) to be
order-strict if, for any ordered completion ((X*,
)’ j) in standard form,
From Proposition 3.4, it follows that the ordered completion of
!
*
CI)
2’ 2
I 2"
Theorem 3.6
is order strict.
COROLLARY 3.7. For an o.c.s.
,
(X,
C), the following statements are
equivalent.
,
,
()
(x,
(2)
(X, O, ) satisfies conditions (COC)I (COC) 2, and (COC) 3.
C) has an order-strict ordered completion.
(X,
)
(X* O* *) is an ordered completion of (X
(3)
(4)
c0cs
fi)
,
,
Without repeating the relevant definitions here, we remark that, in the
terminology of
(X*, *, C*)
[5],
the functor
K: COCS
OCON, defined by
K(X,
C)
is an order-strict completion functor, and consequently that COCS is
a completion subcategory of OCS.
as the fine ordered completion of
,
We shall henceforth refer to
(X,
C), and
K
((X*, @*, C*), j)
will be called the fine
ordered completion functor.
4.
UNIFOMIZABLE ORDERED CAUCHY SPACES.
Let
,
(X, O, )
be a poset
(X, @)
equipped with a T
.
uniformity
Then
2
if there is a quasiuniformity
(X,
) is a uniform ordered space (Nachbin, [9])
g U g -I and
g on X such that
g; such a quasiuniformity is said to
determine (X,
). The set of -Cauchy filters is denoted by
,
D. C. KENT AND R. VAINIO
492
(UOC)
(3x3)
(g(3) U
and so
(g-l(3)
(UOC) 2
x
(,))
S
x 3
-
,
x 3
.
in
(X,
g.
D
__D
x
g.
9__ g
g()
x
-I
__D S U S
=
__D S
(x,y)
,
)
(X,
g
N
y.
x
(X,
is a uniform ordered space, then
X
that determines
,60,C
3 x
c_
Conditions (UOC)
’2
2
and
(OC)
(X,
,
,
is
).
Let
Clearly g generates
and therefore
(UOC) 3
thus imply
follows from
(UOC)I
and the
by Theorem 2.9.
o.c.s,
(X, @, C) is uniformizable if there is a T
2
uniform ordered space
C
such that
THEOREM 4.3.
Similarly,
This implies that
,)
An o.c.s.
D
x)
g(,.
D
since
But
conditions (OC) and (OC) 3, respectively.
2
Thus (X,
is an
fact that g ! <O>
, %{)
g
-I(3)
S-l)) U
(3 x @ )- S"
S
D
Then, by Proposition 2.5,
Thus
(
g() x
(S (,39 x
D
a quasiuniform convergence structure coarser than
$
3 N
g D__ g
3 x 3
Likewise,
gl())
be a quasiuniformity on
Let
PROOF.
x
x x
If
g.
3 __D
S
D
then
%(
g and
x
THEOREM 4.2.
x
g-1
S U
g ()
and
(3) x g(3)
x (S(@) U
S() __c
an o.c.s.
<
g
and so 3 x
(UOC) 3
3
_=
3 E
__c
x _< y.
then
g-l(,))
g(3))
x
_
c
(UOC)
PROOF.
g
()
g
If
g(3)
C
.
cq-l(3)E
then g (3) U
E
If
2
(UOC) 3
But
Then the following conditions hold.
C,
If 3 (
(UOC)
be a uniform ordered space determined by a
(X, ,)
Let
PROPOSITION 4.1.
quasiuniformity g
Each uniformizable o.c.s, has an ordered completion.
(X, ), C)
,
7) be a
compatible uniform ordered space; let g be a quasi-uniformity which determines
Let
PROOF.
(X, ,).
C
Let 3
g (3)
g
!
and
g(3)
preceding proof that
be a uniformizable o.c.s., and let
U
x
Then
3
$
J
x, and consequently
< U
@($), and therefore g() ! @(3)
(3) c
8
(3) 0 < L 3 >
forward argument shows that
follow from (UOC)
2
and
Thus
g()
(UOC)3,
(COC)
and it follows from the
x
!
3
(X,
U>
g()
!
>
Similarly,
follows from
<
(UOC)I.
Also
A straight-
whenever 3 <
!
respectively.
and so (COC) and (COC)
2
3
The conclusion follows by
Theorem 3.6.
It is natural to ask whether the fine completion functor K, when restricted
o.c.s.’s,
to uniformizable
the answer is no.
preserves uniformizability.
As the next example shows,
The same example shows that the fine completion
preserves
neither regularity nor total boundedness.
EXAMPLE 4.4.
partial order:
are
E-related
E
Let
E
be the Euclidean plane with the usual topology and
{(a,b), (c,d)):
a
c and b _<
d}.
iff they lie on the same vertical line.
Note that the elements in
{L n N} U
Let Y
n
E
493
ORDERED CAUCHY SPACES
{(0,0)},
{(
Ln
where
};
y _<
y): 0
(Y, O’, 4’)
let
be the compact, T
z-
ordered topological space (considered as an object in OCON) with order and topology
inherited from E.
Let
Cauchy structures which
,
tively.
(X,
X
N}
n
T2-ordered
is a compact,
respec-
(X, O, ),
completion of
((X*, 0., *), j)
an obvious correspondence between the sets
Let
be the Frechet filter on
be the corresponding filter on
X*.
is non-convergent relative to
Y
(X, O, ).
of
There is
relative to which the
Y
X* and
S
correspond te the subset
X*
equivalence classes of non-convergent filters in
Y.
The order and
and
is a uniformizable o.c.s.
Now consider the fine completion
of
Y- S.
X
and let
are denoted by O
Y
inherits from
(Y, 0’, ’)
Since
)
{(
S
), and let
(
converges to (0,0) in (Y, 4’),
p ( X* is the equivalence class
of the sequence
Although
(X*, *).
If
of filters converging in (X, ) to the origin, then the closure of the
p-neighborhood filter in (X*, *) is nonconvergent, and so (X*, *) is not
regular, and consequently not uniformizable.
(Y, ’, ’)
From the fact that
is totally bounded (meaning that every ultrafilter on
converges in
ultrafilter finer than
(X,
is compact, it also follows that
X
).
is in
,
)
Since no
(X*, *), we see that total boundedness
is
not preserved by the fine completion functor.
Although no definition of "regularity" has been given for ordered Cauchy
(X, O, 4)
spaces, the space
as an ordered Cauchy subspace of a
T2-ordered,
compact
topological space, would be "regular" by any reasonable definition of that term.
(X*, G*, )
Since
convergence space sense) no
is not regular (in the usual
reasonable notion of regularity is preserved by the fine completion functor.
TOTALLY ORDERED CAUCHY SPACES.
5.
We begin with a simple condition which is sufficient for the existence of an
ordered completion of an o.c.s.
An o.c.s.
DEFINITION 5.1.
)
i
<
U,
satisfies Condition (A)
]
nonconvergent, there is
is
i
,
(X,
such that
O-I
(I) c__
if, whenever
<
> and
L
>
If (X, @, g)
PROPOSITION 5.2.
is an o.c.s, which satisfies Condition
(A),
then:
(COC)
(3)
(COC) 2
(OC) 2
(COC) 3
(OC) 3
(X, @, ) has an ordered completion.
(5)
PROOF.
(I)
-l(y),
and
on
coincide.
(OC)
Suppose
<
and
U <
>
L
follows
<
reasoning shows that
implies
<
If
exists.
intmediately.
Condition (A) guarantees that
Q
<
The relations
(2)
(4)
L
Q
(I)
U
-I()
whenever
U
If
converges to
exists, and so
<
U
>
y, then
is nonconvergent,
exists.
<
Similar
Thus it is clear that
The converse is always true.
Statements (2), (3), (4), and (5) are easy corollaries of (1).
|
D. C. KENT AND R. VAINIO
494
(X, ), C)
x* x x*:
Note that if
{([], [])
o*
< }.
(X, ,C )
PROPOSITION 5.3.
(F) 0 G
-I(F)
() U
PROOF.
~<
~<
But
A
L.
-I ()
t.o.c.s,
-I
c_
Condition (A).
every ordered completion of
() D
exists, and so
[], @()
C be nonconvergent.
L>
(X,
U
If
exists, which implies
This, along with the dual argument, proves
, ,
C) has an ordered completion, and
(X,
C) is totally ordered.
The existence of an ordered completion follows immediately from
PROOF.
Proposition 5.2 and 5.5.
(X*, *, C*)
The completion
is totally ordered by
Proposition 5.3 and the remark following Proposition 5.2.
@*
.
satisfies Condition (A).
Every t.o.c.s.
COROLLARY 5.6.
0 G
[].
be a t.o.c.s., and let
< 5, and since
<
()
Thus
then
)-I(F))
is impossible.
assumed, so by (OC) 2,
(X, @, C)
Let
PROOF.
x
E
fails to exist, then
is
PROPOSITION 5.5.
such that
be a t.o.c.s, and
exists, or else
() D
If
X N G
X, and
G
and
((F) U
which means that
-I(F)
(X, O, C)
Let
COROLLARY 5.4.
,
G
(F) U
is nonempty,
then either
(X, @),
are filters on a totally ordered set
and
-1()
and
F
If
is a total order.
U
exists.
exists, or
If both fail to exist, there is F E
PROOF.
Since
for which @
(5)!J
then either
(abbreviated t.o.c.s.) will be used
space
The term totally ordered Cauchy
for any o.c.s.
(A), then
is an o.c.s, satisfying Condition
Since the total order
is the smallest allowable order for an ordered completion in standard form, it
is the only possible order for an ordered completion in standard form.
6.
AN ORDERED COACTIFICATION.
Every
compactification of an ordered topological or convergence
T2-ordered
space can be regarded as the completion of a certain totally bounded o.c.s., and so
the entire subject of
T2-ordered
present investigation.
compactifications lies within the scope of our
Our immediate goal, however, is rather modest; we shall
formulate the ordered convergence compactification constructed in
[8]
as an
ordered Cauchy space completion, thereby gaining some further insight into its
properties.
Let
(X,
the notation
,
+)
be an ordered convergence space (i.e., an object in OCON);
x"
"7
relative to
X
consisting of all convergent filters
[8]) let X’ be the set of all nonconvergent,
(X, O, +). In [8] an ordered convergence space
and (following
maximal convex filters on
(X, O, +)
is defined to be strongly
dual are satisfied:
If
x,
T2-ordered,
X’
and
With each ordered convergence space
structure
Let C
converges to x in this space.
indicates that
be the complete Cauchy structure on
C
U
F(X):
there is
if the following property and its
()
exists,
-I()
c
(X, @, +), we associate the Cauchy
6
X’
is the finest Cauchy structure compatible with
such that
G c
}.
Note that
which is both totally
495
ORDERED CAUCHY SPACES
bounded and locally convex relative to
(X,N)
A Cauchy completion of
If
X*
the filter on
[8]
PROPOSITION 6.1.
is a T
2
T2-ordered
(X,
)
F(X), let
If
Let
F(X*):
(X, +).
,
(X,
A convergence ordered space
PROPOSITION 6.2.
iff
5}.
}.
A
and
For any ordered convergence space
compactification of
,
X’
{: F
generated by
$ c
}.
such that
$
in standard form is constructed as follows.
X*:
j(X) U {[]
A c X, let
(X,
be
there is
,
-),
(X*,
is strongly
is an o.c.s, which satisfies Condition (A).
(X, (9, ) is an o.c.s., then (X, (9, +), which is its associated
convergence space, is clearly an ordered convergence space. It is also clear that
)) implies the strongly
Condition (A) applied to (X,
property for
PROOF.
If
,
T2-ordered
(x, (9, +).
<
Conversely, assume
(X,
relative to
,
), where
-,
are both
A
convex, and either which is non-convergent is assumed to be maximal convex.
straightforward argument based on the strongly
T2-ordered
property leads to exactly
one of the following conclusions:
(I)
(2)
(3)
(4)
X
x, y
There are
There is x
X
There is y
X
Both $ and
such that
such that 5
such that
(X,
<
I($I).
L>
8/ y,
is nonconvergent, and
() c
(OC)I, (OC) 2, and (OC)
,
N)
is an o.c.s.
@() ! l, or
,
Finally, if
and
of
(X,
,
*
it follows that
(X*, *, *)
which implies
((X*, @*,
), j)
is an order-strict,
(X, 8, -)
which is the fine completion of
[8],
we obtain:
+) is strongly
T2-ordered
compactification
4"
is finer than
If
"*
is
also has an extension of the
From Theorem 3 6
(X, @, )
totally bounded, then
is also a compactification, and indeed
T2-ordered
of
)*
in this case.
Under the assumptions of Theorem 6.3,
compactification of
When this happens,
(X, (9,
It is shown in
[8] that,
(X*, @*,
(X, @, +)
iff
N*
(X*, *, *)
j) is
is totally
* ), j)[8]. coincides with the compactification
constructed in
under the assumptions of
is the largest "relatively regular"
There is not, in general, a largest
T2-ordered
,
(X,
If an ordered convergence space
PROPOSITION 6.4.
the largest
(X*, *,
are easily
-).
(X* (9*
bounded.
3
is nonconvergent,
I(),
x
equivalently
Under the assumptions of Theorem 6.3,
form
(9()
This, along with the dual argument establishes Condition (A).
THEOREM 6.3.
T2-ordered,
c
then statement (2) above
x
Combining the preceding results and Proposition 2.8,
then
()
are nonconvergent, and
is the maximal convex filter coarser than
gives the result
()
is nonconvergent, and
With the help of these results, conditions
verified, and so
y, and
x,
x,
T2-ordered
T2-ordered
Theorem
6.3, (X*, *,
(X, @, +).
compactiication of
compactification of a strongly
convergence space as the concluding example shows.
D. C. KENT AND R. VAINIO
496
This example is based on Example 4.4, and the same definitions
EXAMPLE 6.5.
,
/) be the T2-ordered topological
Let (X,
is the relativization to X of
(X, ,); in other words,
is the finest locally convex
the usual topology on E. It turns out that
+). The compactotally bounded Cauchy structure on X compatible with (X,
and notation will be used here.
space compatible with
,
(X*, @*
tification
while the fine completion
*
Thus, in general,
(X*, *, *)
,
and
structure on
(X, @,
@’)
is not totally bounded.
There is no largest
T2-ordered
(I,0) instead of (0,0). Then (Y, ’, C")
(X, @, +) which is neither larger nor smaller
filters are assigned to converge to
than
of
are distinct.
’
). Indeed, suppose C" is the complete Cauchy
(X,
Y which agrees with ’, except that the filter
and all finer
compactificatlon of
is a
(Y,
is equivalent to the compactification
T2-ordered
compactification of
(Y’, @’, ’).
By altering the order structure in the preceding example, one can get quite
Suppose
different results.
by
unchanged, but replacing
(X,
(X,
+)
i’ /)
@i’ the
is obtained by leaving
X
and
It
usual order of the Euclidean plane.
and that
is again the finest
compatible totally bounded locally convex Cauchy structure.
In this case the fine
turns out that
(X*,
completion
,
Proposition 6.4,
[8],
@I’
is strongly
T2-ordered
*) of (X, @I,
(X*, @, *) coincides
and is the largest
T2-ordered
is totally bounded.
Thus, by
with the compactification constructed in
compactification of
(X,
I’ +)
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Trans. Amer. Math. Soc. 259 (1980), 357-392.
I.
BALL, R.N.
2.
BALL, R.N.
Cut Completions of Lattice Ordered Groups by Cauchy Constructions,
Groups, M. Dekker, Lecture Notes in Pure and Appl. Math. Vol. 62
(1980), 81-92.
Ordered
3.
Cook, C.H. and FISHER, H.R.
Uniform Convergence Structures, Math. Annalen 173
(1967), 290-306.
4.
FLETCHER, P. and LINDGREN, W. Quasi-uniform Spaces, Lecture Notes in Pure
and Appl. Math. Vol. 77 (1982).
5.
FRIC, R. and KENT, D.C. Completion Functors for Cauchy Spaces, Internat.
J. Math. & Math. Sci. 4 (1979), 589-604.
6.
GHLER,
W.
Grundstrukturen der Analysis, Band I, Akademie-Verlag, Berlin
(1977).
7.
KELLER, H.H. Die Limes-Uniformisierbarkeit der Limesrume, Math. Annalen 176
(1968), 334-341.
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KENT, D.C. and RICHARDSON, G.D.
9.
NACHBIN, L.
10.
REED, E.E. Completions of Uniform Convergence Spaces, Math. Annalen
(1971), 83-108.
A Compactification for Convergence Ordered
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194