I nt. J. Mh. & th. Sci.
Vol. 5 No. 3 (1982) 513-527
513
UNIFORMIZABLE CAUCHY SPACES
EVA LOWEN-COLEBUNDERS
Vrije Universiteit Brussel
Oepertement Wiskunde, F7
Pleinlaan 2, 1050 Brussel
(Received April
13,
1981)
ABSTRACT. A family C of filters on a set X is uniformizable if there
is a uniformity on X such
that C is its collection of Cauchy filters.
Using the theory of completions and Cauchy continuous maps for Cauchy
spaces, we obtain characterizations of uniformizable C,auchy spaces.
In pati-.ular, given a Cauchy structuYe C on X we investigate
F
what conditions the filter u(C)
F x F
is
under
a uniformity and
C
is
FEC
its collection of Cauchy filters.
This problem is treated
usinE Cauchy
coveinE systems.
-,
KEY ’ORS AN PHRASES. Uniformity, Cch pace, completion, Cauchy
ompctne Cchy covering ytem.
@inu
1980 MAThEMATiCS SUBJECT CLASSIFiCATiON CODES. 54A20, 54E15.
INTRODUCTION
As background for eading this paper, the mathematician unfamilia with Ceuchy spaces might consult the paper Completions of uniform convergence spaces
by E,
Rmed
I.
A Cauchy structure C on a set X is a coiIection of fiIters on X
satisfying the conditions that for every point x of X the fiIter gene-
514
E. LOWEN-COLEBUNDERS
{x} belongs to C.
rated ty
Ig a
gilter
belongs to
C,
then
so does
every finer filter and, if two fllters in C have a supremum, their
intersection beiongs to C.
In |2 KeIier has shown that these proper-
ties are necessary and sufficient in order to have a uniform conver-
gence structure on X in the sense of
collectlon of its Cauchy fiIters.
in
the sense of
to be
(4]
uniformi.za,ble.
with
C as
3
such that
C
is exactly the
If there exists a unlgorm structure
its family of Cauchy fllters,
This notion was introduced in
5.
C is
said
The main pur-
pose of this paper is to flnO supplementary conditions on a Cauchy
space in order to characterize the unlformlzabillty.
When
C
the
is
collection of convergent fiIters of a topoIoglcal space, the problem
reduces to the complete unlformlzabillty of the topology.
Two diffe-
rent attempts wIII be made in studying the unlformlzabIiity. The first
method uses Cauchy continuous maps. Barring measurable cardlnais, we
obtain that a Cauchy space
(X,C) is unlformizable if and oniy
ig
there
exists a colIectlon of realvaiued Cauchy continuous maps such that C
is
the family of aiI filters having a Cauchy image on
IR
for every
map in the colIectlon.
A second method uses completlons. We show that (X,C) is unlformlzable if and oniy if there is a dense Cauchy embedding in the coIIec-
tlon of convergent fllters of a completely unlformlzable topological
space, more specifically if and only if its Kowalsky or ultrafllter
compietlon
Barrln
(1
is a completely unlformizable topologlcaI space.
measurabie cardlnals the previous
the realcompactness of the completlon.
tions on
C can
condition is equivaIent to
From this property some condi-
be derived.
Any uniformity on X having C as its family of Cauchy filters is
coarser than the fiIter u(C)
converent
F
F x F.
filters of a topoIogicaI space,
hood filter of the dlagonal.
When C is the coilectlon of
then u(C)
is
the neighbor-
In the second part of the paper we shaI1
UNIFORMIZABLE CAUCHY_ SPACES
515
investlgate if u(C) ls a unlformlty and If so,
C
is its collectlon
of Cauchy Tilters.
under what conditions
A very usefuI notion to treat
these problems is that of a Cauchy covering system, a notion introduced in
6]
and extending the definition of covering system of a
convergence space
7]. Using Cauchy covering systems
we characterize
the property of u(C) being a uniformity. We show that u(C)
C
its family of Cauchy filters if and only if
has C as
unlformizabie.
is
Exten-
dlng the notion of paracompactness we introduce Cauchy paracompactness
for Cauchy spaces. We show that for every Cauchy space wlth minimum
elements In its equlvalence classes, which
Is
Cauchy paracompact and
u(C) Is a uniformity with C as its family of Cauchy filters.
round,
In the flnal section we treat the speclal case of a totally bounded
Cauchy space.
PRELIDIINARIES
2.
We Eecall some of the fundamental definitions on Cauchy spaces
needed in the paper.
6
8)
9
oY
For addltionai Informatlon. we refer to
10). Let X
be a set and F(X)
the collectlon
ters on X. A subset C C F(X) is a Cauchy structure if’
{i)
FoE every x E X the filter
{II)
If
F E C
and G
(iii)
If
P E C
and
F
G E C
The filters in C aye
then
and
G
F vG
generated by
{x} belongs to C
C
exists then
_Ca.uchy filters
F
and the space
Every uniform space {X,U) in the sense of
41
G
C
(X,C) a _Cjauch s_pace.
induces a Cauchy struc-
ture c(U) which is its collection of Cauchy filters. A Cauchy structure induces a convergence structure
The Hausdorff property,
F
x if and only if
the Cauchy
E C.
closure operation, and continuity for a
Cauchy space are defined in the induced convergence space.
theer
F F
Throughout
pace (X,C) used in the formulation of the theo-
rems is supposed to be Hausdorff. A map f between two Cauchy spaces
(X,C) and (Y,D) is Cauchy continuous if the image f() of every
in
E. LOWEN-COLEBUNDERS
516
belongs to
D. Here fCF)
Cauchy continuous maps are continuous.
Cauchy embed-
Cauchy subspaces,
are defined in the obvious way. For
and Cauchy homeomorphisms
dings,
F E F}.
{fCF)
the filter generated by
is
Cauchy continuity of maps between uniform spaces, we use the induced
Cauchy spaces When C and
C
then
D (
(
coarser,
is
(,) to (X,C)
D
are Cauchy structures on the same set X,
D
is
finer)
A Cauchy structure is totall_y
Cauchy continuous.
is
bounded if every ultrafilter is Cauchy
of its convergent filters is
(Y,P)
it is
if every
c__o.m.piete
For every convergence space the collection
Cauchy filter converges.
(Y,D,K)
if the identity from
if and only
of a Cauchy space
a complete Cauchy space.
(X,C)
consists
of a complete Cauchy space
Y such that K(X) is dense in Y.
X
and a Cauchy embedding k
A completion
Two filters in C are equivalent if their intersection belongs to C.
For a filter F in C, let <F> be its equivalence class. X
of all equivalence
6]
8]
9]
classes an j
10]
{o E E
A C X,
O
let A
o
X
.
C
Finally
(X
,C ,j)
is
o(p)}. For
.
a filter
on
X
for every p
is the collection
let
ro(F)
(X,C)
o
E
is
be the filter
either B or
{A
F <
XIA
be
X
p for p
and
on X,
let
o
be
converges to some p
X
P such that
of all convergent filters on
the KowalsKy completion of
X
F
E, there is a filter F
and B subsets of X, we have the relation A <
and,
5]
F}. A corvergence structure on
F
A filter
for every o
o(pi
such that
x for x E X}. For each o
O
generated by {F
is defined as follows
if and only if,
X
A
F(X)
X
(J(x))
{p E X
the filter on X
Let I
will be used explicitely in the text.
the collection of all maps
I]
In
Eiven to construct completions. The
methods are
Ko____wals_____Ky com_p=l_e=ti_o._n.
and let I
maps x to <x>.
X
X
is the set
E
(X,C). For
o
and A
B if and only if A C B
belongs to o(p).
A for some F
r
For F E F(X),
F}. A Cauchy space
caiied round if it is closed for the operation r
for every
UNIFOIIZABLE CAUCHY SPACES
517
WEAK CAUCHY STRUCTURES
3.
{f1"
i E
Let F
C
Let
for i E I,
This
oefines
I} be a collection of maps
if and only if g.
1
.
a Cauchy structure on X0
called the weak Cauchy
structure
qf
P.
(F}
(X,C), let F U (X,O)
It
(X,.C)
is uniformizable if and only if
(1) There exists a nonempty collection ( C
(2)
structure of @,
C
the weak structure of
is
PF’OOF. If C
is
(X,C) the collection
(XC).
one of the folowin equivalent conditios are
weak
For a
continuous,
the collection of ali Cauchy con-
be
of all Tealvau_ed Cauchy c.ont.%nqou.s ma.ps on
A Cauchy spa.ca
1.
1
is the coarsest Cauchy
tinuous maps from (X,C) to uniform spaces and
T,EOREK 3,1.
X-(Yi,Di)
fi
/or every
(not necessarily Hausdorff),
structure on X making all the maps in @ Cauchy
Cauchy space
where
Fulfilled
F U (X,)
C
such tr. at
is
the
U
uniformizable and
consider i the identit_y
U
is
a uniformity
fr (X,) to (X,Q)
on X such that
and let
@
{i}.
Hence (I) is satisfi_ed,
et @
To shcw that (I) lmplSes (2]
Cauchy structure of
U
is
(x,C)
C.
and the
@. Then
atter
is
U
(x,C) such that C
is fineT than
ls the weak
th.e weak str6cture of
flne. r than the weak struoture of @ which
Hence, they a.l co%ncde and (2) %s fulfil_led,
To complete the proof, suppose C %s the weak Cauchy structure of
u(X,C),
Let U be the weak unifermity of
u(X,C),
sest uniformity on X makin all the maps in
tinuous.
Then we have c(O)
THEOREM 3,2, If a Cauchy
uniformity of
C
and
C
is
r U (X,C]
uniformly con-
uniformizable.
same {X,C) %s
F U (X,C) is the finest
which is the coar-
uniformizab%e,
.uniformity with
of Cauchy filters and among the uniformities with
C as
its family
C as family of
Cauchy fiIters it is the only uniformity makin all maps in
uniformly continuous.
the weak
u
(X )
518
E. LOWEN-COLEBUNDERS
PROOF. Let U De the weak uniormlty of F (X C)
U
the previous
C. Let
theorem it is shown that c(i/)
uniformity on X such that c()
r U (X
(X,W) belongs to
C. Then
C
C). Hence
In the proof
be another
the identity from
(XC) to
I W in addition is
supposed
to make all the maps in F (X C) uniformly continuous
U
since
U
is the
then
coarsest such structure.
In the next theorem we suppose X has non measurable cardinaiity,
which is a very light restriction to make since no measurable cardlnals are known.
With this restriction complete uniformlzabillty og
111.
a topoIogical space Is equlvalent to realcompactness
THEOREM 3.3. A Cauchy space (X,C) is unlformizable if and only if
one of the following equivalent conditions is fulfilled
.
(1} There exists a nonempty collection
the weak Cauchy structure of
(2)
C
is the weak Cauchy
structure of
@ C
F]
FIR (X,C)
such that
C
is
[X,C).
PROOF. Let U be a uniformity on X such that c(i/)
C.
Suppose
F
is a filter on X such that for every
]]R
Let (X
Cauchy
of X in X
X,
and
CIR
be the completion of
(X x I/x)
CiR (X
mlty of
I/K)
CIR (X
hoJ(F)
on X
the embedding
is IR-Cauchy for
But then j(F) Is a Cauchy filter
I/x)
J
the collection of all real valued conti
nuous maps on [X 1/x). It follows that
h
[X,U)
or
every
the weak unlfor
Since we are barring measurable cardinals
thls uniformity is complete and so
J{F) converges. Hence F E C.
The
other implications in the theorem follow trlvlaIly from (3.2).
4.
COMPLETIONS
In this paragraph we investigate the relation between the uni?or-
mizability of (x,C) and properties of its completions.
THEOREM 4.1. A Cauchy space (X,) is uniformlzable if and only if
it is embedded in a [dense) Cauchy subspace of a completely uniformizable topological space.
UNIFORMIZABLE CAUCHY SPACES
519
PROOF. If (X,C) is uniformizable by U, then the completion (X
,U )
(X,C)
provides e completely unlformizable topological space in which
is densely Cauchy embedded.
If there is a Cauchy embedding j of (X,C)
having a complete uniformity
W,
then
j-1
into a topological space Y
x
j-l(,)
is a uniformity on
X for which C is the collection of Cauchy filters.
(1|
In
several methods are given to construct completions of a
given Cauchy space. Some of them preserve unlformlzability, for
example the KowalsKy completion and the ultrafilter completion.
fact, when
In
(X,C) is unlformlzable, these completions are Cauchy
homeomorphic to the Cauchy structure of the 8ourbaKi completion.
From
the previous theorem, we now immediately have the following property.
EOREM 4.2. A Cauchy space is uniformizable if and only if its
KowalsKy completion (ultrafilter completion) is a completely uniformizable topological space.
Ws will apply this theorem for the case of the Kowalsky completion
in order to find explicit conditions for unlformizability.
tel property that will be needed is the
A fundamen-
.r.e.ular _ext_enelon prppert
1]. It means that every Cauchy continuous map
from a Cauchy space
(X,C) to a complete and regular Cauchy space has a unique (Cauchy)
continuous extension to the Kowalsky completion. In the special case
of IR
FIR (X
this implies
X
that there is a
C x) mapping f to
its extension
is a ring isomorphism and a
p E X
x,
the value of
fX
f--F--.
fx
from
fir
For f E
in p is the limit of f(F) where
This
(X,C) to
This bljection,
lattice isomorphism.
filter arbitrarily chosen in p.
A
biJection
in fact,
F
F
(X,C) and
is a Cauchy
limit is unique and equals
We shortly write lim f(F).
FEF
THEOREM 4.3. For nonmeasurable cardinals, a Cauchy space (X,C) is
uniormizable i and only if each of the following conditions
(2),
(3)
and
(4)
hold.
Condition
(4)
(1),
can be replaced by the equivalent
520
E. LOWEN-COLEBUNDERS
(4’).
conditlon
(I)
Every equivalence class in C has a minimum element.
[2)
The minimal Cauchy filters have an open base.
(3)
For every minimal Cauchy filter
and for every M E
M
exists a Cauchy continuous map f such that llm f()
lim f(f)
(4)
{(f)
the subset of X
F
fir (x,C)}
f E
and
X\M.
0 for every Cauchy filter containing
In the collection (x,C)
there
where (f)
is
consisting of those p for which there exists an
p with lim f(F)
O, every subcollection, maximal for the
finite intersection property and with the countable intersection
property, has a nonempty intersection.
Let
4
(rf) fEr
rl (x,c).
g
be a collection of real numbers indexeO by
(x, C)
If for every finite subset I 0 C
> 0 there is an x E X such that
f E I O,
If(x)
then there is a Cauchy filter
for every f E
r]R
F
r (x,C)
rfl
and for every
for every
<
such that
lim f(F)
rf
(x,C).
PROOF. Suppose (X,C) is uniformizable by a uniformity O. Then
(1)
and
(2)
hold.
and p E X X\j(x)
Let U be the map on X
to the minimum element in p. Then, for p E X
the neighborhoodfilter in the topology
filter
U
where
the minimum in p.
is
T
of the completion is the
Condition
(3)
then follows
immediately from the complete regularity of this topology.
r]
f-1(O),
(X,C), we have (f)
zerosets for
x T)
on (X
C]
(X
T ),
Condition
x
.
(x,C), let p
x,
for p
and let o
(4)
X
(x,C)
is nothing but the
is
the collection of
x,C )
be
+
the map p(x)
N be the map O(P)
Then o is the extension of p
T
realcompactness of
be the product of real
(x,C)
(X
so
For
the collection of realvelued continuous maps
In order to prove (4’), let
fir
,
to F(X) which maps J(x) to
lines indexed by
(f(x))fEr3
(fX(P))
x T
Since (X
for
q (x,c)
fr
is
X, C)
realcom-
pact, the map o is a homeomorphism onto o(X x) which is closed in
.
UNIFORMIZABLE CAUCHY SPACES
521
Hence o[X x) D -(-X-). Condition (4’] follows.
Suppose conditions (I),
(2).
(3)
(4)
the Kowalsky completion is a completely
Let
gical space.
are fulfilled. We show that
re&ular
realcompact topolo-
be the map as defined above. Then
.
belongs to
Z.
It is clear that a filter converges to p for the Kowalsky structure
if and only if it is finer than
is open, so is M
M
where
M
is the minimum of p.
If
It follows that the Kowalsky completion is topo-
logical and (3) and (4) clearly imply that it is completely regular
and realcompact.
{X
,C*)
If [4’) is fulfilled instead of (4), then. since
is completely regular, we have that the map o defined above
o(X}.
is a homeomorphlsm onto
It follows from (4’) and from the
P-
continuity of o that o(X
$o 0
is closed and (X
x Cx
is
realcompact.
s.
c__.ucHY_ COy!N..G. _Sy__I_Em_s_ AND,..H__ .fl_T_ R _p.,)
Fo a Cauchy space (X,C), let u(C)
{F x F
F
6
C}.
u(C) clearly is a filte with a symmetric base and whose elements all
contain the
dlaonal. In
this section, we investigate under what con-
ditions u(C) is a uniformity and, if so, whether
of Cauchy filters, i.e. c(u(C))
When
space,
C
C is,its
collection
C.
is the collection of convergent filters of a topological
u(C) is the neiBhborhood filter of the diagonal in X x X.
There exist completely unlformizable topological spaces which are not
normal [11]. The neighborhood filter of the diaonal in such spaces
is not a uniformity. Hence the unlformlzablllty of
C
does not imply
that u(C) is a uniformity.
Conversely for the open ordinal space [0.)
filter of the
(121. Hence
is evident
diaBonal
if u(C)
f.i.,
the neighborhood
is a unique uniformity and it is non complete
is a uniformity
C need not be uniformizable. It
that when u(C) is a uniformity we always have
C C c(u(C)).
THEOREM 5.1. u(C) is a uniformity if and only if it is the weak
522
E. LOWEN-COLEBIYNDERS
uniformity of
r u (x,C).
Proof. The if part is trivial. For the only if part suppose
u[C) is a uniformity. Then the identity from [X,C) to (X,u(C)) is
Cauchy continuous So u(C) is coarser than the weak uniformity of
r U (x
c)
On the other hand
let f be a Cauchy continuous map from
(x,C) to a uniform space (Y,). For W E
such that f(F) x f(F) C W. Then, for V
F E C,
and
U
F
choose F
F x F, we have
FEC
f x f(V) C W. It follows that u(C) makes f uniformly continuous and
so u(C) equals the weak uniformity of F (X,C),
u
From (3.2) and (5,1) we now have the following property,
COROLLARY 5.2. If (X,C) is a Cauchy space and if u(C)
fortuity, then we have c(u(C))
C
If the Kowalsky completion (X
if and only i
,C )
C
is
is a
uniformizeble.
of (X,C] is divisible, which
means that the neighborhood filter V(A x] of the diagonal in X
is a uniformity,
u(C) is the trace of
with c(u(C))
(Xx,C x)
V(A ) on X
and if
unt-
is completely untormtzable,
x
X
then
x X, and so it is a uniformity on X
C. In order to prove the converse of this statement,
we make a preliminary assumption on (X
borhoods of the dlgonal in X
x.,X
,C ),
namely ,that the neigh-
have a close base.
In
13] this
property is called A-normality and the precise position of A-normality with regard to normality conditions end other separation axioms
is settled there
THEOREM 53 If u(C) is a uniformity and C is untformtzable, then
(X
C)
is divisible if and only if
PROOF If
(Xx,C x)
(X
C R)
is A-normal
is divisible then it clearly is A-normal,
prove the converse suppose (X x ,C
is 8-normal,
u(C)
is a uniformity.
and C is untformlzable,
nal of X
M
x M
P
Let V be a closed neighborhood of the diagoFrom theorem 4 3 it follows that or every p E X x we
x X
can choose an
To
Mp
E
Mp
such that,
C V. Then we have
U
pEX
M
(with the notations of 4.3),
P
x M
P
E u(C) and so
UNIFORMIZABLE CAUCHY SPACES
cl
Xx X *
U
M
pEx
P
x M
523
C V. But then V(A X) is the uniformity
P
of the completion of (Xou(C)).
DEFINITION 5.4. A collection of subsets of a Cauchy space (X,C)
__Cauchy cov_ertng syt_em
is a
the collection
filter in
it is a basic Cauchy covering
C there are
C contains a member of
if every filter in
_system
if for every
a finite number of members of the collection
whose union belongs to the filter.
The first notion was introduced in
[8]. Both definitions
in
(5.3)
extend the notions of covering system and basic covering system of
a convergence space introduced in
7]. The condition in the following
theorem extends the characterization by means of open covers given in
14] of
the property that the neighborhood filter of the diagonal of
X x X is a uniformity. The proof is straightforward.
THEOREM 5.5. u(C) is a uniformity if and only if, for every
Cauchy covering system
ring system
y belong to
B
{Bj
U
Bj BK,
A
j E
{A t
t E I}, there exists a Cauchy cove-
J} such that. if
Bj
N
k
#
and if x and
there is an i E I such that x and y both belong
to A I
In order to prove an extension of the theorem that every paracompact space is completely unlformizable by the neighborhood filter
of the diagonal, we need some preliminary definitions.
DEFINITION 5.6. A Cauchy covering system of a Cauchy space is
Cauchyftnite if every Cauchy filter contains a set intersecting only
finitely many members of the system.
DEFINITION 5.7. A Cauchy space [X,C] is [basic) Ceuchy_pa.recompact
if for every Cauchy covering system, there is a (basic)
Cauchy cove-
ring Cauchy finite refinement.
THEOREM 5.8. If (x,C) is a (basic) Cauchy paracompact space which
is
round and has a minimum Cauchy filter in each equivalence class,
then u(C) is a uniformity and c(u(C))
C.
54
E.
PROOF, Let U E u(C).
an A E
such that A x
LOWEN-COLEBUNDERS
Fo every
E U. Let A be
^
constructed in this way. Let
m, nlma! element
the following way.
A E A
,
,
(
C
x to F(X) which maps
ls round, we have
rX()
We make a neu Cauchy cover A’ in
Cauchy filter
such that A’
covsnE
<
and for
A. Ths Cauchy
ne
auehy
covetn
,
enement
and we let
we hom A B E A such that B
A B x A B U (XIB] x (X]B]. Then V B belongs to u(C). Let
f V n let
be a minimal Caohy flte. There exists an
B
Fo B E
V
,
For every minimal
(bae) Cauhy
sy=tee, h=
VB
in
we choose A’ E
the collection of sets A
be the map on X
p to its minimum element (p). Since
for every
we choose
minimal Cauchy filter
M E
<X AB
Intersecting only fn!tely sony members of
h{VB
M x M C
0 E
,
But thn V(x) C
VB(X
have
B h M- }. !t follows that V E u(C}. For
a B E
E C an so there exists
x E X, thm flltec
B. We
C
AB
end so V(x)
B
containing
x.
x V(x) C U. Slnce V ls symme-
trlo we have V,V C U,
.
It remains tO be lhown that c(u{C)} c C. Let F E c(u(C)). Suppose
F
M E
.
foc every minimal Cauchy fllter
M
F,
and let
A
in
C. For
every
choose
be the Cauchy covering system consist!rig of
Sets M onstructed In this way. In the flrst part of the proof.
it is shown that a V E u(C) exists much that
(V(X))xE x
refines
There I= an
Choose F 0 E F such that F 0 x F 0 C V end choose x 0 E
M O. A contradiction
and F 0
and an M 0 ( n 0 (h A wlth V(x O) C
FO
MO
ollows,
THEOREM
5.g.
A Cauohy space (x.C) has minimum elements in each
equlvalenoe class, is round, and Cauchy paraoompact if and only if
the Kowalsky completion ls a paracompact topological space.
PROOF. From [4.2) and (5.8). it follows that when C has minima.
is round, end Cauhy paracompact,
be the map as in
(4.3)
p
then {0
X
is a topological space.
0 C X,
open} is a
Let p
base for the
UNIFORMIZABLE CAUCHY SPACES
open sets of X
open.
Then
m.
{0 i
Let
{Oi
i E
I}
j E
{Oj
J}
be
an open cover
is a Cauchy cover of X.
a Cauchy finite refinement,
Then
I}
i
525
o
X
wlth
Ol
C X
E
J}
be
J
{gj
Let
This collection can be chosen open
is an open
(4.3).
locally finite refinement of the given
collection.
Suppose X
minima (4,3).
open, then {A
.
m is a paracompact topological space. Then (X,C) has
If
p
A
cover of X which may be chosen to be
is a Cauchy
A E A} is an open cover of X
finite open refinement.
Then the collection
Let B be a locmlly
{JI(B)
B E B} is a
Cauchy cover of X which is a Cauchy finite refinement of the given
collection,
Since
is paracompact and Hausdorff,
X
9]
It follows from a corollary in
and
ZM
satisfies
for the map
introduced in
7. if and only if
REMARK
5,iO.
Z where T.
has an open base, we have
the filter generated by
is
Therefore every
M E }
{7.M
M E F for some F E p}.
{p E X
On the other hand since
X
regular.
that the Kowalsky completion
coincides with the completion constructed there,
minimal Cauchy filter
it is
rim
M E M} and M l
{M
.
MX
MH
where
MEX(p)}
{p E X
(5.B), Finally it is easily checked that
It follows that X is
If u(C) is a uniformity and c(u(C))
round,
C, then
clearly C has minima in lts equivalence classes and C is round, But
C
need not be Cauchy paracompact.
Corson
15]
This follows from a result of
that a completely uniformlzable topological
which the netghborhbodftlteb Of the diagonal
space for
ts a uniformity
need
not be paracompact.
In the final section we treat the special case of a totally
bounded Cauchy Space,
THEOREM 5.11. A totally bounded Cauchy space is basic Cauchy
paracompact,
PROOF. Let A be a Cauchy cover in a totally bounded Cauchy space
526
E. LOWEN-COLEBUNDERS
(X,C). For every
that there exist A
8
An }
{A
there is an A E A N U. It follows
n
U A
X. Let
such that
i
ultrafilter on X,
A n in
A
I=I
then this is a basic covering Cauchy finite refine-
ment.
THEOREM 5.12. For a totally bounded Cauchy space the following
properties ere equivalent.
(I)
C
(2) X x
is uniformizable
is a compact Hausdorff topological space
(3]
C
[4)
u(C) is a uniformity and c[u(C))
[5)
There is exactly one uniformity with C as its collection of
is round and has a minimum in each equivalence class
C
Cauchy filters
PROOF. (1)
(5.11)
{4)
"
"
(2)
(5)
1]
From (4)
(2)
"
(3)
it follows
and so it has a unique uniformity
(3]
(5.9)
(5)
that X
=
(I)
"
(4)
(5.8) and
is compact Hausdorff
trivial.
ACKNOWLEDGEMENT. I would like to thank Professor D.C. Kent for the
many fruitful suggestions on the subject.
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