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1970
Completeness and related topics in a quasi-uniform
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John Warnock Carlson
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COMPLETENESS AND RELATED TOPICS
IN A QUASI-UNIFORM SPACE
by
JOHN WARNOCK CARLSON, 1940 -
A DISSERTATION
Presented to the Faculty of the Graduate School of the
UNIVERSITY OF MISSOURI - ROLLA
In Partial Fulfillment of the Requirements for t h e Degree
DOCTOR OF PHILOSOPHY
in
MATHEMATICS
1970
ii
ABSTRACT
Completions and a strong completion of a quasiuniform space are constructed and examined.
It is shown
that the trivial completion of a To space is T 0
•
Ex-
amples are given to show that a T 1 space need not have a
T 1 strong completion and a T 2 space need not have a T 2
completion.
The nontrivial completion constructed is
shown to be T 1 if the space is T 1
structure is the Pervin structure.
and the quasi-uniform
It is shown that a
space can be uniformizable and admit a strongly complete
quasi-uniform structure and not admit a complete uniform
structure.
Several counter-examples are provided concerning
properties which hold in a uniform space but do not hold
in a quasi-uniform space.
It is shown that if each mem-
ber of a quasi-uniform structure is a neighborhood of the
diagonal then the topology is uniformizable.
iii
ACKNOWLEDGEMENTS
The author wishes to express his deep appreciation
to Dr. Troy L. Hicks for his inspiration and guidance
during these years of graduate study.
The value of his
many penetrating suggestions and consistent patience in
the preparation of this thesis can not easily be measured.
The author also wishes to express his appreciation
to his wife and family for their understanding and
encouragement.
iv
TABLE OF CONTENTS
Page
ABSTRACT
•••••••••••
o
•
•
•
•
ll
o •••
•••••••••••••••••••
o
•
•
lll
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
•o••·····
3
COMPLETIONS OF A QUASI-UNIFORM SPACE . . . . . . . . . . . . . .
5
A.
Trivial Completion and Some Examples..........
5
B.
Constructions of a Completion.................
9
c.
A Completion for a Pre-compact Structure
and Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . .
16
D.
Cauchy F i l t e r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
E.
A Cour..'cer-example. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
F.
General Properties for a Completion . . . . . . . . . . .
25
SOME THEOREMS AND EXAMPLES REGARDING
QUASI- UN IF 0 RM SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
A.
Neighborhoods of 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
B.
U/\U- 1 . • • • • . • • . • • • • • . . . . . • . . . . • . • . . . . • . . • • • • .
34
c.
Saturated Quasi-uniform Spaces . . . . . . . . . . . . . . . .
38
D.
0-Complete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
E.
q-Ti Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . .
45
F.
A Counter-example in (F,W) . . . . . . . . . . . . . . . . . . . .
51
SUMMARY, CONCLUSIONS AND FURTHER PROBLEMS . . . . . . . . .
53
BIBLIOGRAPHY . . . . . . . . . . . . . ·. · . · · · · .. · · ·. · · · · · · · · · · ·
55
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
o
•
o
•
o
•
o
•
o
ACKNOWLEDGEMENT ..
I.
II.
III.
IV.
V.
o
•
o
•
o
•
o
•
•
•
•
•
•
•
•
•
•
•
•
REVIEW OF THE LITERATURE . . . . . . . . . . . . . . . . .
•
•
•
•
•
l
I.
Let (X,U)
INTRODUCTION
be a quasi-uniform space.
The primary
problem in Chapter III is to construct a completion
for
(X,U).
It is shown that it is impossible in general
to construct a completion which preserves the Hausdorff
separation property.
A trivial completion for any
(X,U)
is given which preserves the T 0 separation property, but
the trivial completion is not T 1 •
Several nontrivial
completions are given for special classes of quasi-uniform
spaces.
One of these constructions preserves the T 1
separation property when U is the Pervin quasi-uniform
structure.
A parallel discussion of strong completions
is also considered, and an example is given to show that
not every T 1 space need have a T 1
strong completion.
A
sufficient condition for the concepts of complete and
strongly complete to coincide is given.
An example of a
uniformizable space which does not admit a complete uniform structure is shown to admit a strongly complete quasiuniform structure.
In Chapter IV we consider some well-known properties
of uniform space which fail to carry over for quasiuniform spaces.
It is of interest to have conditions
on U that will guarantee that U is compatible with a uniform structure.
It is shown that if each U
borhood of the diagonal in X
with a uniform structure.
~
x
E
U is a neigh-
X then U is compatible
necessary and sufficient con-
2
dition for U
sented.
~
_l
U
to be a quasi-uniform structure is pre-
A class of separation properties which is depend-
ent on the particular quasi-uniform structure under consideration is defined and some of their properties are
studied.
Finally, an example is given to show that a sequence
of quasi-uniformly continuous,
(continuous)
functions
may converge quasi-uniformly to a function g and g not
be quasi-uniformly continuous,
(continuous).
The topological terminology found in this thesis is
consistent with the definitions found in Gaal [11].
The
definitions of concepts related to quasi-uniform spaces
can be found in Murdeshwar and Naimpally [16], with the
following exception.
exists z
EX
such that
By U o V we mean {
(x,z)
E
U and
(x,y)
(z,y)
is the definition of U o V found in Gaal
E
V}.
[11].
there
This
3
II. REVIEW OF THE LITERATURE
In 1955, Krishnan [14] essentially showed that every
topological space admitted a compatible quasi-uniform
structure.
In 1960, Csaszar [3]
showed this more explicit-
ly and in 1962, Pervin [19] gave a useful and simple construction of a compatible quasi-uniform structure for a
given topology.
Fletcher [7] gave a construction for a
family of compatible quasi-uniform structures.
In 1960, Csaszar [3] extended the notions of a Cauchy
filter and completeness to a quasi-uniform space.
[13]
Isbell
noted that the convergent filters were not necessarily
Cauchy, so Sieber and Pervin [20]
in 1965 proposed the def-
inition of Cauchy filter which is now in use.
They defined
a space to be complete if every Cauchy filter converged.
We will call a space strongly complete if it has this property.
In this paper, they showed that every ultrafilter
is Cauchy in a pre-compact space.
They obtained the
following generalization of the Niemytzki-Tychonoff
Theorem [18].
A topological space is compact if and only
if it is strongly complete with respect to every compatible quasi-uniform structure.
In 1966, Murdeshwar and Naimpally [16] continued
to use the definition of Cauchy filter proposed by Sieber
and Pervin [20] but defined a quasi-uniform space to be
complete if every Cauchy filter had a nonvoid adherence.
4
By making use of the fact that every ultrafilter in a
pre-compact space is Cauchy, the corresponding generalization of the Niemytzki-Tychonoff Theorem carried over
for this new definition of completeness.
Sieber and Pervin [20] proposed the question, does
every quasi-uniform space have a completion?
[22]
Stoltenberg
in 1967 showed that every quasi-uniform space had a
strong completion.
However his construction left open the
question of whether every Hausdorff quasi-uniform space
had a Hausdorff completion.
The techniques used by Liu
[15]
motivated the completions developed in this thesis.
In 1965, Naimpally [17]
showed that if (X,U)
and
(Y,V)
were quasi-uniform spaces and Y is T 3 and V the Pervin
structure then U and C are closed in (F,W), where F
= YX
and W is the quasi-uniform structure of quasi-uniform convergence and u
tinuous
(C)
is the set of all quasi-uniformly con-
(continuous) mappings from
(X,U)
to (Y,V).
5
III.
A.
COMPLETIONS OF A QUASI-UNIFORM SPACE
TRIVIAL COMPLETION AND SOME EXAMPLES
DEFINITION 1.
Let X be a nonempty set.
A quasi-uni-
form structure for X is a family U of subsets of X
x
x
such that:
(x, x)
{
( 1)
/:,
( 2)
if u
( 3)
if U,
( 4)
for each u
v
=
0
u
E
v
:
X
E
and uc
u
E
u for each u
X} C
v, then v
f
then un
E
u
v
E
E
E
u.
'
u;
u.
'
there exists V
E.:
u
such that
v cu.
DEFINITION 2.
If U is a quasi-uniform structure for
a set X, let tu = { A c X : if a s A there exists U
that U [a]
C
A}.
Then tu
lS
E
U such
the quasi-uniform topology on
X generated by U.
DEFINITION 3.
Let (X,t) be a topological space and
let U be a quasi-uniform structure for X.
patible if t
If
(X,t)
Then U is com-
= tu.
is a topological space and 0 s t, let
S(O) = 0
X
0
u
(X-0)
In [19], Pervin showed that { S(O)
X
X.
: 0
E
t}
lS
for a compatible quasi-uniform structure for X.
refer to this as the Pervin structure.
a subbase
We will
The following def-
inition is due to Sieber-Pervin [20] and is equivalent to
the usual definition of a Cauchy filter in a uniform space.
DEFINITION 4.
Let (X,U) be a quasi-uniform space
6
and F a filter on X.
F is U-Cauchy if for every u
there exists x
such that U[x]
=
x(U)
DEFINITION 5.
E
E
U
F.
A quasi-uniform space
(X,U)
is
strongly complete if every U-cauchy filter converges.
(X,U)
is said to be complete if every U-Cauchy filter has
an adherent point.
DEFINITION 6.
Let (X,U) and (Y,V) be quasi-uniform
spaces and f a mapping from X toY.
The mapping f is said
to be quasi-uniformly continuous if for each V
exists U
E
E
U such that (a,b)
E
E
V there
U implies that (f(a) ,f(b))
V.
DEFINITION 7.
(Y,V)
Two quasi-uniform spaces (X,U) and
are said to be quasi-uniformly isomorphic relative
to U and V if there exists a one-to-one mapping f of X
onto Y such that f and £DEFINITION 8.
(X,U)
1
are quasi-uniformly continuous.
A completion of a quasi-uniform space
is a complete quasi-uniform space (Y,V) such that
X is quasi-uniformly isomorphic (relative to U and V)
to a dense subset of Y.
In [22], Stoltenberg proved that every quasi-uniform
space has a strong completion.
The proof is long and in-
volved and it is not clear if any separation properties
the space may possess carry over to the completion.
The
following construction shows that every quasi-uniform
space has a rather simple completion.
7
Let (X,U) be a quasi-uniform space
CONSTRUCTION 1.
s
and put X* = X U { S} where
=
S(U)
Then B = { S(U)
U U
: U
t:
structure U* for X*.
Us U, and S(U) [x]
=
F on X* converges to
Also,
u
= { U*
n
X
l
X
:
{
~
X.
(S,x)
X
1 1
'-J
U,
let
X*}.
Note that S(U) [13] =X* for each
U[x]
s.
if x
t:
X.
Clearly, every filter
Hence (X*,U*) is strongly complete.
X : U*
(X,U)
-+
is a completion of
and t u. * = t u.
E
t:
U} forms a base for a quasi-uniform
U*} and
E
(X*,U*)
is a quasi-uniform isomorphism.
(X*,U*)
For U
Since X is dense in X*,
(X,U)
In fact, X* is compact
{X* } .
Fletcher [7], working independently, also discovered
the above construction.
If X is T 0
,
then X* is T 0 •
X* is
never T 1 since the only open set containing S is X*.
Thus
the following question naturally arises.
quasi-uniform space have a T 2 (T 1 ) completion or strong completion?
EXAMPLE 1.
We give an example of a Hausdorff quasi-
uniform space that does not have a Hausdorff completion
and consequently does not have a Hausdorff strong completion.
Let X denote the real numbers and A = { 1/n : n
1,2, ••• }.
=
Set
t = { E-B : B c A and E is open in the usual topology}.
(X,t)
is Hausdorff but not T 3
•
Let U denote the Pervin
quasi-uniform structure generated by t.
Suppose (X*,U*)
8
is a completion for
n x
and U = { U*
(X,U).
X : U*
x
We may assume that X = X*
U*}.
E
Let F be the filter on X
generated by { A,{1/2,1/3, ... }, {1/3,1/4, ... }, ... }.
Then
there exists an ultrafilter M
Since
X such that M ~ F.
~n
'V
U is pre-compact, M is a Cauchy filter.
trafilter on X* generated by
M.
U
E
E
U so that there exists x
Clearly U* [x]
"'M.
E
Hence
u*
If
Let M be the ul-
U*,
E
u*
n x
X such that u [x]
M is
Cauchy.
E
x =
x
/d.
Since (X*,U*)
lS
'V
complete, there exists x* s X* such that M converges to x*.
Thus every open neighborhood of x* must meet every set of
the form { 1/n, 1/ (n+1), ... },
(n
vve show that
N).
E
x* can not be separated by disjoint open sets.
u = (X-A
Then there exists U*
n
U*[o]
that
s
>
0
o
0 1 and X*
E
u (A
02
•
Let
X) .
X
n
U* such that u = U*
E
X
X
X.
Now
o ~ x*.
Therefore,
A= [0.
E
X-A)
X
and
o
n
Then 0 1
such that ((-E,E)- A) C
0
X
t
E
~ow
()X.
1
so there exists
there exists
positive integers N and k such that
1/N
Since 1/k
~
02
E
and 1/k
£
n
X
(1/k- 8,1/k + 8)
[ (- E
Therefore,
I
E)
-
t
A] ()
X
n
1 /N ' 1I (N+ 1 ) ( . . . 1 .
there exists 8
-A C 0 2
01 n 02
EXAJ'vlPLE 2 .
E
n
02
E
n
[ ( l /k ·-
~~and
X,
0'
o
>
such that
Clearly
1 /k
(X*,U*)
+ 8)
-
A]
~
0.
is not Hausdorff.
We give an example of a quasi-uniform
structure U for the set N of natural numbers such that:
(1)
tu is the discrete topology,
and
9
(N,U) does not have a T 1 strong completion.
( 2)
u
n
= { (x,y) : x =
y or x
~
n},
B
= {
U
n
n
Let
N}, and let
E
U denote the quasi-uniformity generated by the base B.
Suppose (N*,U*)
is a strong completion
assume that N
N* and
u=
U*
n
N
X
~or
N.
(N,U).
If F
=
We may
F is a
{N},
"v
Cauchy filter on (N,U).
on (N*,U*)
Let F
generated by F.
complete, there exists n*
n*.
Clearly, n*
E
~enote
Since
E
N* - N.
(N*,U*)
N* such that
N
=
N.
to
n
E
F and 0*
N
E
F.
For each n
E
N, we have n
E
0* for every open
set in N* containing n*.
B.
P converges
Then 0*
containing n*.
n
is strongly
Let 0* be any open set in N*
'V
0*
the Cauchy filter
Thus N* is not T 1
Thus
•
CONSTRUCTIONS OF A COMPLETION
LEMMA 1.
Let
(X,U)
be a quasi-uniform space and S
Then a filter F converges to x if and
a subbase for U.
only if for each S
E
S we have S[x]
~
F.
Throughout this section we will let
(X,U) denote a
quasi-uniform space with a base B such that for any V
we have that V o V
= V.
E
B
If U has a subbase with this
property then the base generated by the subbase also has
this property.
It is clear that the Pervin quasi-uniform
structure has such a base as well as the class of quasiuniform structures introduced by Fletcher in [7].
will say that two Cauchy ultrafilters M1 and 1\l
are equivalent provided U[x]
E
M1
2
We
on X
if and only if U[x]
E
M2 •
10
Certainly this is an equivalence relation on the set
of all Cauchy ultrafilters on X.
We will denote the
equivalence class containing ,\! by 1'·.!.
= {
Set !\
M
=
on X} and X*
M is a nonconvergent Cauchy ultrafilter
D(V)
XU!\.
will denote the set of all
mappings 6 from!\ to X such that V[6(M)J
f
Since AI is Cauchy, D (V)
~
M where V
c
for each V c B.
c
F~
For V
B.
B
and 6 c D(V), we set
=
S(V,6)
V U 6 U {
= {
S*
LEMMA 2.
(/d,y)
S(V,6)
,\j
: V
t:
and y
II
t:
B,
c')
c
c V [ 6 ( ,\!) ] } •
D(V)}
forms a
subbase for a quasi-uniform structure U* on X*.
It is clear that S* f
PROOF.
~.
Let S (V, ())
Then S (V, cS) ::;) 6 and we show that S (V, cS)
Suppose
(x*,y*)
If x*
= x
(x,y)
c
X,
c
V,
S(V,6)
c
then y*
(y,z)
c
=
and
(y*,z*)
S (V, cS) C
S(V,6).
c
= z c
c X and z*
y
o
S*.
1:
S (V, 6) •
Case
X.
(a).
Thus
V and since V o V = V, we :1a- 2
(x*,z*)
7
h
(X , Z )
have
z*
=
S (V 1 6 ) .
E
z
c
c V and
z*
s
=
=
(x*,z*)
z
(V, cS)
X,
y
'
c V[6(i\,j)]
s
(V, cS)
PROOF.
M
E
c V [ 6 (,\1) ] ,
consequently
0
(b ) .
(y*,z*)
THEOREM 1.
where
CaSe
and
s
=
X*
\J
(V, 6) •
( y, z)
and
( 6 (i\1), z)
(x*,z*)
If y*
f, •
c
If y*
= "
.L
V.
t:
(:\f,z)
E
) V
S(V,cS).
,lj
we
X, then
( 6 (t.l) , y)
Iience
c V since V
=
E
=
=
V.
Then
Therefore,
c s (V, 6) . I I
(X*,U*)
is complete.
Let F be a Cauchy filter on X*.
is an ultrafilter.
If Jl! converges,
adherent point and U* is complete.
Then F C
F has an
Suppose tl is not
.\l
=
11
We show that X s M.
convergent.
Let S(U,6)
U*.
s
s (U, 6) [x*]
Since M is Cauchy, there exists x* s X* such that
E
M.
"'
= M1
If x*
A,
s
then
A
M1 implies there exists S(V,6)
M does not converge to
s (V, 6) [ M1 ]
such that
~
S*
s
Since M is Cauchy, there
M.
A
exists z* f. M1 such that S (V ,y) [z*]
s
Now if z* s X
,\{.
A
there is nothing to show; so we suppose that z*
A
= M2
s
A.
A
Then M1 f. M2 and we have
A
u
(V[y(M2)]
A
(U[6(Ml)]
{.\!2}) (l
u
{,\{1})
EM.
M since
Thus X s
n
X~ V[y(M2)]
M0 = { M
E
U[6(Ml)]
M : M C X} is an ultrafilter on X.
We show that M0 is U-Cauchy.
with V CU.
Let 6 s D(V).
exists x* s X* such that
have V[x*]
If U s U, there exist V s
Trien S(V,6)
s (V, 6) [x*]
M and thus U[x*]
s
EM.
s
M0
s
s
AI.
If x*
•
U* so there
If x* s X we
_M s
then
A,
A
V[6(;\,{)]
U
{M}
s
and since
,\1
Consequently, U [ o UA) ]
s
M0
s M,
X
we have V[6(M)]
Thus M0 is Cauchy on X.
•
Either M0 is convergent on X or it is not.
converges to x.
V [x]
s
A! 0
•
Let S (V, 6)
Thus S (V, 6) [x]
S*.
s
converges to M0
•
Case (2).
If S(V,6)
0 ]
M0
=
,\! 0
Then M0 s A and we show that M
E
S*.
A
S(V,6) [M
(1)
and we have that M con-
,\-f
s
Case
Then S(V,6) [x]
verges to x which is a contradiction.
is nonconvergent on X.
sM.
= V[o (M 0 )]
U
{M
0
}.
B
12
Now V[o(M 0 )]
s
and consequently S(V,o) [J\,1 0 ]
M0
sM.
Therefore, M converges to M0 and this is a contradiction.//
THEOREH 2.
By theorem l,
PROOF.
Ms
A and U*
S(V ,o )
n
n
(X*,U*) is a completion for
U*.
E
n
n
l
= {M}
l
Since V.[o.(M)]
n
n
l
-:~
v. [o. (M)J
l
U V. [ 6 . ( ,1{) ]
l
n
A
c u*.
for i
=
1
Let
), ••• ,
We have
1, 2 , •.• n .
= 1,2, ... ,n, we have
Now
¢.
l
u* [ MJ
l
M for eacn i
s
l
cv.l , ol. )
s
l
A
S ( V . , o . ) [ M]
is complete.
Now there exists S(V 1 ,o
S* such that
c
(X*,U*)
(X,U).
x =>
n
en s cv. , o . ) ) [ MJ n
A
l
l
n
n
=
S(V. ,o,) [M]
l
n
x
X
l
n
=
Thus
M
E
l
X and X= X*.
l
Let U'
uniform structure of U* on X.
V s B with V CU.
v = s cv, o)
n
x
x
-:1¢.
nv.[o.(M)J
denote the induced quasiIf U s U there exists
Let o s D(V).
x.
Thus v
U'
E
Then S(V,o)
and hence u
s
E
U* and
u· .
Suppose U s U'.
Then there exists U* c U* such that
n
Since U* c U* there exists S (V 1 , o 1 )
u = u*
x
x.
x
S(V ,o ) s S* such that U*
n
n
~
n
n
(n s (v , o ) ) n x
l
n
(1
'l'herefore,
that U
= U' .
mapping i:
l
V
i
S(V. ,o.).
1
x c u* n x
x
Consequently,
1
x
x
u.
l
U and hence U
E
Lf.
Thus we have shown
From this i t follows that the identity
(X,U)
isomorphism.//
C
·~
, ••• ,
n
(X,U* tl X
x
X)
is a quasi-uniform
13
One can let A = { F
filter on (X,U) }.
F is a nonconvergent Cauchy
Then using the same construction we
0
get a strong completion of
0
It is clear that
(X,U).
Denote it by
0
(X,U).
0
(X,U)
is not in general the trivial
strong completion given in construction l.
A space
EXA.l'\iPLE 3 .
not T 1
(X,U) may be T 1 and
Consider the space
•
(X*,U*)
(N,U) described in example 2.
It is clear that U o U = U for each U in the base B.
n
n
n
n
There
Thus we may construct the completion (N*,U*).
M containing the filter base
exists an ultrafilter
Since U is pre-compact, \·Je have
{ {n,n+l, ... } : n s N}.
;ll
E
Since M is nonconvergent,
M is a Cauchy filter.
that
N*
.
Let S(U ,6)
Clearly,
S*.
s
n
6(\l)
>
n.
There-
fore,
s (u ,
n
= u [ c5
c5 ) [ MJ
n
=
Hence
(N*,U*)
1s not T
THEORE1'1 3.
Let
1
N
UA ) J U Ul}
U { ;\1}.
•
(X,t)
be a T
1
topological space and
Then (X*,U*)
U the Pervin quasi-uniform structure.
a T 1 completion for
that
X
E
0, y
have O,W s
S*.
(X,U).
~
If x,y s X, x
PROOF.
t*.
E:
W,
Let
lS
~ W,
X
11A
1
~
y, there exists O,W s t
and Y ~ 0.
Since X s t*, we
1\l 2 be points in/\ and S(U,cS)
Then
{ M1
}
U U [ o (,\1 1 )
such
]
and ,
s
14
Suppose x s X and M s A.
filter.
(X -
X and x
s
1
Now X -
{x} s M since M is an ultra-
Set
u
Let x
M.
~
x, we have {x}
Since M does not converge to
{x})
Define 8 by 8(M) = x
X -
{x}
8
D ( U)
{x})
U
{x}
X.
x
x; if X = {x} we would have X* = X.
~
1
(X
x
for each M s A.
1
Clearly, U[x 1 ]
N for each nonconvergent ultrafilter N.
E
~
E
and S ( U 1 8)
S*.
E
Now s ( u '
~
=
6 ) [ M]
Thus
u (X
01}
(X* ,U*)
is T 1
DEFINITION 9.
-
{
X } )
M.
and X is a neighborhood of x that does not contain
Therefore,
=
./I
(X,U)
1s R 3 if given x s X and u s
there exists a symmetric V
s
U such that
(V
o
G,
V) [x] C U[x].
The above condition was introduced in [16].
It might
be described as a local symmetric triangle inequality.
In [16, p.41]
it was shown that if
1
the Pervin structure is R 3
THEOREM 4.
space.
Then (X*,U*)
f
there exists 0
fl
is regular, then
•
be a T 1 and R 3 quasi-uniform
(X,U)
is a T 1 space.
It suffices to show that for each x s X and
PROOF.
1IA
Let
(X,t)
1 ,
0 7
t:
t* such that x
< 0 1
M s
1
02 ,
~
x ~ 0
7 ,
and M
4
01.
Let x and
AI
be given.
does not converge to x, there exists U
U[x]
[x] C
~
11A.
U such
M
that
There exists a symmetric V s U such that (V o V)
M is Cauchy implies that there exists x 0 s X
U[x].
with V[x 0 ]
E
Since
f~
M.
We show that x ~ V[x 0 ] .
Then (x,x 0 )
E
Suppose x
V and
(x 0 ,a)
E
s V.
15
Hence a c:
U(x]
(V
V) [x] C U[x].
o
But this implies that V[x J C
0
M.
~
which is impossible since U[x]
There exist
A
o
D(V) with o
E
A
~
X
= x0 •
(;\!)
V[x 0 ]
U {Af}
Then
=
A
S (V, o) [,\!]
while M ~ X, an open neighborhood of x.
THEOREi-1 5.
properties of
Thus
Listed here are some easily verified
(X*,U*).
(l)
X is an open dense subset of X*.
(2)
(X,U)
(3)
If
is T 0 if and only if
(X*,U*)
(X*,U*)
is T 0
•
has property P and P is an open
hereditary property, then
(4)
(X*,U*)
(X,U)
has property P.
A is closed in X* and the subspace topology
on A is the discrete topology.
(5)
If
(X*,U*)
then
(X,U)
is pre-compact and Hausdorff,
is completely regular.
0
These properties also hold for
PROOF.
and S(U,o)
By theorem 2, X= X*.
(l) .
c:
U*.
is open in X*.
0
(X,U).
U[x] C X.
Then S(U,o) [x]
( 2) •
Let x
r
X
Hence X
Since T 0 is a hereditary property
the sufficiency is clear.
Suppose
(X,U)
is T 0
,
then since
X is open in X* i t suffices to consider the case where
x*
+
A
,vr
A
E
A,
x*,
~~~
E
X*.
Let S(U,o)
[
U* and we have that
A
M ~ S(U,cS) [ x*] .
Hence X* is To.
from
A
(l) .
S(U,cS)
E
( 4) •
Statement
( 3)
follows
is closed in X* by (l) , and for any
U* we have that {l.f}
=
f\
n
S(U,cS) [:1.1]
for any
16
M E A.
(5).
Therefore the subspace topology on A is discrete.
If
(X*,U*)
is pre-compact and Hausdorff then since
i t is complete i t must be a compact Hausdorff space.
Therefore the subspace
C.
(X,U) must be completely regular.//
A COHPLETION FOR A PRE-COMPACT STRUCTURE AND FURTHER
EXAMPLES
Let
(X,U)
be a pre-compact quasi-uniform space.
~-
= U U
Define X* as before and set S(U)
andy E U[x]
(M,y)
B* = { S(U): U E U} forms a base for a
LEMMA 3.
quasi-uniform structure for X*.
f
B*
Suppose S(U)
U {
M for some x E X}.
E
PROOF.
~
~
and S(V)
c s(u;
~
and
0.
Denote i t by
for each S(U)
belong to 13*.
Then S(U
B*.
s
n
V)
U
n
B
s
and we show that
n
S(U () V) C S(U)
Let
(x,y)
E
n
S(U
If
V).
X
S(V).
E X, then
(x,y)
E
V and
~
thus
(x,y)
which case
E S(U)
(x, y)
such that y E (U
y
E
V[z]
B*.
E
E
n
If x =
S(V).
V) [z]
E M.
S(V).
fore
If x
(x,z)
~
A, then y
s
(x,y)
Thus y E U [ z]
E S(U)
Then there exists V E U such
o
M
S(V)CS(U), let
X then
n
S(V).
th~t
E V o V CUC S(U).
= M then ( x, z)
(y,z)E S(V) C
and
E
We
(y,z)
(y,z) E V and there-
Now if
S(U).
1\l and
V o V CU.
(x,y)E S(V)
(x,yJ E V and
f:
Let S(U)
X=
,q
~
y
ln
S (U) () S (V), or there exists z "' X
Hence
AL
show that S(V)
E
n
If x
f
!\and
= s: and
17
y
E
X,
(y,z)
v [ s]
then there exist s
E
•
Hence
V.
(s,z)
o
X such that y
V
o
V[s]
l
V C U, that is,
,\J
z ,
and
~
U[s]
"M and we have that (x,z) = UJ,z)
Thus U[s]
Therefore S(V)
E
s
,_ S(U).
and we have that B* is a base
S(V) C S(U)
for a quasi-uniform structure on X*.//
'\,
(X*,U)
THEOREM 6.
Suppose
PROOF.
is complete.
(X*,~)
is not complete.
Then there
exists a nonconvergent Cauchy ultrafilter M* on X*.
Using
the same argument as in the proof of theorem l we have that
M C X} is an ultrafilter on X.
is pre-compact i t follows that
Case
M converges to x
(1).
U[x]
E
M*, and U[x]
E
M*.
Thus x
E
~
M*
lim
is Cauchy [ 16, p.
(X,U)
51] .
''v
E
X.
such that S(V) Cu.
exists S(V)
Hence V[x]
1\l
Since
Let U
U.
r
Then there
Since x c lim ,\1, V [x]
S(V) [x]
;\j.
= V[x] implies that
Case
but this is impossible.
'\~
lim
( 2) •
S (V) C
M = ¢.
U.
Now S(V)[,\1]
Hence S (V) [M]
M
E
Then :vl
M
E
E
=
X.
00 U
Let U c U,
then there exists
( U ( V[y]: V[y]
and therefore U [,\!]
lim M which is a contradiction.
r-:
c
,'•{}).
Consequently
,\l.
Thus there are no
''v
that is,
nonconvergent Cauchy ultrafilters on (X*,U);
'\,
(X*,U)
is complete.//
'\,
(X*,U)
THEOREM 7.
PROOF.
is a completion for
(X,U).
'"
By the previous theorem (X*,U)
1s complete.
'0
It is clear that X= X*.
~
I
then there exists v
X
there exists
w
E
We show that U
E
= U .
~ such that u = v
U such that S (H) C
Let u
X
n
X
X
X.
Now
V, hence W C U and we
18
'\,
have that U
X
X
X
X
E
'\,
u.
:s.
Let u
E
ux.
u,
then S(U)
E
u
and u = S(U)
Thus the identity mapping
(X, U)
n
l
a quasi-uniform isomorphism.//
lS
We note that the Pervin quasi-uniform structure
lS
pre-compact.
A proof similar to the proof of theorem 4
shows that if
(X,U)
'\,
THEORE.:'1 8.
is T 1 and R 3
Let (X,U)
then
,
(X*,U)
is T 1
•
be a pre-compact quasi-uniform
space.
'\,
(l)
(X*,U)
is pre-compact if and only if A is
'\,
finite if and only if
(2)
(X*,U)
is compact.
X* is T 3 implies that X* is compact.
'\,
(l)
PROOF.
follows from the definition of U and
the fact that completeness and pre-compactness are equivalent to compactness.
Since every ultrafilter on X
lS
Cauchy, i t must converge to a point in X* and hence
(2)
follows from theorem 4.17 in
EXAMPLE 4.
Let N
[16] .;;
= {1,2,3, ... }
and let U denote the
quasi-uniform structure generated by the base B
n c: N}, where Un = { (x,y)
x = y or y
~
x
~
be an ultrafilter containing {N,{2,3,4, ... },
... }.Then Mo
EA.
homeomorphic to
of
We show that
(N 00 ,t 00 ) ,
A
c=
{,\10}
and
n}.
{ un
Let
,\j 0
{3 , 4 , 5 ' . . . },
(::*,tu*>
lS
the one-point compactification
(N, t) .
Let tv!
be a nonconvergent Cauchy ultrafilter on N.
Note that Un [k]
if k .?. n.
= {k} if k -< n, and Un [k] = {k, k+', ... }
It follows that Un[k]
s
:.1 if and only
19
Therefore, M
~
M0 and A= {M 0 }.
~ Noo be the identity on Nand i(n
are continuous at each n
E
N.
0
= oo
)
Let i
N o'v
and i- 1
i
We show that i
: N*
is continu-
A
ous at M0
Let 0
•
{oo}
0) U
(X -
pact, there exists k such that n
E
Since 0 is com-
E
t
0
implies n
<
Now
k.
{k,k+l
V{Mol-
(u k
1
8 is defined by 6(M)
8 ) [ .~~ 0 ] ) =
{k
1
k+l
1
•
•
•
}
v {
00 }
= k for each ME A.
0
C
•
•
•}
Now i(S
.-1
'0Je show that
•
00
1
l
A
is continuous at oo.
Let S(Uk,6) be given.
Now o(A1 0 )
k,
.:.::.
A
=
so let 0
{1,2, ..• ,8 (M )-l) }.
and S(U 1 ,6) [M 0 ]
{oo} U
0
(X
1
)
= 1, then k = 1
Let
(N- 0).
EXAMPLE
/', U {
0
= N* and there is nothing to show.)
= S(Uk,6)
(N- 0)
(If 8 (,"i
0
Y)
Let I denote the set of integers and Un
5.
:
X
[M 0 ] .
=
Y
1
Y .::. X
?_ n
1
Or
Y _2 X
-n} •
<
Let U de-
note the quasi-uniform structure generated by the base
{ U
n
:
n
=
0,1,2, . . . }.
and U is pre-compact.
Then I has the discrete topology
Let M be an ultrafilter containoo
ing the f i 1 ter base { { 2, 3, 4, ... } , { 3, 4, 5, ... } , ... } and AI -oo
be an ultrafilter containing the filter base {{-2,-3, ... },
{-3,-4, . . . }, . . . }.
Then Moo and 1\l_co are nonconvergent Cauchy
ultrafilters.
A
Hence M "I M
00
-00
Now by considering the various cases, as
in example 3, we have that A= {M
compact.
,M -co }.
Thus
Now we show that I* is Hausdorff.
(I*,U*)
lS
Since I is
discrete and open in I*, i t is clear that distinct points
20
in I are separated by disjoint open sets in I*.
k
I.
E
Choose n
>
k and -m
such that cS
1
k.
<
There exists cS
<~C!)
=
n,
[Moo]
=
=
cS
1
(tC!
-co
Now let
1
E
D(U )
=
-2.
n
)
A
=
m, and cS 2
(M -co )
There exists o
-
m.
Now
{k} (I
S(Un,o
{k} (\
S(Um,o 2 ) [M_oo)
1
)
D(U ) where cS(M
2
E
)
0 and
= 0.
=
A
2 and cS(M
00
Then
A
S(U2,cS) [M]
(l S(U2,cS) [M
1
oo
-oo
({M 0
}
u
{2,3,4, ... })
n
({M -co }
u
{-2,-3,-4, ... }) =f.
Hence I* is Hausdorff.
D.
CAUCHY FILTERS
In a uniform space the adherence of a Cauchy filter
equals its limit.
The following example shows that this
is not necessarily the case in a quasi-uniform space.
EXAMPLE 6.
X
<
Let X
Since W o W
y}.
W we have that
a quasi-uniform structure.
Set F
W[2)
E
=
{X,{2,3,4,5}}.
F.
and~={
{1,2,3,4,5}
{~}
(x,y)
forms a base for
Denote the structure by U.
Then F is a Cauchy filter since
It is clear that lim F
=
{1,2} while adh F
=
X.
Hence we have a cauchy filter whose limit is not equal to
its adherence.
It is natural to wonder if there are quasi-uniform
spaces, that are not uniform spaces, in which the limit
of every Cauchy filter equals its adherence.
The follow-
21
ing theorem shows that such spaces do exist.
THEOREM
9.
Let (X,U) be a R 3 quasi-uniform space.
If F is a Cauchy filter then lim F = adh F.
PROOF.
Let F be a Cauchy filter.
prove that adh F C
Since (X,U)
(V
a
V
o
E
E
C
U [x] .
X such that V[a]
Since
X
E
E
Vis symmetric,
E
V.
F.
E
~ill
We
V.
Let c
adh F and U
(b,a)
E
Therefore c
COROLLARY.
U.
E
U such that
E
V.
E
Thus
Let (X,U)
show that V[a] C U[x].
V[x]
E
V [a] .
E
n
V[a].
Hence
Then (a,c)
(x,b)
E
V,
E
E
F.
Thus x
V.
(b,a)
(v o V o V) [x] C U[x].
V[a] C U[x], and we have U[x]
(X,U)
E
Since F is Cauchy there exists
adh F there exists b
V and (a,b)
(a,c)
Let x
is R 3 there exists a symmetric V
V) [x]
o
lim F.
It suffices to
(x,b)
Since
E
V, and
Hence
lim F.//
E
be a R 3 quasi-uniform space.
is complete if and only if it is strongly complete.
PROOF.
The result is obvious by theorem 9. and
the
definitions.//
In a complete uniform space if lim F
follows that F is Cauchy.
+¢
then it
It is natural then to ask if in
a complete quasi-uniform space we have a filter F such that
+ ¢,
adh F
does it follow that F is Cauchy.
The follow-
ing example shows that such a filter need not be Cauchy.
EXAI-iPLE 7 .
Set Un = {
(x,y)
Let X denote the set of positive integers.
: x
y, or x
l and y
2 andy= 2n + 2k + l, where k = 1,2, ... }.
u3 =
6
U{(l,S),
(1,10), . . . }
u
{(2,9),
2n + 2k, or x
For example,
(2,11), . . . }.
22
Let B
= { un : n = 1,2, ... }.
To show B is a base for a
quasi-uniform structure it suffices to prove that un
cun for each n = 1,2,.0.
U
and
n
x
f
( y, z)
s U .
n
y then either x
2n + 2
Un .
>
2.
Hence z
Let Un be given, and
:£ x = y, then (x,z) = (y,z)
= 1 or x = 2.
Suppose x
and therefore y
= z.
Thus
(x,z)
~
= (x,y)
~
(x,y)
f:
u .
If
n
=
(x,y)
2n + 2 + 1
E
un
= 1, then y
= y and we have that (x,z)
On the other hand, if x = 2 then y
o
Un.
>
~
E
3
Therefore
B generates a quasi-uniform structure which we will denote
by u.
The topology generated by U is compact since given
any open cover there exists 0 1 containing 1 and a
taining 2.
Clearly 0 1 U
0~
con-
0 2 contain all but at most a
finite number of members of X.
Hence
(X,U)
lS
compact and
therefore strongly complete.
The sets of the form { n,n+1, ... } (n s X)
generate
a filter F and 1 s adh F, but F is not Cauchy.
In a uniform space i t is well known that the neighborhood filters are minimal among the Cauchy filters.
This does not hold in general for quasi-uniform spaces
as the following example indicates.
EXAHPLE 8.
Consider the space in example 6.
Let
N(4) denote the neighborhood filter of 4.
Clearly N(4)
is the collection of super sets of {4,5}.
,~J
collection of all super sets of {5}.
and thus N(5)
( 5)
Now N(4)
is the
~
N(S)
is not minimal among the Cauchy filters.
23
E.
A
COUNTER- EXAI:1PLE
The following theorem shows that:
(1)
A uniformizable space can admit a strongly complete quasi-uniform structure and not admit a
complete uniform structure.
(2)
A space can be uniformizable and admit a strongly
complete quasi-uniform structure and not be real
compact.
(3)
A countably compact space may admit a quasiuniform structure that is not pre-compact.
(This can not happen with uniform structures.)
Let W denote the ordinals less than Q, the first
uncountable ordinal.
W.
t will denote the order topology for
It is well known
p. 74]
[12,
that
(vl,t)
is normal,
countably compact, not metrizable, and not real compact.
Dieudonne
[5]
showed that
uniform structure U and
(W,t)
THEOREM 10.
(W,t)
(W,U)
admits a unique compatible
is not complete.
admits a strongly complete quasi-
uniform structure.
Let P denote the Pervin quasi-uniform struc-
PROOF.
ture
U ~
for t.
n
V
Suppose U
CU.
(xI
y)
X
I£ AsS, then A~~.
P}.
then U
= {
Set L
n
E
P and
L c S,
Thus V () L
~-
(U
n
L)
n
(V
y}.
:::.
If U
n
n
=
L)
Let
s
=
L s Sand V
(U
nv) n
L
u
n
n
L s S,
£
L
S.
then there exists V c P such that V o V
S
and
(V () L)
o
(V () L) C U () L.
For
24
suppose
Also
(a,b)
(a,b)
Hence
n
L and
that is,
(b,c)
~
L implies a
c
c;
a~
V
c
b and
(a,c)
v n
c
(b,c)
L.
c
L.
Then
(a, c)
L implies b
c
Therefore
(a,c)
Thus S is a base for a quasi-uniform structure.
the generated quasi-uniform structure by U.
= t u.
that t
Since P
~
Then there exists L
U) [x] C
Now U[x]
t.
0.
L[x]
=
=
[I,x]
respect to t.
n
u
we have that t
[x]
(\
=
U[x]
=
(W, U)
exists w c W* such that w
[ 1 , x]
M.
Case 2.
c
M*.
E
X
(L
E
n
() U) [x]
(L
Hence 0 '- t
Let W* = [ 1
is complete.
Now
Therefore
vv.
( x, Q]
Q
, o~]
Let ,\! * denote the
lim
M*.
Case l.
=
st.
If w c
vv,
Since :d is Cauchy
= [I,x]
is a neighborhood of \2 and
we have that
(x, st]
c
(W,U)
,\!*, but this
is complete.
(W,U)
lS
complete can be
M be a Cauchy ultrafilter on W.
there exists x
and
Since W* is compact there
Suppose w
An alternate proof that
Let
such that
Hence w c W and by case l, we have that
is impossible.
given.
We will show
there exists x c W such that L[x]
since 1'-'l* converges to
w c lim M.
Denote
tu.
generated ultrafilter on W*.
Hence
L.
is also a neighborhood of x with
and let A1 be a Cauchy ultrafilter on
and L c U,
~
tu.
with respect to t.
X
We now show that
lim
u
is a neighborhood of x with respect to
[I,x+I)
is a neighborhood of
E
u
E
c
c.
Thus
L
then w
~
U we have that t
and x c 0.
u.
t.:
E
W such that [l,x]
=
L[x]
c
Since L
M.
t.
U
25
be the trace of M on [l,x], then M1
[l,x].
Since [l,x]
that z slim M1
lS
an ultrafilter on
is compact there exists z
is R
3
Let L
•
n
u
Then there exists x s vJ such that (x,w] C
Let 0 = [ 1 ,x], S = (x,w], and T = (w,rt).
are open.
Then V 1
,
[l ,x]
E
u
such
Hence z slim M.
•
We now show that (X,U)
W.
r
(L n
and w
U)
~
[w].
Then 0, S, and T
Set
V2
v1 = o
x
o u (W-o)
x
w
v2 = s
X
s u
(W-S)
X
w
v3
X
T
u
(W-T)
X
w
T
V 3 belong toP and hence to U.
,
It is easy to
see that
z = v 1 n v2 n v 3
(0
Thus Z s U,
c
(L
z
n U) [w] .
X
0)
(S
is synunetric, and
Therefore
(W,U)
theorem 9 we have that (W,U)
F.
u
S)
X
z
o
u
(T
z = z.
X
T).
Now Z[w]
=
(x,w]
is R 3 and by the corollary to
is strongly complete.//
GENERAL PROPERTIES FOR A COMPLETION
It is apparent that not all of the properties of a
completion for a uniform space carry over for a quasi-uniform space.
In this section we note that irregardless of
the definitions of "Cauchy" filter and "completeness" not
all of the pleasant properties of a "completion" are going
to be preserved in a quasi-uniform space.
We would like a definition of "Cauchy" filter and
26
"completeness" that would satisfy the following conditions.
(a)
The definitions would reduce to the ordinary
definitions on a uniform space.
(b)
There exists at least one completion for every
quasi-uniform space.
(c)
If
(X,U)
is
(d)
If
(X,U)
is compact, then (X,U)
(e)
If f
(Y,V)
~
(X,U)
uous where
then there exists a
T2 ,
(Y,V)
T7
completion.
is complete.
is quasi-uniformly contin-
is complete and T 2
,
then there
exists a quasi-uniformly continuous extension
'\.,
f
:
(X*,U*)
tion of
(f)
If
(g)
is any comple-
(X,U).
(X,U)
where
(Y,V) where (X*,U*)
~
is pre-compact then
(X*,U*)
(X*,U*)
is any completion of
is coP1pact
(X,U).
Every convergent filter is Cauchy.
It is clear that each of these conditions hold 1n a
uniform space.
(c)
THEOREM 11.
spaces
and
(f)
can not both hold
(for all
(X,U)).
PROOF.
Let
(X,t)
be a
T
2
space that is not T 3 and
U the Pervin quasi-uniform structure for
t.
By
(c)
there
exists a T 2 completion (X*,U*)
and by
(f)
(X*,U*)
1s compact.
Hence
THEOREM 12.
space
(X,U)
(d)
and
for
(X,U)
is T 3 which is iP1possible.//
(e)
can not both hold
(for all
(X,U)).
PROOF.
Let X
= ( o, 1l and
U the Pervin quasi-uniform
27
structure associated with the usual topology.
[-1,1]
and let V be the Pervin structure associated with
the usual topology on Y.
(d)
it is complete.
Now f
Let Y =
(y
1
Define f
v)
is
X~
~
2
and compact and by
Y by f(x)
=sin 1/x.
is continuous and since U and V are the Pervin
structures it follows that f is quasi-uniformly continuous.
Now X*= [o,11
is compact with the usual topology.
Let U* be the Pervin structure for X*.
that
(X*,U*)
Now by (e)
rv
sion f
: X*
(d) we have
is complete, hence it is a completion of
(A,U).
there exists a quasi-uniformly continuous exten~
Y, but this implies that f has a continuous
extension to [o,I] which is impossible.
(e)
By
Therefore {d)
and
can not both hold.//
THEOREM 13.
PROOF.
If
(d)
holds then (b)
holds.
The trivial completion is compact and by
(d)
it is complete.//
Considering our usual definition of Cauchy filter we
obtain the following summary.
Complete
(or Strongly Complete)
(a)
holds
(b)
holds
(c)
does not hold
(d)
holds
(e)
does not hold (See theorem 12.)
(f)
does not hold
(g)
holds
28
To see that (f) does not hold consider the space in
example 2.
Since every filter is Cauchy in this space i t
is clear that there are infinitely many nonconvergent
Cauchy filters.
Let A denote this collection.
0
Now the
0
construction of the strong completion (X,U) carries over
0
with A redefined as above.
If
0
(X,U)
is compact then i t
must be pre-compact and hence A would be finite, which i t
is not.
29
IV.
SOME THEOREMS AND EXAMPLES REGARDING
QUASI-UNIFORM SPACES
A.
NEIGHBORHOODS OF
~
One of the points of interest is the following.
Given
a quasi-uniform structure U for a set X, when is U compatible with a uniform structure?
By definition U is compat-
ible with a uniform structure if and only if the topology
generated is uniformizable.
Other sufficient conditions
will be obtained.
DEFINITION l.
A quasi-uniform space
have property P if each U
X
E
(X,U)
is said to
U is a neighborhood of
~
in
X with respect to the product topology.
x
It is well known that a compact uniform space has property P.
The following example shows that this need not be
the case in a quasi-uniform space.
It also demonstrates
that a quasi-uniform structure may be compatible with a uniform structure and not have property P.
EXAMPLE l.
Let X=
[0,2] with the usual topology and
let U denote the Pervin quasi-uniform structure.
(
1
3
2'2 )
E
t
SO
U
a neighborhood of
1
l
( --E
2
, -+E)
2
X
0
X
~,
0 u (X-0)
X
X
E
u.
then there exists
Now 0
Suppose that U is
E
>
o such that
Now p =
Hence U is not a neighborhood of L.
The following example shows that a quasi-uniform
structure can have property P and not be a uniform struc-
30
ture.
EXAMPLE 2.
set Un = { (x,y)
Let N denote the positive integers and
: x
s:
or n
y
x
s:
y} .
forms a quasi-uniform base.
1,2, ••• }
Then { Un : n
=
Let U denote the
quasi-uniform structure generated by this base.
It is clear
that U is not a uniform structure and the topology generated is the discrete topology.
Hence 6 is open in X
x
X, and
therefore each U s U is a neighborhood of 6.
In the above example we note that U is compatible
with a uniform structure.
The following theorem shows that
if U satisfies property P then this is always the case.
Let (X,U) be a quasi-uniform space.
DEFINITION 2.
Then
u- 1 = { u- 1
u
E
is a quasi-uniform structure on
U}
X and i t is called the conjugate quasi-uniform structure of
U.
U V
u- 1
denotes the smallest quasi-uniform structure
-1
which contains both U and U
THEOREM 1.
Let (X,U)
.
be a quasi-uniform space satis-
Then U is compatible with a uniform
fying property P.
structure.
PROOF.
Let t
and s denote the topologies generated
by the quasi-uniform structures U and U V
ly.
It is clear that U
that t
.: .:. s.
u- 1
V
u
uniform base for
u v u- 1 •
n U- 1 SUCh that
n
u- 1
where
I
u
E
(U
n
U-
1
)
It is
<
t.
E
U, form a
Let 0 s s and x s 0.
X
respective-
1s a uniform structure and
Thus it suffices to show that s
clear that sets of the form
exists U
u- 1 ,
[x] C 0.
rrhen there
Since U is
31
a neighborhood of
Q
~
t
such that x
b
with respect to t
E
Q and Q x Q
E
Q C
CU.
x
t, there exists
Hence
Q
X
Q
c u- 1 •
Therefore,
X
=
U[X]
n u- 1 [x]
u- 1 )
(U {')
[x]
Co
Hence 0 s t and thus t = s.//
COROLLARY.
space
~ith
Let
be a complete quasi-uniform
(X,U)
property P.
Then there exists a compatible
complete uniform structure.
In the above proof we showed that U V u- 1
PROOF.
a compatible uniform structure.
u - uv
u- 1 i t follows that
lS
Since U is complete and
u- 1 is co~plete.//
uv
Every closed subspace of a complete quasi-uniforn
A complete subspace of a
space is complete.
Hausdorff
The following theorem gives an
uniform space is closed.
analogous result for a complete Hausdorff quasi-uniform
space that satisfies property P.
'rHEOREM 2.
Let
be an arL,itrary r:ausdorff
(X,U)
If Y C X, and
quasi-uniform space with property P.
(Y,Uy)
1s strongly complete then Y is closed in X.
Let a
PROOF.
Y : U
E
-
l.
Y.
Now U[a]
n
U} forms a filter base on Y.
ter on Y generated by B.
If u
u,
E
Y
f 0,
and B = { U[a]n
Let F denote the fil-
n
then v = u
y
Since u is a neighborhood of A, there exists 0 s t
a
f
0 and 0
x
0 CU.
Now 0
n
y
f
,0 and 0
n
y
X
X
y
E
uy.
such that
0 f1 y c
v.
32
Let b
Y.
0 (\ Y.
E:
Then V [b]
n
~ 0
Y
and hence F is Cauchy on
Since Y is strongly complete there exists c s y such
that c s lim F.
Then c
c
lim F'
Let F' be the filter on X generated by F.
and a s lim F'
have that a= c s Y.
DEFINITION 3.
will say that
Hence Y is closed in X.//
Let (X,U) be a quasi-uniform space.
(X,U)
lection { V[x]
and since X is Hausdorff we
We
has property S if for each x, the col-
: V s U, V is symmetric} forms a fundamental
system of neighborhoods for x with respect to the topology
generated by U.
If U has
The following question arises naturally.
-l
property P, then does U
have property P?
Since this type
of question will be of interest later, we make the followlng definition.
DEFINITION 4.
A property Q will be called a quasi-con-
jugate invariant if a quasi-uniform structure
has ;?roperty
Ll
Q implies that u-l also has property Q.
The space considered in example 2, Chapter III, clearly
has property P since its topology is discrete.
suppose u~ 1
is a neighborhood of
product conjugate topology.
2
c:
with respect to the
exi~ts
0
F
tu-l with
ox o c u- 1 •
Thus taere exists positive integer
-''
which
Hence (2,k) s u r:
such that {2,k,k+l," .. } C 0.
0 and
k
Then there
6
_l
U ~'
Now
;)
ls impossible.
Therefore U-l does not have property P, that
is property P is not a quasi-conjugate invariant.
THEOREH 3.
Let
(X,U)
be a quasi-uniform space satis-
33
Then U- 1 satisfies property P.
fying properties P and S.
Let
PROOF.
u- 1
E
U-1
Since U has property P, we
•
have that for each x c X there exist V(x)
V(x) [x]
V(x) [x]
X
each x c X.
c
u.
Hence V(x) [x]
for each x
is symmetric i t follows that T(x)
U { T (x) [x]
: x s x}
U such that
V(x) [x]
c
u- 1 for
By property S there exists a symmetric T(x)
U such that T(x) [x] C V(x) [x]
Thus
X
s
c u- 1 •
x
x c
T (x) [x]
E
U-1
X}C
E
Since T(x)
X.
for each x
U { V(x) [x]
Hence u- 1 is a neighborhood of
A.
X.
f
x
V(x) [x]
with res-
pect to the product conjugate topology and therefore
u- 1
has property P.//
COROLLARY.
Let (X,U)
be a quasi-uniform
satisfies properties P and S then
u- 1
s~ace.
If l/
is compatible with a
uniform structure.
The result lS an immediate consequence of
PROOF.
theorems l
and 3.//
Theorem 1.47 in [16]
shows that if tu is weaker than
the conjugate topology then the conjugate topology is uniformizable .
.
U
Slnce
=
(U-l)-1
it follows that if tu is
stronger than the conjugate topology, then tu is uniformizable.
THEOREM 4.
Let
(X,U)
be a quasi-uniform space.
1
(U- ) has property S then tu_1
PROOF.
(tu)
lS uniformizable.
Suppose U has property S.
than the conjugate topology,
If U
Then tu is weaker
for let 0 c tu and x c 0.
Since U satisfies property S we have that there exists a
34
symmetric V
U such that x
E
E
V[x]
C 0.
Hence V
E
U
-l
Then by theorem 1.47 in [16] we have that t
V
E
t
E
u_-l
U-1 with
that V
With
X
X
E
U and thus 0
E
quently
Then there exists a symmetric
0.
V[x ] C 0.
E
u_-1
Now suppose that u-l has property s and
is uniformizable.
let 0
and
E
t
u
Since V is symmetric we have
.
~
Therefore tu._ 1
tu. and conse-
we have that tu. is uniformizable.//
If
COROLLARY 1.
(X,t)
is T 3 then it has a uniformiz-
able conjugate topology.
PROOF.
By theorem 3.17 in [16] U, the Pervin quasi-
uniform structure, has property S. tu._ 1 is uniformizable by
theorem 4.//
Let (X,U) be a quasi-uniform space such
COROLLARY 2 •
that U and U
-l
have property S.
Then t
(,(_
t
and t
u- 1
u
is
uniformizable.
PROOF.
since
u- 1
Since U has property S we have tu.
has property S we obtain tu.-1 ~
and thus t,u. = t u
V
t u-l = t u v u_--l •
tu..
~
tu._ 1 and
Hence tu_
=
Therefore tu. is
uniformizable.//
B.
-1
U 1\ U
DEFINITION 5.
on X.
Then U
~
THEOREM 5.
on X.
V =
Let U and V be quasi-uniform structures
{
U
Let U and V be quasi-uniform structures
If U A V is a quasi-uniform structure on X then
U A V is the greatest lower bound of the quasi-uniform
35
structures U and V.
It is clear that U A V
PROOF.
U and U A V
~
V.
~
Now
suppose that S is a quasi-uniform structure such that S
and S :;: V.
It
s
If
S, then s
E
ul
well known that if
lS
ul
tures for X then
ture.
u2
"
u n v.
~
Hence S
~
U
U A V.//
and U;o are quasi-uniform struc-
need not be a quasi-uniform struc-
The following example shows that even
u
;\
u-J need
not be a quasi-uniform structure.
Let X denote the natural numbers and set
EXAMPLE 3.
Un = {
(x,y)
: x = y or x =
1
andy~
n}.
Let U be the
quasi-uniform structure generated by { U
: n =
n
J ,), ... }.
Now
U = {
(X
1
y)
:
=
or y
E
E
E
Now
(t,J)
E
U A U
= l and
c
4 }
-l
c V
such that
EX
such that
u-l
there exists m
and
U which is inpossible.
j_'
U 1\ U-
(J,k)
(l,t+J)
t:
_:::.
1
11
•
V
c V
r;
V C
X such that
F
. .:.
14
anr-t therefore
V
Hence U A U-
<:;'
,_,lnce
U.
for each k
Then t
Lett= max{m,n}.
V for each k .: :. m.
v cu.
E
V
U there exists n
Similarly, since V
X
and x .: :.
Suppose there exists
V
= y Or
X
~
n.
(k,l)
since
(t,t+!)
does not form a
quasi-uniform structure.
If
LEMMA l.
U ~ u-
1
u A u-l
is a quasi-uniform structure then
is a uniform structure.
Then 0
PROOF.
u- l .
Also, U
E
U
-l
-l
and thus U
E
[;
F
.
u
and hence
Therefore
u- 1
E
36
DEFINITION 6.
A quasi-uniform space
have property
* if for each symmetric U
symmetric V
U such that V o V CU.
E
(X,U)
is said to
U there exists a
E
It is apparent that each uniform structure possesses
*
property
Not every quasi-uniform structure has property
* as the space in example
1 demonstrates.
It is clear that
* is a quasi-conjugate invariant property.
property
an easy exercise to show that property
It is
* does not imply and
lS not implied by any of the usual separation properties.
However, as the next theorem shows it does characterize
u- 1
those structures U for which U A
is a quasi-uniform
structure.
Let (X,U)
THEOREM 6.
U A
u- 1
be a quasi-uniform structure.
is a quasi-uniform structure if and only if
(X,U)
satisfies property *·
Suppose (X,U)
PROOF.
u
"
u-1
$5 and if u
=t=
and v::> u implies v
Now U, v
u
(\
v
u
~
u- 1
E
u 1\ u
E
E
E
u ;\ u-1
u and v
u
/\
E
U and T is symmetric.
a symmetric v
ric, We haVe V
u- 1 •
E
impiies u nv
Let u
E
E
/\ u-1.
u
and u
E
u- 1
u-1
Then
f_:
u 1\ u-1
u- 1 •
u 1\
n v E u- 1 •
Hence v
E
E
u'
and T
'l'hus
=
By hypothesis there exists
Since V is symmet-
U such that V o V CT.
E
Now u
then u ::> [',.
u
u-1 •
Clearly
satisfies property *.
and thUS V
E
LJ !\
LJ-
1
•
Hence U !\ u-l
lS a quasi-uniform structure.
Now suppose that U
Let U
E
A u- 1 is a quasi-uniform structure.
U and U be symmetric.
Then u
E
u A u- 1
and there
37
u- 1
exists V s U A
v
s
U A
such that V o V Cu.
u- 1 •
Hence V- 1
u- 1 •
Let S
s
and more over
(X,U)
Hence
v
n
v- 1
v- 1 •
s =
o
(V
u- 1 , an ct th ere f ore v -
s
1
s
Then S s U and S is symmetric
n v- 1 )
(v n v- 1 ) c
0
v
v c u.
o
A U2 is a quasi-uniform structure, and
denote the topology i t generates by t.
1
U and
s
Let U1 and U 2 be quasi-uniform structures
Suppose U1
where t
v
satisfies property *.//
THEOREM 7.
on X.
u,
cG
Now
and t
Then t
= t
1
t
1\
2
,
are the topologies generated by U1 and U 2
2
respectively.
Then there exists U s U 1
PROOF.
and V s U 2 such that x s U[x] C 0 and x s V[x] C 0.
Therefore 0 s t.
U U V s U 1 1\ U 2 and x s
(U U V) [x] C 0.
Now suppose x s 0
Then there exists U s U 1
that x s U[x] C
0
E
t 1
and 0
E
t.
E
Since u
0.
t
E
and hence 0
2
u1 and u
E
t
1
A t
Then
E
2 •
~
U2
such
U2, we have that
Therefore, t
=
L2t G denote the family of all quasi-uni-
COROLLARY.
form structures which generate the topology t on the set X.
Then if U1 and U2 are in G and U1 A U2 is a quasi-uniform
structure,
then u1
Let
THEOREM 8.
fying property
( i)
A
u2
E
(X,U)
G.
be a quasi-uniform space satis-
*
If U A U- 1 generates t
u.
then t
u.- 1
is uniformiz-
able.
(ii)
If U A U- 1 generates tu._ 1 then tu. is uniforrniz-
38
able.
PROOF.
( i)
~
By hypothesis tu
tu_ 1 and from theorem
1.47 in [16] we have that tu-1 is uniformizable.
hypothesis tu-1
~
( ii) .
By
tu and by the remark following theorem 3
we have that tu is uniformizable.ll
THEOREM 9.
Let
(X,U)
be a quasi-uniform structure.
U V u-l generates t
if and only if there exists a compatu
ible uniformity stronger than U.
PROOF.
The necessity is obvious.
a compatible uniform structure
E
u-l
then u
I
E
u
and thus u-1
Now let t denote the
Suppose there exists
v
such that
E
v.
u v . Let u -l
u -< u v u-1 ~ v.
u v u-1 . Then
~
Hence
generated by
topolog~
Hence U V u- 1 generate
Let
THEOREJVI 10.
fying property *·
(X,U)
be a
quasi-uniform space satis-
Then there exists a weaker compatible un-
iform structure if and only if U A u-l generates tu.
patible uniform structure such that V
V and hence V
have that V-]
E
V
E
u-l.
A
u-~.
U and V
Suppose V is a com-
The sufficiency is clear.
PROOF.
C
Thus
v
c
U
E
~
If V
U.
U and V_l
E
U.
E
V, we
That is,
A u- 1 and we have that
Denote the topo 1 ogy genera t e d b y U
1\
V ~ U
U-l by t.
since V ~ U A u- 1 ~ U and the hypothesis
u
Thus L1 1\ U- 1 generates tu. II
that V generates t .
u
Then t
C.
u
~ t
~ t
SATURATED QUASI-UNIFORM SPACES
39
DEFINITION 7.
A topological space
(X,t)
is called sat-
urated if for each x s X there exists a minimal open set
containing x.
That is, for each x s X there exists an open
set Ox containing x such that if 0 s t
and x s 0, then Ox C
0.
It is clear that a topological space
(X,t)
is saturated
if and only if every intersection of open sets is open.
DEFINITION 8.
A quasi-uniform space
u : u
n{
quasi-saturated if
E
U}
is called
u.
E
A quasi-uniform space
LEivWlA 2.
(X,U)
(X,U)
is quasi-satu-
rated if and only if there exists a unique base
for U con-
sisting of a single set.
PROOF.
E
U} •
The sufficiency is clear.
By hypoth2sis V s U.
show that V o V C
v.
= {V}
.
n{ U
:
U
It suffices to
Since V s U there exists U s U such
But V C U and thus V o V c U o U C V.
that U o U C. V .
Hence B is a base for U.
LEMMA 3.
Set B
Let V =
If
(X,U)
The uniqueness is obvious.//
is quasi-saturated then tu is a
saturated topology.
PROOF.
lemma 2·
W[x]
Let
=
[W} be the base for
Let x s 0, where 0 s t
:.L
.
U developed in
Then x s W[x] C 0, and
is the rninimal open set containing the point x.//
THEOREM 11.
lS
B
If
(X,U)
is quasi-saturated then
(X,U)
strongly complete.
PROOF.
Let F be a Cauchy filter.
base for U consisting of a single set.
Let B
=
{W} be the
Then, since F is
40
Cauchy, there exists x
each U
x
E
~
U we have U[x]
E
X such that W[x]
E
W[x]
1
E
F.
Hence for
and therefore U[x]
F and
E
lim F.//
THEOREM 12.
space.
Let
(XIt)
be a saturated topological
Then there exists a strongly complete compatible
quasi-saturated quasi-uniform structure.
PROOF.
x.
Set W
Let Ox denote the minimal open set containing
= { (x y) : y
Ox 1 x
E
1
X} .
E
Then
{\'J}
forms a
base for a quasi-uniform structure which we will denote by
u.
w.
It suffices to show that W o W c
(b 1 c)
Then b
W.
E
Oa and c
E
E
Ob.
Let (a 1 b)
Since Ob is the mini-
mal open set containing b we have that c
(a 1 c)
If 0
VV.
E
Thus t
W [x] •
E
t
and X
-< t u .
~
C 0 and hence tu
If 0
E
E
0 then X
tu_ and X
E
E
E
ObC Oa·
Hence
=
OX C 01 but OX
0 then X
By theorem 11 (X 1 U)
t.
W and
E
E
= W[x]
OX
is strongly com-
plete . / /
Fletcher has shown in [7] that if a topological space
has property
Q
then i t admits a compatible strongly complete
Fletcher's theorem is a clear gen-
quasi-uniform structure.
eralization of theorem 12.
The following example shows that metacompact and paracompact are not necessary conditions for a topological space
to admit a strongly complete structure.
Let N denote the natural numbers and t
EXAMPLE 4.
{N
1
~
1 {
1}
1 {
l
ical space.
17} 1 {
1
121
Set Oi
3}
1
=
{1
•
•
•} •
1 2
(N 1 t)
1 ••• 1
i}
1
=
is a saturated topologthen { Oi : i =
1
1
2
1 •
•• }
41
is an open cover for
(N,t).
refinement of { Oi:
i = 1,2 ••• }.
Let { Qa
: a s
Q}
be an open
Suppose that
is con-
1
tained in only a finite number of the Oa, say Qa , ... , Qa .
+N and
Clearly each Q,~
"'
mum element, say r.
n
1
n
further the set U Qa. has a maxi-
Now r
1
l
is contained in some QB
+ Qa·
l
fori=
1, •••
,n.
o6
Then {1,2, ••• ,r} C Q , that is 1 s
6
which is a contradiction.
Hence
(~,t)
is a saturated
space that is not metacompact.
THEOREM 13.
Quasi-saturated is a quasi-conjugate in-
variant.
PROOF.
Suppose
(X,U) is a quasi-saturated quasi-uni-
Then there exists W s U such that if U s
form structure.
It is clear that {W- 1 }
U then W CU.
and hence by lemma 2,
Let
THEOREH 14.
u- 1 is quasi-saturated. I I
(X,U) be a quasi-saturated quasi-
uniform space with a base {W}.
open set in tu containing x.
Let Ox denote the minimal
Then W is a neighborhood of
that is, U has property P if and only if W = U { Ox
/',.;
PROOF.
The sufficiency is evident.
borhood of /',. then
then b
X
u- 1
forms a base for
E
E
W (a] C
~-J
0a •
::::> U { Ox
Hence
x
Ox : x
(a,b)
E
Oa
x
If W is a neighIf
(a,b)
s
X}.
X
oac U{ Ox
X
E
It¥,
Ox:
X}.//
THEOREM 15.
structures on X.
PROOF.
Let U and V be compatible quasi-uniform
If V is quasi-saturated then U
Let U s U and {V} the base for V.
~
If
V.
(x,y)
s
42
V, then y
E
V[x] C U[x].
This follows since V[x]
smallest open set containing x.
have V
C
U and thus U
COROLLARY.
If
E
Hence
(x,y)
E
is the
U and we
V.//
(X,t)
is saturated then there exists
one and only one compatible quasi-saturated quasi-uniform
structure.
PROOF.
The result is an immediate consequence of
theorems ll and 15.//
Fletcher [6] working independently has obtained slmllar results for finite topological spaces.
D.
0-COMPLETE
DEFINITION 9.
F will be called an open filter
A filter
if it has a base consisting of open sets.
DEFINITION 10.
Let (X,U) be a quasi-uniform space.
(X,U) will be called o-complete if every open Cauchy filter
(X,U) will be called strongly
has a nonempty adherence.
o-complete if every open Cauchy filter has a nonempty limit.
DEFINITION 11.
A topological space
(X,t)
is called
generalized absolutely closed if every open cover { 0
has a finite subcollection Oa , ... ,0
l
DEFINITION 12.
(X,U)
each u
A filter
an
a
}
such that X=
F in a quasi-uniform space
is said to contain arbitrarily small open sets if for
E
U there exists x
C U[x] and Ox
E
F.
E
X and Ox
E
tu such that x
E
Ox
43
It is clear that if (X,U)
plete),
is complete,
then it is o-cornplete,
LEMJI.1A 4.
(strongly com-
(strongly o-cornplete).
Let (X,U) be a quasi-uniform space such
that every Cauchy filter contains arbitrarily small open
sets.
Then (X,U)
is strongly o-cornplete implies that (X,U)
is strongly complete.
PROOF.
= { 0 : 0
Let F be a Cauchy filter and let B
s F and 0 is open}.
Let G be the filter generated by B.
Since F contains arbitrarily small open sets we have that
G is an open Cauchy filter.
The result follows from the
fact that lim G = lim F.//
It is an easy observation that if
(X,t)
is a general-
ized absolutely closed topological space and U is any cornpatible quasi-uniform structure then (X,U)
is o-cornplete.
This is true since in a generalized absolutely closed space
every open filter has a nonempty adherence.
If
LEMMA 5.
(X,U)
is pre-compact then every open ul-
trafilter is Cauchy.
PROOF.
Let M be CJ.r open ultrafilter on X; that is, 1\l
is a maximal element in the class of all open filters on X.
Let U
E
U, then there exists V
E
U such that V
Since U is pre-compact there exists x
1 , •••
o
V CU.
,Xn s X such that
n
i
1,2, •.•
,n.
Hence
u
oi =X and so there exists ok
E
u
1
since
M is an open ultrafilter.
Consequently
Now U[xk]
~
M is Cauchy.//
Ok and there-
44
LEMMA 6.
Let (X,U)
is pre-compact and
(X,U)
be a quasi-uniform space.
o~complete
If
then X is generalized
absolutely closed.
PROOF.
It suffices to show that every open ultrafil-
ter has a nonempty adherence.
Let
M be an open ultrafilter,
then since U is pre-compact M is Cauchy by lemma 5.
Since
is a-complete we have that adh A1 =f fJ.//
(X,U)
Let (X,U)
THEOREM 16.
be a uniform space.
generalized absolutely closed if and only if
(X,U)
(X,U)
is
is pre-
compact and a-complete.
Lemma 3 is a generalization of the sufficiency.
PROOF.
Suppose
(X,U)
is generalized absolutely closed, then (X,U)
is a-complete by a previous comment.
is pre-compact.
(X,U)
We now show that
Let U s U, then there exists a sym-
metric V s U such that V o V CU.
a neighborhood cover of X.
Now { V[x]
Since X is generalized absolute-
ly closed we have that there exists x
1 , •••
,xn s X such that
n
l
z
f
U V[xi].
X
<:
V[y]
n
: x s X} is
V[xi].
'I' hen
( y, z)
E
V
is symmetric we have that (xi,z)
fj.
Let
s V and since V
1
s V and (z,y)
s V.
Thus
n
(xi,y)
s V o v C U or y s U[xi].
therefore
(X,U)
Hence X
= U U[xi] and
l
is pre-compact.//
Fletcher and Naimpally [10] working independently obtained analogous results to those found in this section.
They call a-complete, almost complete and generalized absolutely closed,
almost compact.
They defined a quasi-un-
45
iform space
to be almost pre-compact if for each u c
(X,U)
U there exists x
, •••
1
,xn c X such that X=
a U[x.J.
l
Using
l
these definitions they obtained the following generalization to lemma 6.
A topological space is almost-compact
if and only if every compatible quasi-uniform structure
is almost complete and almost pre-compact.
E.
q-T.
l
SEPARATION AXIOMS
Let (X,U)
be a quasi-uniform space.
only if given x
x
~ U [ y)
f
if and
0
y there exists U c U such that either
or y ~ U [ x] •
¢ U[y]
U[y]
=
~-
~
andy
f
larly, X is T 2 if and only if given x
n
f
X is T 1 if and only if given x
there exists U c U such that x
U such that U[x]
X is t
U[x].
y
Simi-
y there exists u c
The following example shows
that this characterization does not carry over for an arbitrary T
quasi-uniform space.
3
EXAMPLE 5.
Let X denote the natural numbers and let
U be the quasi-uniform structure generated by the base consisting of all sets of the form Un =
x
n}, for n c X.
~
{
(x,y)
Now tu is the discrete topology, so
F = {2,4,6, ... } is closed in X.
U [2n) =X, so U[F] =X.
n
If U c U,
Thus
DEFINITION 13.
3
if given x
that U[x]
n
U[F]
¢
there exists
(X,U)
there does not exist U c U such that U[l]
q-T
: x = y or
~
A quasi-uniform space
is T 3
U[F]
(X,U)
=
,
but
~-
is called
F, F closed, there exists U c U such
=
~-
46
It is clear that if a space is g-T 3 then it is T 3
Example 1 showed that a space can be T
but not q-T
3
will show that every uniform space is q-T 3
THEOREM 17.
Then (X,U)
Let
is q-T 3
PROOF.
(X,U)
(V
•
be a R 3 quasi-uniform space.
~
Let F be closed in X and x
F.
n
U such that U[x]
E
Since X - F
F
= 0.
Since
is R 3 there exists a symmetric V s U such that
o V) [x] C
exists f
U [x] •
Suppose a s V[x]
F such that (f,a)
s
is symmetric we have
(V o V) [x] C U[x].
F
we
•
•
is open there exists U
(X,U)
3
•
= 0.
Hence V[x]
COROLLARY.
(x,a)
s V and
V[F].
(x,a)
v.
s
But this is impossible since U[x]
n
V[F]
V.
Since V
s
s
(a,£)
Then there
Thus f
n
V and
n
s
= 0.//
Every uniform space is q-T 3
DEFINITION 14.
•
A quasi-uniform space is called g-T 4
if for every pair of disjoint closed sets F and G there
exists U s U such that U[F]
n
U[G]
= 0.
It is clear that every q-T 4 space
every g-T 4 + T
0
space is a q-T
3
T 4 and moreover
lS
space; that is, quasi-
normal implies quasi-regular.
Let (X,U)
LEMl"iA 7.
F closed in X.
PROOF.
Then
be a q-T
(F,UF)
4
quasi-uniform space and
is a q-T 4 space.
Let G and H be closed disjoint subsets of F,
then G and H are closed and disjoint in X.
q-T4 there exists u
v = u
n
F
X
F.
Then
u
E
v
E
such that U[G]
UF and V[G]
n
n
Since
U[H]
V[H]
= 0-
= 0-
(X,U)
Let
Hence
lS
47
A space can be T 4 and not q-T 4 ; consider the space 1n
=
example 5 with F
=
{2,4, ... } and G
DEFINITION 15.
{1,3, ... }.
A quasi-uniform space
(X,U)
is called
q-T 5 if for every pair of separated sets F and G there
exists U s U such that U[F] 0
= 0.
U[G]
Clearly every q-T 5 space is T 5
1 is T 5 but not q-T 5 •
The space 1n example
•
It is also evident that each space
is a q-T 4 space.
LEMMA 8.
If
(X,U)
is q-T 5 then every subspace is
q-Ts.
PROOF.
Now (clX F)
Let Y C X and F and G separated in (Y,Uy).
n
we have that
G
n
~
Y
ely F.
(ely F) (J G
Y
n
(ely F)
Since F and G are separated 1n Y
G = 0.
0.
~
n
Now (clX F)
Similarly
n
(clX G)
n
G C (clX F)
F
= 0.
Thus
F and G are separated in X and hence there exists U s U
n
such that U[G)
Uy and V[G]
n
= 0.
U[F]
V[F)
THEOREM 18.
= 0.
Let
Let
Hence
(X,t)
v = u
(Y,Uy)
n
y
(X,U)
be a topological space and U
If
For 1
=
3
Let F, G be disjoint closed sets in X.
x
0 U
(X,t)
is T. ,
l
the result follows by theorem 17,
since the Pervin structure of a T 3 space is R 3
0
E
is q-Ti for 1
PROOF.
0, Q s t
v
Then
is q-T 5 . / /
the Pervin quasi-uniform structure for t.
then
Y.
X
such that F C 0, G C Q and 0
(X-O)
x
X and T
=
Q x Q U
•
=
Let i
Then there exists
n
(X-Q)
Q
= 0.
x
X.
Let S
Then set
~
4.
48
u
n
s
[0
Now U[F]
(X, U)
T
u
0
X
Q
X
= 0 and U[G]
q-T 4
lS
Q]
u
[X -
Q)
= Q, hence U[F]
The proof for i
•
u
(0
=
5
n
X
X]
U[G]
Hence
follows in the same
manner.//
LEI1l'1A 9.
(X,U) be a saturated quasi-uniform space.
Let
If X is Ti then i t is q-Ti f o r i = 3,4,5.
PROOF.
If U is quasi-saturated then there exists a
w.
base consisting of a single set
F where F is closed.
such that
X
W[F] C: Q.
E
~
Then there exists open sets 0 and Q
n
0, F C Q, and 0
(X,U)
Hence
Suppose X is T 3 and x
is q-T 3
Q =
~.
Now W[x] C 0 and
The proofs for q-T 4 and
•
q-T 5 follow in a similar manner.//
Let X denote the natural numbers.
EXAMPLE 6.
Let U
be the uniform structure generated by the base consisting
of the sets Un = {
(x,y)
: x
=
y or x
~nand
y
~
topology generated is discrete and hence T 5 and T 4
is clearly not q-T 4 and hence not q-T 5
a uniform space may be T 4
Let
THEOREM 19.
Then
space.
(X,U)
that for each u
U
exists x
ly x
i
E
U}
~
adh B.
F ()G.
(X, U)
(q-T 5 ) .
be a compact q-T 3 quasi-uniform
is q-T 4 .
u
we have U[F]
is a filter base.
E
The
Thus we see that
(T 5 ) and not q-T 4
(X,U)
•
Let F and G be disjoint closed sets.
PROOF.
U [G]
•
n}.
n
U[G]
+ ~-
B
= {
Suppose
U [F] ()
Since X is compact there
We may suppose that x ~ F, since clear-
By hypothesis X is q-T 3 so there exists
49
U
E
U such that U[x]
=
~'
n
U[F]
~-
=
n
Then U[x]
but this is impossible since x c adh B.
(U[F]
n
U[G])
Therefore
is q-T 4 • I I
(X' u)
COROLLARY.
Then (X,U)
Let (X,U) be a compact uniform space.
is q-T 4
•
The following characterization of q-T 3 seems complicated but it proves a helpful tool in the following theorem.
(X,U)
L E i''L.'VlA 1 0 •
is q-T 3 if and only if for each open
set 0 and x c 0 there exists U c U such that U[x] C
X -
U[X- 0].
PROOF.
The proof follows immediately from the defini-
tion of q-T 3 .11
THEOREf--1 2 0 •
is q-T3 for each l .
PROOF.
X
0 where 0
E
E
l
= n ul..
Sufficiency.
is x.l except for i
For each i = i
n.o. c o .
l
and 0·l
u
u.1
1
, •••
= i
+
E
U[x]
n
that y c U[a].
t
If a
t
ik and
Then there exists a
c
k
Let U
u.
E
E
= TIU.
l
Suppose
X - 0 such
0 then there exists ik such that
c U1· [x.1 ] and y 1·
c U[ai ] .
Thus Yik c
k
k
k
k
k
U· [x· ] C X·
- U. [X.
- 0 1· ] .
But Y·
c U[a. ] CU.
lk lk
lk
lk
lk
k
lk
lk
lk
Therefore U[x] n U[X - 0]
[Xi
- o. ] which is impossible.
k
lk
= ~ and hence U[x) C X - U[X - 0] and by lemma 10 we have
0 1·
k
•
Thus y 1·
U[X - 0].
1 ' ••• '
,ik there exists Ui
such that u.1 [x·1 ] c: Xi
- u.1 [X.1
- oi ] .
k
k
k
k
k
k
k
where ui =xi X xi for l
il, ... ,ik.
Then u
that y
Let
Then there exists rrioi where
lS open in X.
lS open in t·l
0·l
X
Let X = nxi and
50
that (X,U)
q-T 3
lS
Necessity.
•
Let xi s Oi, where Oi is open in Xi.
Let
x be any point in X with the ith coordinate equal to xi.
Then there exists Us U such that U[x] C X- U[XThere exists nkuk C U where uk = Xk
Xk except for
x
ni 1 (Oi)]
~
fi-
nite number of indices and Uk s Uk for each k.
Then
Suppose y.
l
Ui[xi]
n
Ui[Xi - Oi].
that
(a.,y.)
(yk)
s
l
l
-1
(Oi)
each k
(a.)
J
J
+i
J
J
J
Now (a,y)
Then a s X-
s nkuk since for
= (yk,yk) s Uk and for k =
(ak,yk)
Therefore
Then y =
(rrkuk) [X- TTi (Oi)].
+ i.
by a. = y. for each j
+ i.
_l
and therefore y s X-
since ai s Xi- oi.
(a.,y.)
l
l
(Oi).
Then there exists ai s Xi - Oi such
Choosey,= x. for all j
l
(ITJ.;.Ykl [x]
Define a=
11i
s u ..
s
l
we have
EX- TT.-1
(a,y)
l
1
Hence y s rrkuk[X- TTi (Oi)] which is a contra-
diction.
Therefore
U l · [x l·]
n
U l· [X·l
-
Thus Ui [xi] C Xi - Ui [Xi -- Oi]
each factor
spac~
0 l·]
=
~.
and by lemma 10 we have that
is q-T3.//
It is easy to see that the product of a family of a
family of q-T 4 spaces need not be q-T 4
•
family of T 4 topological spaces such that
Let (Xa,ta) be a
(rraxa,nata)
is not
51
Let U
rl' 4 •
a
be the Pervin structure associated with t
a
.
(X a' Ua ) is q-T by theorem 18.
4
Then each factor space
Hence
if n a Ua were c-T 4 then n a t a would be T 4 which is impossible.
j_
F.
A COUNTER-EXAMPLE IN
DEFINITION 16.
spaces.
Y.
Let
(F,0)J
(X,U)
and (Y,V) be quasi-uniform
Let F denote the set of all functions from X to
For each V s V we let
W ( V)
= { (f , g )
s
F
F :
x
( f ( x) , g ( x) )
The collection of all such W(V)
V
s
for each x
s
X} •
forms a base for a quasi-
uniform structure which we denote by W.
is then a
(F,(v)
quasi-uniform space and W is called the
quasi-unifor~
structure of quasi-uniform convergence.
The following example shows that neither the set of
all continuous mappings from X to Y nor the set of all
quasi-uniformly continuous mappings from
(Y,V)
Let X and Y denote the integers.
EXANPLE 7.
Set 8
to
(F,W).
need be closed in
u n = { (x,y)
(X,U)
c~
X
= { Un: n =
X
x
:
1,~,
y or x =
x
l
andy
~
Let
n}.
Then G is a base for a quasl-
. . . }.
uniform structure on X, which we will denote by U.
'i'he set of all
vn
where n
=
= {
(x,y)
1,2, ••• ,
s
Y x
Y
:
=
x
y
or x .:.: n}
forms a base for a quasi-uniform struc-
ture for Y, which we will denote by V.
is discrete.
Define f :
(X,LI)
~)-
(Y,V)
The topology on Y
by f(x)
= x for each
52
X
E
The function f is not continuous since f- 1 (1)
X.
{1} ~ tx, and hence not quasi-uniformly continuous.
For each natural number n, define
fn :
fn(x)
(X,U)
~
(Y,V)
by,
= x for x
<
n
=
2:
n.
1
for x
We will show that each fn is quasi-uniformly continuous.
Let fn and Vm E V be given.
(a,b)
E Unthen
fn(b)
and thus
It suffices to show that if
(fn(a),fn(b))
(fn(a) ,fn(b))
EVm.
E Vm.
If a= b, then fn(a)
If a
+ b,
then a
and b .:::. n.
=
1
Thus
each fn is quasi-uniformly continuous and by theorem 1.24,
in [16]
i t is continuous.
Let F denote the set of all functions from X to Y and
W the quasi-uniform structure of quasi-uniform convergence.
Let U C F denote the set of all quasi-uniformly continuous
functions from
(X,U)
to (Y,V),
and let C c F denote the col-
lection of all continuous functions from (X,U) to
We have shown that each fn E
that f
c:
Consider
s
U.
U C
Let Vn be given.
(f(x),fn(x)); i f x
<
C,
and f
~
C.
(Y,V).
We now show
We claim that fn E W(Vn) [f]
nthen
(f(x),fn(x))
Vn' and if x.::: n then (f(x) ,fn(x)) = (x,1)
E Vn.
= (x,x)
Hence
fn E W(Vn) [f]
and since Vn was arbitrary, we have that f
IT c c.
~
But f
C.
c:
Hence neither U nor C is closed in (F,W)
53
V.
SUMMARY, CONCLUSIONS AND FURTHER PROBLEMS
We noted that a quasi-uniform space need not have
a T1 or T 2 strong completion nor a T 2 completion.
It is
of interest to know if an arbitrary space has a T 1 completion.
Our construction showed that for certain classes
a T1 completion exists.
No one as yet has exhibited a
topological space that does not admit a complete or strongly complete quasi-uniform structure.
Our example demon-
strates that a space can be uniformizable, admit a strongly complete structure and not admit a complete uniform
structure.
It seems that the present definition of Cauchy
filter may admit too many filters.
ample 2, we saw that F
In Chapter III, ex-
= {X} was a Cauchy filter.
Since
the topology on X is discrete this seems a bit unreasonable.
Thus the study of other classes of "Cauchy" filters
would appear to be worthwhile.
Since Section F, Chapter
III showed that regardless of the definition of "Cauchy"
filter and "completeness" not all of the pleasant properties of the completion of a Hausdorff uniform space
could be preserved for a quasi-uniform space, it seems
logical for one to decide which of these properties
should be preserved, if possible, before a new definition of "Cauchy" filter is proposed.
On the other hand it can be noted that for special
classes the completion constructed in Chapter III, Section
54
B, has many of the desired properties.
In Chapter IV several topics were considered.
i'1any
other areas in quasi-uniform spaces also remain open to
investigation.
No one has as yet characterized the uni-
versal quasi-uniform structure associated with a given
topology,
and no necessary and sufficient conditions are
known for when a topology admits a minimal quasi-uniform
structure.
gl'ven for
Necessary and sufficient conditions were
u
1\
u- 1
to be a quas1-un1
.
'f orm structure.
showed that U and C need not be closed in
Naimpally [17]
showed that if
(Y,V)
(F,W)
We
and
1s T 3 and V is the
Pervin structure then U and C are closed in (F,W).
More
general conditions to insure that U and C be closed or
complete seem desirable.
The q-T.
separation properties seem to this author
l
to at least be an interesting concept when one is more
concerned about the particular quasi-uniform structure
than the generated topology.
55
BIBLIOGRAPHY
l.
Barnhill, C. and Fletcher, P., Topological Spaces with
a Unique Compatible Quasi-uniform Structure, Archiv
der Math., to appear.
2.
Carlson, J. W. and Hicks, T. L., On Completeness in a
Quasi-uniform Space, J. Math. Anal. Appl., to
appear.
3.
Csaszar, A., Fondements de la Topologie Generale,
Gauthier-Villars, Paris, (1960).
4.
Davis, A. S., Indexed Systems of Neighborhoods for General Topological Spaces, Amer. Math. Monthly 68
(1961)' 886-893.
5.
Dieudonne, J., Un Exemple d'espace Normal Non Susceptible d'une Structure Uniform d'espace Complete,
C. R. Acad. Sci. Paris 209 (1939), 145-147.
6.
Fletcher, P., Finite Topological Spaces and Quasi-uniform Structures, Canad. Math. Bull. 12 (1969),
771-775.
7.
Fletcher, P., On Completeness of Quasi-uniform Spaces,
submitted.
8.
Fletcher, P., Note on Covering Quasi-unifor~lties and
Hausdorff Measures, submitted.
9.
Fletcher, P., Homeomorphism Groups with the Topology of
Quasi-uniform Convergence, submitted.
10.
Fletcher, P., and Naimpally, S. A., Almost Complete and
Almost Precompact Quasi·-uniform Spaces, submitted.
11.
Gaal, s. A., Point Set Topology,
New York and London.
12.
Gillman, L. and Jerison, M., Rings of Continuous Functions, (1960), D. Van Nostrand, Princeton, New
Jersey.
13.
Isbell, J., Math. Rev. 22
14.
Krishnan, v. s., A Note on Semi-uniform Spaces, J.
Madras Univ. B 25 (1955), 123-124.
15.
Liu,
(1964), Academic Press,
(1961), 683.
c., Absolutely Closed Spaces, Trans. Amer. Math.
Soc., 130 (1968), 86-104.
56
16.
Murdeshwar, M. G. and Naimpally, S. A., Quasi-uniform
Topological Spaces, Noordhoff Series A, (1966).
17.
Naimpally, S. A., Function Spaces of Quasi-uniform
Spaces, Indag. Math. (Akad. Wetenshcappen) 68
(1965) 1 768-771.
18.
Niemytzki, V. and Tychonoff, A., Beweis des Satzes, das
ein Metrisierbarer Raum dann und nui dann Kompact
ist, Wenn er in Jeder Metric Vollstandig ist, Fund.
Math. 12 (1928), 118-120.
19.
Pervin, W. J. Quasi-uniformization of Topological Spaces,
Math. Ann. 14 7 (19 6 2) , 316-317 .
20.
Sieber, J. L. and Pervin, W. J., Completeness in Quasiuniform Spaces, Math. Ann. 158 (1965), 79-81.
21.
Sian, M. and Willmott, R. c., Hausdorff Measures on
Abstract Spaces, Trans. Amer. Math. Soc., 123
(1966)
275-309.
1
22.
Stoltenberg, R. A., A Completion for a Quasi-uniform
Space, Proc. Amer. Math. Soc., 18 (1967), 864-867.
23.
Weil, A., Sur les Espaces a Structure Uniforme et sur
la Topologie Generale, Gauthier-Villars, Paris,
(1937).
57
VI'l'A
John Warnock Carlson was born on November 10, 1940
at Topeka, Kansas.
He graduated from Blue Valley High
School at Randolph, Kansas in May 1959.
He entered
Kansas State University that fall, where he received his
Bachelor of Science degree with a major in Mathematics in
June 1963, and his Master of Science degree, with a major
ln Mathematics and a minor in Statistics, in August 1964.
From September 1964 until June 1967 he was employed
as an Instructor of Mathematics at Washburn University at
Topeka, Kansas.
He attended two N.S.F. Summer Institutes
in Computer Science.
The first was during the suwmer of
1966 at the University of Missouri-Rolla, and the second
during the summer of 1967 at Texas A
&
1'1
University.
Since
September 1967, he has been employed as an Instructor of
Mathematics at the University of Missouri-Rolla, where
he has been continuing his graduate training.
He was married to the former Sara Ann Hollinger on
September 9, 1961.
and David.
They have two children, Catheryn