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Retrospective Theses and Dissertations
1971
On bitopological spaces
Marcus John Saegrove
Iowa State University
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SAEGROVE, Marcus John, 1944ON BITOPOLOGICAL SPACES.
Iowa State University, Ph.D., 1971
Mathematics
University Microfilms, A XEROX Company, Ann Arbor, Michigan
THIS DISSERTATION HAS BEOi MICROFILMED EXACTLY AS RECEIVED
On bitopological spaces
by
Marcus John Saegrove
A Dissertation Submitted to the
Graduate Faculty in Partial Fulfillment of
The Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major Subject:
Mathematics
Approved:
Signature was redacted for privacy.
In Charge oàMajor Work
Signature was redacted for privacy.
Signature was redacted for privacy.
Iowa State University
Of Science and Technology
Ames, Iowa
1971
PLEASE NOTE:
Some pages have indistinct
print. Filmed as received.
UNIVERSITY MICROFILMS.
ii
TABLE OF CONTENTS
Page
I.
II.
III.
IV.
V.
INTRODUCTION
1
SEPARATION PROPERTIES
4
AN INTERNAL CHARACTERIZATION OF PAIRWISE
COMPLETE REGULARITY
l8
PRODUCTS AND BICOMPACTIFICATIONS
26
GENERALIZED QUASI-METRICS AND QUASIUNIFORM SPACES
58
VI.
LITERATURE CITED
49
VII.
ACKNOWLEDGEMENTS
5%
1
I.
INTRODUCTION
A bitopological space
topologies,
P
and
Q,
(X,P,Q)
on it.
is a set
X
with two
Bitopological spaces arise
in a natural way by considering the topologies induced by
sets of the form
form
B
metrics on
B^^^ = fy I p(x,y) < e]
= (y i g(x,y) < e],
X
and
where
p
q(x,y) = p(y,x).
and sets of the
and
q
are quasi-
Quasi-uniform spaces,
which are generalizations of quasi-metric spaces, also induce
bitopological spaces.
Kelly [8j was one of the first to
study bitopological spaces.
Later work in the area has been
done by Fletcher [3] et al., Kim L9j, Lane [10], Patty [13],
Pervin [14], Reilly [16] and others.
Reilly [I6] discusses separation properties of bitopo­
logical spaces at some length in his Ph.D. dissertation.
In
Chapter II of this thesis, we identify a new separation
property which we call weak pairwise
T^
and we attempt to
organize the separation properties into what appears to be
two natural types, weak and strong.
We also generalize to
bitopological spaces some results concerning regularity given
by Davis [1].
We include in Chapter II a number of defi­
nitions and results previously given.
definitions has been changed slightly.
continuous function into
(A,R,L)"
The wording of some
The words "pair
(which will be defined
in Chapter II) have replaced the words "P-lower semi-
2
continous
Q-upper semi continuous function" since the two
are equivalent.
A "P-lower semi-continuous
Q-upper semi-
continuous function" is a function from a bitopological
space
(X,P,Q)
into
A
(the real line) with the usual
topology such that inverse images of open right rays are
P-open and inverse images of open left rays are
Frink
[4]
Q-open.
and Steiner [I7] have given internal charac­
terizations of complete regularity.
In Chapter III we gener­
alize this result to bitopological spaces using a generali­
zation of Steiner's method of proof.
In Chapter IV we define products of bitopological
spaces and obtain a bicompactification of a weak pairwise
T 1
soace.
To the author's knowledge products in bitopo-
logical spaces have not previously been defined.
Fletcher [3]
and Kim [9] have given definitions of pairwise compactness
in bitopological spaces, but under their definitions products
of pairwise compact spaces are not pairwise compact, nor are
real valued pair continuous functions bounded.
We give a
different definition of bicompactness which has these de­
sirable properties.
Kalisch [6] has defined a generalized metric space and
has proven that any generalized metric space is a uniform
space and that every uniform space is isomorphic to a gener­
alized metric space.
In Chapter V we define a generalized
5
quasi-metric space and show that a generalized quasi-metric
space is a quasi-uniform space and that every separated
quasi-uniform space is quasi-uniformly isomorphic to a
generalized quasi-metric space.
Since it has previously
been shown that topological spaces are quasi-uniformizable,
we have essentially shown (if we disregard separation) that
topological spaces are generalized quasi-metric
spaces.
The following figure indicates some of the inter­
relationships that exist among these structures.
metric spaces
quasi-metric
spaces
1
uniformizablegeneralized
spaces
metric spaces
topological
spaces
generalized
quasi-metric
spaces
quasiuniformizable
spaces
1
bitopological
spaces
4
II.
SEPARATION PROPERTIES
In this chapter we consider various properties of
bitopological spaces.
Some effort was made toward complete­
ness and the systematic presentation of the usual degrees
of separation. We begin with a rather weak form of sepa­
ration due to Reilly [16,Definition 1.1], who calls it pairwise
T^.
We prefer to call this property weak pairwise
Definition 2.1:
pairwise
T^
The bitopological space
(X,P,Q)
T^.
is weak
iff for each pair of distinct points, there
is a set which is either
P-open or
Q-open containing one
but not the other.
Definition 2.2:
pair
(x,y)
(X,P,Q)
is pairwise
of distinct points of
P-open set containing
open set containing
x
y
but not
but not
T^
X,
y
iff for each
there is either a
or there exists a
Q-
x.
Definition 2.2 was given by Fletcher, Hoyle, and Patty
[3,Definition 1].
It is easy to see that
topology implies weak pairwise
T^
topology implies pairwise
Paiirwise
TQ
set
T^.
and
T^
T^
T^
in either
in either
does not imply
in either topology as can be seen by considering the
X = {a,b,c}
with
0 = {0jX,{b,c},[a]},
neither
P
nor
Q
P = {0,X,{a,b},{c}}
where pairwise
is
T^.
Pairwise
T^
T^
and
is clear, but
implies weak
5
pairwise
but the following example shows the converse
is not true.
Example 2.1:
and
Consider
R = {(x, o o )
I
X €
where
and
R]
both topologies are
But if
a < b,
tains
a
tains
b,
so
L
R
(RjR,L)
Definition 2.3:
{ ( - o o ,x) 1
(&,R,L)
every
and every
L =
R
is the real line
x €
is weak pairwise
open set containing
open set containing
is not pairwise
(X P,Q)
b
a
iff for
x,y,
set
such that either
y
X
U
and a
and
Q-open set
y€V, x;^V
V
or
also con­
T^.
each pair of distinct points
U
T^.
also con­
is weak pairwise
J
Since
R}.
there exists a
x€V, y/v
P-open
x € U,
and
y € U,
/ U.
Reilly [l6,Definition 2.1] defined the following
property as pairwise
Definition 2.4:
T^.
(X,P,o)
is pairwise
pair of distinct points
x,y,
U
such that
and a
Q-open set
V
iff for each
there exists a
x € U, y
P-open set
U
and
y € V, X X V.
Clearly weak pairwise
and pairwise
implies weak pairwise
implies pairwise
is equivalent to
Obviously, pairwise
T^.
Also pairwise
in each topology [l6,Theorem 2.2].
implies weak pairwise
but
6
Example 2.1 shows that weak pairwise
to pairwise
ing
a
b
for if
also contains
must also contain
a
b.
a < b,
and any
where
P
weak pairwise
R-open set containing
is not equiva­
as any bitopological space
is discrete and
T^,
L-open set contain­
Weak pairwise
lent to weak pairwise
(X,P,Q)
any
is not equivalent
Q
is indiscrete is
but not weak pairwise
Reilly [16] states that Kim called the following
property pairwise
Definition 2.5:
T -, .
(X,P,0)
is weak pairwise
Tg
(or weak
pairwise Hausdorff) iff for each pair of distinct points
X
and
set
or
V
X
y
in
X,
there is a
disjoint from
U
P-open set
such that either
U
and a
x € U, y € V
€ V, y € U.
Definition 2.6:
(x, P,Q) is pairwise
(or pairwise
Hausdorff) iff for each pair of distinct points
of
X,
0-open
there is a
disjoint from
U
P-open set
such that
U
x € U
and a
and
x
Q-open set
and
y
V
y € V.
This definition was originally given by Weston [I8].
Pairwise
Tg
Tg
implies pairwise
implies weak pairwise
implies weak paiirwise
Tg,
are not equivalent, for if
T^.
and weak pairwise
Clearly pairwise
but Example 2.1 shows the two
a < b,
any
L-open set contain­
7
ing
b
also contains
also must contain
pairwise
pairwise
a
b.
and any
R-open set containing
Reilly [l6,Example 3-2] shows that
is not equivalent to pairwise
Tg
Tg
and that a
space is not simply a pair of
spaces [16,Example
topological
The following example shows weak
pairwise
is not equivalent to weak pairwise
Example 2.2:
Consider (O) where
topology and
Q
R - {x}
contains
y
but not
x
x
is the cofinite
R.
but not
. (R,P,0)
however, since any
T^.
For
y,
P-open set which contains
Example 2.1 shows weak pairwise
T^.
and weak pairwise
Example 2.2 is pairwise
Tg
T-j^
Example 2.3:
Let
points
T^,
1,2.
y.
does not imply
(as it is
T^.
in both
The following
does not imply weak pairwise
T^.
X = {1,2,3], P = {0,X,{1,2},{3]} and
Q = {0JX,[1], {2,3}}pairwise
x
does not imply pairwise
topologies), but is not weak pairwise
example shows pairwise
so
is not weak pair-
must contain points of any interval which contains
pairwise
x p y,
and
contains
is weak pairwise
Tg,
P
is the usual topology on
^x - ^ ^ ^ ^ ^ J X + • ^^ ^ ^ ^
wise
a
It is easy to check that
but not weak pairwise
T^,
(X,P,Q)
is
for consider the
8
Davis [1] has given for topological spaces regularity
axioms
such that
i = 0,1,
or
and
2.
together are equivalent to
We extend this to bitopological
spaces.
Definition 2.7:
given a
a
Q-closed set
P-open set
(Q)R
Q
(X,P,Q)
U
is
A
P
and
(Q)R
lent to pairwise
T^.
(X,P,Q)
x ^ A.
open set
such that
and
x ^ U U :
y€A y
We define
iff it is
T^.
For each
y € A,
(Q)R .
P.
R^
R^.
is equiva­
Let
A
there exists a
y c U . x ^ TJ . Thus
y
y
thus we have
(P)R^
Since pairwise
we prove pairwise
P-closed and
U
y
A c U.
and pairwise
be pairwise
implies pairwise
iff
there exists
with respect to
Q
Pairwise
Let
and
is pairwise
Proposition 2.1:
Proof:
^ U
x ^ A,
Q
in the same way.
Definition 2.8: (X,P,Q)
Q
with respect to
and a point
such that
with respect to
with respect to
(PjR^
(P)R
be
Q-
U U =) A
ySA ^
follows in a
similar manner.
Conversely, suppose
pairwise
containing
T^,
x
x, y € X, x ^ y.
suppose first that we have a
but not
set, so there exists a
y.
Then
Q-open set
x ^ X - U,
V
Since we have
P-open set
a
such that
U
P-closed
x jc V
and
9
V Z 5 X - U . Y € V, X ^ V
and we have the desired result in
this case.
Now suppose we have a
y
x.
but not
P-open set
y^ U
y ^X - V
U
such that
y ^ U
(X,P^q)
and
X - V C U .
x€U,
such that
U fl V = 0
and if
x ^ {y} ,
then there is a
Q-open set
V
such that
iff for
P-open set
V
{y}
containing
Q-closed so there is a
is pairwise
^ { y t h e n there is a
X
set
which is
V
and we have the desired result in this case also.
Definition 2.9:
if
Q-open set
and
x 6 V
U
and a
and
P-open set
U D V = 0
and
x,y € X,
x € U
Q-open
{y
c U
U
and a
and
c V.
Proposition 2.2:
Pairwise
equivalent to pairwise
Proof:
Pairwise
and pairwise
Tg.
implies pairwise
[y}^ = [ y = [y]
is
in pairwise
and since
spaces, pairwise
R^
is obvious.
Conversely, if
wise
R^
open set
{y}^ c V,
x ^ y,
then
condition, there is a
V
such that
that is
x ^ {y}^
P-open set
U fl V = 0
and
x € U
so by the pairU
and a
Q-
and
y € V.
The next two definitions are due to Kelly [8].
Definition 2.10:
P
Let
(X,P,Q)
is regular with respect to
be a bitopological space.
0
if for each point
x € X
10
and each
P-closed set
P-open set
U
and
and
X
Ç U
and a
A
Definition 2.11:
Rg)
if
P
A
such that
Q-open set
V
x ^ A,
there is a
such that
U fl V = 0
V.
c
(X^P^q)
is oairwise recrular (or pairwise
is regular with respect to
lar with respect to
Q
and
Q
is regu­
P.
Reilly [l6,Proposition 4-3] gives the following
equivalent formulations of Definition 2.10.
Proposition 2.3:
If
(x,P,Q)
is a bitopological space, the
following are equivalent:
a)
P
is regular with respect to
b)
For each point
ing
X,
x Ç X
there is a
and
0-
P-open set
P-open set
U'
U
contain­
such that
X € U' c ÏT^ c u.
c)
For each point
that
X
X ^ A,
€ U
Proof:
and
are
P-closed set
P-open set
U
A
such
such that
n A = 0.
(X,P,Q)
is pairwise
Rg,
then it is
R^.
Let
x, y € X.
Q-open sets
U n V = 0
If
and
there is a
and
Proposition 2.4:
pairwise
x € X
and
P-open and
U
x c V
If
and
V
and
Q-open sets
x;^ [y
there are
P-open
respectively such that
{y}^ c U.
U
and
If
V
x
[y]^,
there
respectively such
11
that
U n V = 0
and
Proposition 2.5:
pairwise
Proof:
X
x € U
If
and
(X,P,q)
Let
A
open sets
be
and
R^,
Similarly,,
(X,P,0)
is
(o)R^
we have pairwise
R^.
Definition 2.12:
(X,P,Q)
R^
x ^ A.
there exist
y G A,
P-open and
fl
A c U U
and
y€A ^
(p )R. with respect to
with respect to
is weak pairwise
(X,P,0)
is pairwise
Proposition 2.6:
Pairwise
Rg
T^.
and pairwise
(X,P,Q)
Let
be pairwise
X = y.
Since singleton points are
spaces,
U
respectively and
and
V
P;
1%
Q.
hence
iff it is
T^.
iff it is pairT^.
and
x, y € X
P-closed and
Thus there exist
which are
P-open and
and
where
Q-closed
x ^ {y}^-
x = fx}^ c v
= 0
is equivalent to pairwise
Proof:
in pairwise
Q-
Then
{pairwise regular) and pairwise
disjoint sets
For
(pairwise regular) and weak pairwise
Definition 2.13:
Rg
then it is
respectively such that
€ V
and {y} c u .
^
^
X JC U U , and (x, P,Q) is
y€A y
wise
is pairwise
Q-closed and
X
pairwise
c V.
R^.
X {yso by pairwise
and
{y}
Q-open
y = {y}^ c U.
12
Definition 2.14: [I5]
(Y,S,T)
from
A function
f
from
(X,P,Q) into
is pair continuous iff the induced functions
(X,P) into (Y,S) and (X,Q)
into
f
(Y, T) are
continuous.
Remark:
Lane [10] proves a number of results for
semi-continuous
P-lower
Q-upper semi-continuous functions which
we restate in terms of pair continuous functions into
(R,R,L):
i)
If
f
and
(R,R,L),
ii)
If
f + g
is also.
and
g > 0
are pair continuous functions
(ft,R,l),
f
and
(S,R,l),
iv)
If
f
then
are pair continuous functions into
then
f > 0
If
into
iii)
g
g
then so is
f • g.
are pair continuous functions into
then
min(f,g)
and
(-f)
Definition 2.15:
is pair continuous into
R
and
A space
regular iff for each
L
f(x) = 1
(X,P,Q)
P-closed set
B
and
(ft,L,R).
is pairwise completely
A
f(A) = {0},
and each point
(ft,R,L),
have been interchanged.)
there is a pair continuous function
closed set
are also.
is a pair continuous function into
(Note that
such that
max(f,g)
y ^ B,
and each point
x ^ A,
f : (X,P, Q) — ([0,1],R, L)
and for each
Q-
there exists a pair
15
continuous function
g(y) = 0
and
as pairwise
g : (X,P,Q)
g(B) = {l}.
([0,1],R,L)
such that
We also refer to this property
R,.
Definition 2.16:
(X ?,q)
J
is weak pairwise
T ^
(or weak
5
pairwise Tychonoff) iff it is pairwise completely regular
and weak pairwise
Definition 2.17:
T^.
(X,P,Q)
is pairwise
T ^
iff it is pair-
5
wise completely regular and pairwise
T^.
It is easily seen that weak pairwise completely regu­
lar implies weak pairwise regular, and weak pairwise
implies weak pairwise
T^.
Also pairwise completely regular
4
implies pairwise regular and pairwise
wise
T,
implies pair-
.
Definition 2.18:
P-closed set
there is a
U
4
T -,
A
(x,P,q)
and
Q-closed set
P-open set
disjoint from
V
property as pairwise
Definition 2.19:
weak pairwise
is pairwise normal iff for each
V
B
disjoint, from
containing
containing
A.
B
and a
A,
Q-open set
We also refer to this
.
(x,P,Q)
is weak pairwise
and pairwise normal.
iff it is
14
Definition 2.20: (X,P,Q)
wise
is pairwise
iff it is pair-
and pairwise normal.
Reilly [16,Proposition 5-2] proves the following
theorem on equivalences of pairwise normality.
Proposition 2.7:
The following are equivalent
a) (X,P,Q)
b)
is pairwise normal.
For each
P-closed set
containing
that
c)
A,
disjoint from
d)
For each
disjoint from
A
and a
such that
4
and pairwise
fl
Definition 2.21:
space and
A
and
T,,
such
and
Q-closed set
A
and
U
Q-closed set
P-open set
Q-open set
B
U
V
B
con­
containing
A
= 0.
implies weak pairwise
implies pairwise
T ,.
4
[11] Let (X,P,Q) be a bitopological
B
pletely separated from
subsets of
B
f(A) = {0}
and
X.
Then
with respect to
exists a pair continuous function
such that
A
there is a
Obviously, weak pairwise
T -,
U
there is a Q-open set
—P
such that U fl B = 0.
A
B
0-open set
V
A,
P-closed set
taining
Q-open set
c V.
P-closed set
containing
and
there is a
A c u c
For each
A
A
Q
is
P-com-
if there
f : (X,P,Q) — ([0,1],R,L)
f(B) = {1}.
A
is
Q-completely
15
separated from
B
with respect to
pair continuous function
that
f(B) = {0}
and
Notice that if
with respect to
from
A
into
([0, 1],R, L),
f : (X,P,Q) -* ([0,1],R,L)
is
then
with respect to
([0,1],L,R).
then
B
P.
is
Also if
1 - f
f
and each
B
Q-closed subset
[8]
If
(x,P,Q)
P
A
of
Q
from each
B
of
X
X
is
from each point
B
is
Definition 2.22:
(X,P,Q)
[11]
into
Q
If
from
f
(R,R, L),
P-zero set with respect to
Q
0-zero set with respect to
P.
with respect to
P
X
Q-closed, then
rated with respect to
spect to
is pairwise normal and
are disjoint subsets of
closed and
from
is pair-
X - B.
Theorem 2.1:
and
(X,P,Q)
P-closed subset
Q-completely separated with respect to
of
is pair continuous
is pair continuous into
It is clear that a space
X - A
B
Q-completely separated
P-completely separated with respect to
point in
such
P-completely separated from
wise completely regular iff each
is
if there exists a
f(A) = [ij.
A
Q,
P
Q
A
is
A
is
P-
P-completely sepa­
B.
is a pair continuous function
then
and
fx 1 f(x) <0}
is a
{x I f(x) >0] is a
We usually call
P-zero sets and
Q-zero sets.
such that
A
P-zero sets
Q-zero sets with re­
16
clearly
are
P-zero sets are
Q-closed.
Lane [10] proves that any
{x I g(x) =0}
also of the form
ous into
P-closed and
(R, R, L)
and
where
g
Q-zero sets
P-zero set is
is pair continu­
g > 0.
The following three propositions are due to Lane [10].
Proposition 2.8:
respect to
and a
a
Q
If
from
A
B,
Q-zero set
is
P-completely separated with
then there exist a
such that
Q-neighborhood of
A
and
B^
If
A
is a
then
A
is
P-zero set
A^ f! B^ = 0
is a
and
A^
is
P-neighborhood of
B.
Proposition 2.Q:
Q-zero set
B,
respect to
Q
from
Proposition 2.10:
regular iff the
sets and the
P-zero set disjoint from the
P-completely separated with
B.
The space
(X,P,Q)
is pairwise completely
P-zero sets form a base for the
Q-zero sets form a base for the
P-closed
Q-closed sets.
17
Figure 2.1 indicates some implications proven in Chapter
II concerning separation in bitopological spaces.
PR
WPT
PT,
Î
PT,
WPT,
I
WPT,
PTg
PT^
Î
PT
PT^
Figure
>
WPT^
2.1. Implications concerning separation
The letters
weak pairwise
T^
PTo
and
WPTo
respectively.
refer to ^
pairwise
T^
o
and
18
III.
AN INTERNAL CHARACTERIZATION OF
PAIRWISE COMPLETE REGULARITY
In this chapter we give an internal characterization of
pairwise complete regularity^ that is, a characterization
which does not depend on an outside space, namely, the real
numbers, as in the original definition and the character­
ization given by Lane.
This generalizes a result of E. F.
Steiner [l?].
Definition ^.1:
5
a family of
sets.
Then
for each
exists
and
A c S"
C € Q
A c X - C
a family of
(X,P,Q)
be a bitopological space and
P-closed sets and
(3,Çj
Definition 3.2:
Q
Let
Q
a family of
Q-closed
is called a pairwise-normal pair iff
and
BEG
D € 5
and
and
such that
such that
A n B = 0,
there
X - c n x - D = 0
B c X - D.
Let
5
be a family of
Q-closed sets.
P-closed sets and
Then
is called a
pairwise-separating pair iff i) and ii) hold;
i)
If
F
A € Q
is
and
P-closed,
B € J?
x / F,
such that
then there exist
x Ç A, F c B
and
A n B = 0.
ii)
If
F
A c 2F
is
and
A n B = 0.
Q-closed and
B f G
x ^ F,
such that
then there exist
x € A, F c B
and
19
Theorem 3.1:
A hitopological space (X,P,Q)
is pairwise
completely regular iff it possesses a pairwise normal pairwise separating pair.
Proof:
We prove that the family
and the family
Q.
of all
5
A
is
P
A € 3, B € Q
P-zero sets
Q-zero sets form a pairwise
normal pairwise separating pair
Let
of all
and
(3,Q).
A N B = 0.
By Proposition 2.9
completely separated with respect to
Q
from
A
and
B.
Thus there is a pair continuous function
f :(X P Q) - ([O 1],R,l)
J
f = 1
J
on
B.
Then
D = -^x I f (x) _<
f
f = 0
C = ^x| f(x) >
€ Q
€ 5
X - C nx- D =0
if
such that
J
and
on
and
are the required sets such that
A c X - C
is pair continuous, then
and
B c X - D,
f -
is also.
since
Thus
is a pairwise normal pair.
Now assume
F
is
P-closed and
space is pairwise completely regular,
separated with respect to
Q
Proposition 2.8 there exist a
zero set
such that
A^ H
from
x ^ F.
P
is
Since the
P
x € X - F.
P-zero set
= 0, A^^
Thus i) of Definition $.2 is satisfied.
A^
F
completely
Thus by
and a
and
Q-
x € B^.
Similarly we
could show ii) is satisfied.
The proof of the converse uses a generalization of the
20
method used to prove Urysohn's lemma.
(X,P,Q) has a pairwise normal pairwise sepa­
Assume
rating pair
Since
(5,Q).
(S*, q)
and
€ Ç
such that
F^ € 3
2
F c X - G,
o
X
he a
x € F^
(3,%)
since
and
F
Q-closed set and
is pairwise separating, there are
F^ n G^ =
G^ € Ç
Let
and
F c G^
x ^ F.
F^ € 5
and
is pairwise normal there exist
such that
X - G^ H X - F^ = #
2
and
2
and
2
G, c X - F,. Thus
1
_i
2
2
€ F^ c X - G^ c F^ c X - Gj^. Now F^ H G^ = 0,
so again
2
2
by pairwise normality of
in
3
and
and
2
(3, Q) there are F^ and G^
4
4
respectively, such that X-F^nx-G^=0
Q,
F^ c x - G^
fI
F^, G^
and
4
such that
G^ c: x - F^.
2
è
I
F, c X - G^,
?
and
¥
4
Similarly we get sets
'
I
G, c: x - F_.
Thus we
now have
€ F_ c X - G, c F, c X - G, c F, c X - G, c F,
o
11
11
1
1
4
4
2
2
4
4
By continuing this process we get collections
X
and
^ Q
between
F
O
and
1)
D
such that for
f : X -» [0,1]
f(x) = 1
for
by
x € G^.
X - G,.
1
^^
is the set of dyadic rationals
i, j € D
d X - G . C F . c x - G . c F . c x - G ,.
11
]
]
1
function
and
0
(where
C
and
i < j,
Now define a
f(x) = inf{t € D | x € X - G^}
We show that
f
is a pair
21
(R,R,L),
continuous function into
on
{x}
and
1
on
We first show
X € f~ ^ ( - o o ,a);
0
F.
f : (X,Q)
then
dyadic rational
which is obviously
is continuous.
f(x) = t < a.
t' € D
Let
Thus there exists a
such that
f(x) = t < t' < a.
U (X - G ) so f"^(-oo,a) c U (X - G ).
t<a
t<a
t€D
t€D
Now if X € U (X - G, ), then x € X - G . j t < a .
t<a
o
t€D
f(x) = inf{t € D 1 X € X - G^} = t^ < a, so x 6 f ^(-oo,a)
X
€ X - G. . c
and
U (X - G.) c f ^(-oo^a).
t<a
t€D
f ^(-00,a) =
U (X - G. ),
t<a
t6D
and hence is
Q-open.
Thus
which is a union of
Before beginning the proof that
Q-open sets
f : (x,P) — (R,R)
is
continuous, we prove the following useful fact:
f(x) < a
If
X
assume
a
iff
€ X - G^
for all
for all
t > a,
t € D
then
such that
f(x) < a;
f(x) = inf{t €D|x €X- G^} > a.
t^ 6 D
such that
contradiction.
a
X € X - G^
Conversely, if
such that
i,i $ D, i < i
t.o > a
and
f(x) < a
x
for
Then there is
X - G^ ,
o
a
and there exists
x ^
^ X - G.
t , then
o
f(x) = inf{t € D I X € X - G^} > t^; for by construction,
if
t_o € D
f(x) > t^ > a
t > a.
iff
and
X - G^ c X - Gj.
But this is a
22
contradiction since
f(x) < a.
From the above we have
Obviously
Pi (X - G. ) c fl (X - G )^.
t>a
^
t>a
^
t€D
t€D
the other way, let
exists
f ^(-M,a] =
s G D
c (X - G,
^
r € D
such that
®
f : (XjP) -* (&,R)
Hence the
is
0
on
X
F
Then there
•p
Pi (X - G. ;
t>a
t€D
Now
is
P
Q
P.
Thus
is continuous also.
we have defined is pair continuous and
1
F
being
respect to
f
and
separated from
to
c X - G^.
^
r > a.
H (X - G
c X - G
t>a
^
t€D
for all r > a, so
H (X - G
c H (X - G ) and we
t>a
r>a
tcD
r6D
have
n (X - G
= 0 (X - G ) = f-l(-œ,a]. This means
t>a
^
t>a
^
t€D
t€D
that
CP
To show containment
be such that
r > s > a.
0 (X - G. ).
t>a
t€D
on
F,
so
{x}
with respect to
Q
is
completely separated from
In a similar fashion, each
Thus
Definition "3-3:
(X,P,Q)
completely
which is equivalent
completely separated with respect to
it excludes.
P
{x}
with
P-closed set
Q
from points
is pairwise completely regular.
[1?] A family of sets is called an inter­
section ring if it contains all finite unions and countable
intersections of its members.
23
Definition 3.4:
Q
a family of
elements of
Let
3
be a family of
Q-closed sets.
3
elements of
A sequence
is called a nest in
iff there exists a sequence
Q
such that
Definition 3.5:
P-closed sets and
3
[X -
of
with respect to
of complements of
X -
c
cx- G^ c F^.
A family
5
of
generated with respect to
Q
iff each member of
intersection of a
Theorem 3.2:
3
is
If
P-nest in
3
and
G
P-closed sets in
P-nest
3
is the
3.
Q
are intersection rings where
P-nest generated with respect to
Q-nest generated with respect to
P,
Q
then
and
(3,Ç)
Q
is
is a
pairwise normal pair.
Proof:
Let
exist nests
A € 3, B € Q
{A^}
B = n E . Let
n
and
and
[B^]
{X - G }
n
and
such that
and
sequences of complements.
Choose a member
Vtfhich forms
U
which forms
V.
and a member
If
X - Gn
^ n X - Am = 0 .
X - F^ n X - B^ = 0.
fX - F }
^
n
Define
V = U {X - F^ n X - A^}.
U n V = 0.
A D B = 0.
Then there
A = n A^
and
be the associated
U=UfX-G^ n x - B^}
We first show that
X - G^ H X - B^
of the union
X - F_
m H X - A.m
n > m, X - G^
n c A^;
m'
of the union
hence
If
n —
< m'
, X - F m^ c B n'
^;
Thus in either case,
(X - G^ n X - B^) n (X - F^ n X - A^) = 0;
hence
hence
24
U n V = 0.
Now we show
U
and
V
contain
A
and
B
respectively.
Let
some
X
n
°
€ A.
Since
such that
implies
x € An
A fl B = 0
x
B ,
"o
for all
Similarly
A c U.
Thus
B = fl
x € X - B .
"o
Thus
Q. and
V
x € U
and we have
U
is the complement of a
is the complement of a member of
n{G^UB^}ÇQ
U
since
the desired result.
Similarly
complement of a member of
is the set of
which is
If
5
V
but
5.
U B^ € Q
is an intersection ring.
Q
and we have
could be shown to be the
5.
is the set of
Q-zero sets, then
3
P-zero sets and
Q
is an intersection ring
P-nest generated with respect to
intersection ring which is
to
Q
is the complement of a member of
Proposition 3.1:
xGA
x € A.n +1 c X - G_n^.
o
o
U = U { X - G ^ n x - B^3 = X - n [G^ U B^},
and so
there is
B c v.
It remains to show that
member of
that is
n,
' so that
X € X - B
n x -G .
^o
"o
Thus
and
Q
and
Q
is an
Q-nest generated with respect
P.
Proof:
Let
functions
Fg € 3.
f^
and
F^ = {x I f ^ f x ) = 0 }
fg > 0
[10,p.21].
f^
and
Thus
Then there exist pair continuous
into
(R, R,L)
such that
Fg = [x 1 f2(x) = 0 }
and
f^ > 0,
Fj^ U Fg = {x 1 f^fxjfgfx) = 0}.
25
Since
is pair continuous [10,p.g],
{x I f^(x)f2(x) =0} is a
P-zero set and
U Fg € 3.
Lane [10,Proposition 2.22] shows that
under countable intersection.
is
F € ?.
respect to
Q
tinuous function
f > 0
F =-!xif(x)<—L
n
L
\ /
n;
to show
X -
[F^}
F = f; F^.
such that
Then clearly
1
•K
1
[F^]
^ îîrfTà; -
Q-2ero set since
(S.,L R)
J
LxO,p.$j.
- f
3
with
F = {x I f(x) = 0}.
F = H F ,
n
Q.
Let
so it remains
Let
Then clearlv
X - G - c F -, c X c F„. If we can show
n-RX
n-rl
n
n
Q-zero set, we have the desired result.
r
in
There is a pair con­
is a nest with respect to
= -.x 1 f(x) <
3
Q.
We wish to find a nest
such that
is closed
It then remains to show
P-nest generated with respect to
Let
3
^
G
n
is a
"hich is a
is pair continuous into
26
IV.
PRODUCTS AND BICOMPACTIFICATIONS
Given a family of bitopological spaces, there is a verynatural way to define a product of these spaces.
Definition 4.1:
ffX ,P ,Q )}
be a family of
^ a a a acA
bitopological spaces. The product of this family.
T
is the bitopological space
where
irP
TT (X ,P )
a€A
for
Let
represents the usual product topology for
and
ttQ
represents the usual product topology
TT (X ,0 ).
oGA G a
The following three propositions are immediate conse­
quences of the definition and corresponding results for
single topological spaces [5].
Proposition 4.1:
(X ,P ,Q )
takes
The proiection
^
TT : ir (X
,0 ) -*
a
a' a'
is pair continuous and pair open (that is
TP-open sets to
P^-open sets and
TT
TrQ-open sets to
Q^-open sets).
Proposition 4.2:
product
A function
TT (X ,P ,Q )
a€A
" ct
a € A, TT 0 f
a
f
from
(X,P,Q)
into a
is pair continuous iff for each
is pair continuous.
^
Proposition 4.3:
Let
tinuous functions where
^a€A
^ family of pair con­
fn : (X
\ ny,P ,Q
*ry') -• (Y
^ rt,S
^ ,Tn ^).
Then
27
TT f :
(X ,P ,Q ) - TT (Y ,S .T )
a€A G o€A °
°
oGA ^ a a
is pair continuous.
Since the product of compact spaces is compact in usual
topological spaces we would hope to get a similar result for
bitopolocical spaces.
Fletcher [5] and Kim [9] have given
definitions of pairwise compactness.
a bitopological space
(X,P,Q)
is called pairwise compact
if every pairwise open cover of
where a pairwise open cover
P U Q
such that
C A P
P
with respect to
C
X
has a finite subcover,
is a cover by sets from
contains a non-empty set and
contains a non-empty set.
of
According to Fletcher
C A O
Kim defines the adjoint topology
V € Q
by
p(v) = {p,x} U {U U V I U € P},
and he defines a bitopo­
logical space to be pairwise compact if
for every non-empty element
V
of
pact for every non-empty element
Q
U
P(v)
and
of
is compact
Q(u)
P.
is com­
It is clear
that the real line with the right ray topology and left ray
topology
(ft,RjL.)
definitions.
is pairwise compact under both of these
However
x (ft,R,l)
is not pairwise
compact under either of these definitions, for consider the
cover of
a
x ft by
[(0,oo)
U [(-œ,n} X (-œ,1)]^^^
x
U
{(-oo,l)
which is pairwise open.
x
Obviously
it can not be reduced to a finite cover, so the product of
pairwise compact spaces is not pairwise compact under
Fletcher's definition.
Now
7rL((0,oo)
x (O^oo)), the adjoint
28
topology of
TTL
with respect to
not compact either so
(0, o o ) x (0, o o ) ,
(R, R,L) x (R,R,L)
compact according to Kim's definition.
clearly is
is not pairwise
Olie definition of
bicompactness we shall give has the property that the product
of bicompact spaces is bicompact.
Our definition also has
the desirable property that every pair continuous function
from a bicompact space into
(R, R,L)
is bounded, which is
not the case for either of the previously mentioned defi­
nitions, as one can see by considering the identity function
on
(R,R,L).
Definition 4.2:
A bitopological space
(X,P,Q)
compact iff every pair continuous function from
into (R,R,L)
Definition 4.3:
is pseudo(X,P,Q)
is bounded.
A weak pair
T -,
space is pair real com-
^2
pact iff it is pair homeomorphic to the intersection of a
7rR-closed subset with a
7rL-closed subset of a product of
(R,R,L).
Remark:
Definition 4.3 is a generalization of a character­
ization of real-compactness given by Engelking [2].
Definition 4.4;
A bitopological space is bicompact iff it
is pseudo-compact and real-compact.
29
Example 4.1: ([0,1],R,L)
ness is obvious as
is bicompact, for real compact­
[0,1] = (-co, 1] fl [0,oo).
If it were not
pseudo-compact, there would exist a pair continuous function
f
into
right.
(fô,R,l)
which is unbounded, assume unbounded to the
Then there exists a sequence
that
f(x^) > n
for each
point
with respect to the usual topology and hence
x^
n.
^^n^n€N
This sequence must have a cluster
will also be a cluster point with respect to the
gy.
f~^(-oo, f(x^) + e)
elements of
^ contradiction.
Let
(X,P,Q)
be bicompact.
is pair homeomorphic to the intersection of a
subset
C _
ttR
(R,R,L )
with a
where
topolo­
will then contain infinitely many
f^n^n€N'
Proposition 4.4:
L
x^
irL-closed subset
D
C _
TTL
Then
(X,P,Q)
irR-closed
of a product of
is contained in a product of
bounded intervals with the left ray and right ray topologies.
Proof:
Let
^ttR
h
be the pair homeomorphism between
^ TT ( R, R, L).
bounded since
(X,P,Q)
For
a € A,
is bicompact.
bound for
the map
Let
oh
a product of bounded
Proposit ion 4.5:
is bicompact, then
and
(X,Q)
(X,P,Q)
are compact.
and
is
be a lower
ir oh and b
an
a
a
n
c TT ([a^,b^],R,L),
a€A
intervals.
If
upper bound.
a^
X
Then
(X,P)
30
Proof:
Let
0
be a net in
homeomorphisin from
Then
ho0
X
to
X.
Let
h
represent a pair
C „ fl C _ c tt ([a ,b ],R, L).
a€A
is a net in
C
FL C _ c TT ([a ,b ],R,L).
TTxv
TTj-i
^/'TV
Co O
a cA.
Since the latter set is compact in the usual topology, there
is a cluster point (with respect to usual topology)
ho 0
in
TT ([a ,b J,R, L).
Since both
irR
and
subtopologies of the usual product topology on
and
are closed in the usual topology.
u Li
y € C „ n C - and y is a
ttR
TTL
and a irL-cluster point for
a
P-cluster point and a
(X,P)
and
(X,Q)
TTL
Thus
are
Thus
ho 0
h ^(y) = x
Q-cluster point for
for
R,
TrR-cluster point for
ho 0.
y
0,
is
so
are compact.
Bicompactness is not equivalent to compactness in both
topologies, however, as the following example shows.
Example 4.2:
of
Consider the subspace of
[-1,0) U (0,1]
(R,R,L)
with the induced topologies.
consisting
This
space is compact in both topologies since any cover by left
rays must contain the whole space as an element and simi­
larly for any cover by right rays.
However, if this space
were bicompact it would be pair homeomorphic, with pair
homeormorphism
h,
TT ([a^,b^], R, L;.
sequence
say, to an intersection
Consider the sequence
{h(-H)}n5N
i" SrR ^ Sri, =
a€A
fninCN"
D C^^ c
51
a cluster point
Hence
y
and the
with respect to the usual topology.
will be a cluster point with respect to the
ttL
consider
in}n€N'
y
topologies also.
x = h ^(y).
x
Thus
y €
fl
so
should be a cluster point for
with respect to both topologies, but if
is not an
TTR
L-cluster point and if
x > 0,
x < 0
it
it is not an
R-cluster point.
We show in this chapter that given a weak pair
space
T -,
there exists a bicompact space
such that
(X,P,Q)
(S,P*,Q*)
of
Q*-dense in
is pair homeomorphic to a subspace
(X^,P*,Of)
where
S
is
4
space
P*-dense and
(X*,P^,Q*).
Theorem 4.1:
A weak pair
T -,
(X, P,Q)
homeomorphic to a subspace of a product of
Proof:
Consider the family
P =
is pair
([0,1],R,l).
of all pair
continuous functions
f^ : (X,P,Q) — ([0,1],R,L). Define a
Wf ; (X,P,Q) - ir ([0,1],R,L) by (f(x)) = f (x).
function
By Proposition 4.2
prove
f""^
Let
have
or
is pair continuous.
is pair continuous and
x,y € X
U € P, V € Q
x^V
f
such that
such that either
x€V, yj^V
X € U, y
U
and
and
f
x 7^ y.
is one-one.
Then there exist
x € U, y / U
y € U, x ^ U.
y € V, x ^ V.
It remains to
and
y € V,
Suppose we
Then there exists
52
a pair continuous
f (X - U) =0.
^o
and
(f(y))
= 0.
^o
To show
and
f
€ F such that f (x) = 1 and
^o
^o
Thus f(x) / f(y) since (f(x))
= 1
^o
f ^
X = f~^(y) €U.
xj^X-U
so there exists a pair
continuous map
f
€ F such that
^1
( X - u ) = 0 . The set
A = {y €
f(x)
TT ([0,1],R,L) I y
and
y = f(x) € A.
y' = f(x')
for some
f^ (x") =0,
TrR-open.
weak pair
If
a contradiction.
4.6:
f (x) = 1
^1
fl f(x)
>
x'€X.
Similarly for
Proposition
U € P, y € f(u)
is pair continuous let
y'
If
is
and
irR-open in
is any other point in
y' ^ f(u), x' ^ U
Thus
V F 0, so
A,
so
y' € f(u), f(u) is
f ^
is pair continuous.
The product of weak pair
T^
spaces is
T^.
TT fx ,P ,Q ), x
v where each
acA a a.
(X_,P ,Q^) is weak pair T,. If x ^ y, there exists
^a a a
1
/
^
some a € A such that x
^ y , so there exist
°
*0
Gg/
Proof:
u^o €
Let
x,y €
-a
and
y
e u , x
j ^ V
Go
Go
Go
€ 0,^
such that
^
or
ê
x^^ €
s o suppose t h e latter.
i%
^
Then
and
53
jz €
X
IT (X ,P ,Q ) \ Z
but not
y
and
weak pair
Proof:
is
TTQ-open and contains
jz € TT (X ,P ,Q ) ! Z
^
a€A
a a
open and contains
y
Proposition 4.7:
1
€ V
but not
x,
is
TTP-
the desired result.
4
The product of weak pair
T -,
spaces is
T .
5?
We have just shown in Proposition
product
€ U }
a^j
IT (X ,P jQ )
a(A = °
TP-closed subset and
is weak pair
x^ ^ F.
4.6
T,.
1
X - F
is
that the
Let
F
be a
TP-open and
0 € X - F. There exists a base element
B 3 Xo € B c X - F. We wish to find a function which is
X
at
X
o
B =
n IT (U )
i=l °'i^ °'i
X
oa-
and
€ U
a.
for each
such that
0
on
where
for each
i,
X - B.
U
°^i
is a base element
are all
P
°^i
open.
) = 1
so by weak pairwise
and
f.(X
F. c 77 : 77-(X jP ,Q ) ~* ([0,1],R,L)
1
a^
^a a a
^
Let
B
T , condition
-2A
2
there exists a pair continuous function f^
f.(x^
i
Since
- U
) = 0.
pair continuous and
ç(x ) = 1
and if
y
1 < j < n
—
—
so
for some
j
Then
is pair continuous.
^
g(x) = min{f. o - (x) I i = 1,2,...,n}.
"i
U
1
Then
y )e B,
f.(y' ") = 0
]^
^
g
then
so
is
54
g(y) = 0,
hence
g(X - B) = 0 .
tinuous function
X - B =3 F,
g
which is
1
Thus we have a pair con­
at
x
and
0
on
the desired result.
Theorem 4.2:
The product of a family
[
bicompact spaces is bicompact.
Proof:
pair
From Proposition 4.7 we have the product being weak
T .
It remains to show that the product is pseudo-
compact and pair real-compact.
Since each
each
a
is bicompact, we can find for
a pair homeomorphism
the intersection of a
such that
Consider the map
h =
pair continuous.
onto
7r(C„
aSA
from
onto
TrR-closed set
closed set
By Proposition 4.5,
h^
ir h
a€A
h
Since
n C T ),
n
defined on
with a
c
[a^,b^],R,L)
TT (X ,P ,Q ).
a€A
° °
is pair continuous and
h
FL-
h ^
is obviously one-one and
is
h
is
it remains to show that
7r( C _
P I C ) is the intersection of a TrR-closed set
a6A
with a TL-closed set in order to show real compactness.
T
a €A
Engelking [2,p.71]
a cA
a €A
ir C _
is TrR-closed and
v C ^
is
aSA.
a€A
Trii-closed in
tt ( rr (R,R,L)1.
a6A X€Aa
If
TT (X ,P ,Q ) is not pseudo-compact, there exists
o€a/ a' a'
55
a pair continuous function
f
on the product which is
unbounded (assume to the right).
fx }
such that f(x ) > n
n ntN
^ n
induced sequence j^h
Thus we can get a sequence
for each
n.
Consider the
" =,rla) =
which is compact in the usual topology since it is the
product of compact spaces.
It has a cluster point
respect to the usual topology, but
point with respect to the
gy, so
y_ € TrC„
°
a€A
^o ^
fl
t( X ,P ,Q )
^ a
a
x^,
hfx^) = y^
and
^^n^n€N
topolo­
x^
is a
^
S)
an obvious contradiction.
is pseudo-compact and hence
a
TTL
Thus there is an
TrQ-cluster point for
contains infinitely many
with
is also a cluster
topology and the
irC^.
a€A
s^-^h that
TTP- and a
Thus
-R
y^
y^
bicompact.
The following corollary is immediate from the Theorem
and Example 4.1.
Corollary
4 .1:
Proposition
7 r ([0,1 ], R, L)
A
4.8:
is bicompact.
P-closed (or
Q-closed) subset of a
bicompact space is bicompact.
Proof:
Let
(X,P,Q).
^
A
be a
(X,P,Q)
^
P-closed subset of a bicompact space
is pair homeomorphic by
RjL).
Thus
h(A)
h
is
to
TrR-closed,
36
so
h(A) = (h(A) N
N
closed set with a
the intersection of a
/rL-closed set.
If
A
TTR-
were not pseudo-
compact, there would be an unbounded pair continuous
function
f
on
A.
Assume it to be unbounded to the right.
Then there exists a sequence
f(x^) > n
for each
n.
Proposition 4.5
(X,p)
P-cluster point
x^.
f~^(-oo, f(x^) + e)
diction.
Thus
Since
^
(X,P,Q)
is compact; hence
Since
A
is
Q^)
such that
is bicompact, by^^n^n€N
P-closed,
contains infinitely many
(A,
Proposition 4.Q:
^^n^n€N
&
x^ € A
x^,
so
a contra­
is bicompact.
The intersection of a
P-closed subset
with a
0-closed subset of a bicompact space is bicompact.
Proof:
Let
Cp
P-closed and
is
(X,P,Q)
pair homeomorphic by
be bicompact and
is
h
A = Cp fl
Q-closed.
to
Since
h(Cp)
is
-R-closed and
(h(Cp) P. C_^) n (h^Cg) n
closed set with a
(X,P,Q)
is
n
h(A).h(CPNCG) = h(Cp)nh(CQ) = (h(Cp) n
and since
where
n (h(Cg) n
^(^ )
Q
is
TrL-closedj
is the intersection of a
—L-closed set and
A
irR-
is real compact.
Pseudo-compactness follows by extending slightly the same
argument used in the proof of Proposition
Definition
4.4:
A bitopological space
called a bicompactification for
4.8.
(X*,P*,Q*)
{X,P,Q)
provided
will be
57
(X*,P*,0*)
is bicompact and
to a subspace
dense in
S
of
X*
(X,P,0)
where
S
is pair homeomorphic
is
P*-dense and
Q*-
X*.
Theorem 4.3:
If (X,P,Q)
is a weak pair
T,
bitopologi-
^2
cal space, then there exists a bicompactification for
(X,P,Q).
Proof:
Since
(X,P,Q) Is weak pair
homeomorphic to a subspace
TT ([0,l],R,l).
a€A
A^^ n A^^,
N
A
A
T
it is pair
of a product
obviously is
—R-dense and
-rL-dense in
and by Proposition 4.9 and Theorem 4.2,
is bicompact in
ir ([0,1],R,L). Thus a desired
a€A
bicompactif ication would be (A^^ fl A^^,7rR,TrL).
The question of whether every pair continuous function
from a pair
T ^
space into
(ft,R,L)
can be extended to a
pair continuous function on the bicompactification is not
known.
58
V.
GENERALIZED QUASI-METRICS AND QUASI-UNIFORM SPACES
The concept of a uniform space [7] is a generalization
of a metric space.
void family K
i)
A uniformity for a set
of subsets of
each member of
'V.
X x x
such that
contains the diagonal
ii)
if
U Ç. M,
then
U ^
iii)
if
U 6 ^,
then
Vov c U
iv)
if
U
and
V
v)
if
U €
"U
and
and
is a non-
X
for some
are members of ^ ,
U c V c: X x X,
V c
then
then
A;
;
U fl V € t/;
V cî/.
A quasi-uniform space is a non-void family "U
sets of
XXX
satisfied.
of sub­
such that i), iii), iv), and v) above are
The symmetry condition ii) is deleted.
A quasi-
uniform space is a generalization of a quasi-metric space.
Recall that a quasi-metric is a metric except that the usual
symmetry condition is deleted.
A uniformity ^
topology
sets
6
U.)
T(
of
X
for
so
^
X
induces in a natural way a
where
T(^)
x € U[x] c G.
is the set of all sub­
x € 9,
there exists
By the symmetry condition,
are the same, where ^
T(^) = T(^^).
= {U ^ 1 U
and
When ^ is a quasi-uniformity,^ ^
need not be the same as %( ;
T(t().
X
such that for each
U € K such that
^ and
for
thus
Thus a quasi-uniform space
natural way a bitopological space
T(tt ^)
(X,t()
may differ from
induces in a
(X, T(t()_, T(tC^)).
59
In [6], G. K. Kalisch showed that spaces possessing a
generalized metric are uniform spaces and uniform spaces are
We generalize these results to
generalized metric spaces.
generalized quasi-metrics and quasi-uniform spaces.
In what follows, we let
I
be a non-void partially
ordered directed set, which means that for all
i,j,k
in
i), ii) and iii) hold:
i)
i < j
and
j < i implies
i = j
ii)
i < j
and
j < k
i < k
iii)
i,j € I
i < k
We also let
and
<
implies
implies there exists
and
G =
and
where
I,
such that
j < k.
R
is the set of real numbers
is the usual ordering on
in both ft
k E I
S2.
Note that we use
_<
relying upon context to distinguish
the usage intended.
Definition 5.1:
For
exists a
j € I
such that
exists a
k f I, k < j,
all
i < k,
i)
ii)
g^ <' g^
g^(i) > ggCj)^
such that
iff if there
then there
g^fk) < g2(k)
and for
g^fi) < ggfi)•
Definition 5.2:
function from
g^^g^ € G,
Let
S x S
S
be a non-void set and
G
such that for all
6(s,t) >'0
6(s,t) = 0 » s = t
6
a
s,t^r € S,
I.
40
iii)
Then
6(s,t) <' 5(s,r) + ô(r,t).
(S,5)
will be called a generalized quasi-metric space.
Now let ^ denote the collection of all sets of the
form
N(e ji^, ig,..•
where
e
= {g € G lg(ij)l < e, j = 1,2,...,n},
is a positive real number and
a finite subset of
define
I.
For
{ij^, i^,..., i^^}
N(e;i^,ig,...,i^} € d.,
ig,...,1^) = {h € G 1 h <' g
g € Nfeji^,ig,...,i^)},
is
and let
8
for some
denote the collection
of all such sets.
We can then form a bitopological space
defining
P
and
Q
as follows;
U^*(s) = {t 1 6(s,t) € N*}
and
For
call
P
by
s € S, N* € iS,
let
U^^(s) = {t I &(t,s) € N*}.
It is easy to show that the collections
{U^^(s)3
(S,P,0)
{U^^(s)}
and
form neighborhood systems for topologies which we
and
Q
Now let
respectively [?].
= {(s,t) i t € U^^(s)}
collection
and consider the
order to show this collection
forms a base for a guasi-uniformity for
S,
it is suf­
ficient to show the following [10,p.65]:
i)
ii)
0 ^ ^^N*^N*€lS
For
^N* ^
iii)
A c
N^,
€ S,
there exists
^ ^N*
for all
N* € i®
N* € IS
such that
41
iv)
For
N* € fi,
o
for some
N| € B.
i) is obvious.
For
N|,N^ € G
N* C= N* n N|.
that
satisfied.
clearly there is
Thus
H
c
iii) is obvious.
N*
such
and ii) is
Before proving iv), we prove
a lemma.
Lemma 5.1:
Proof:
p <' q
Suppose
and
r <' t
k < j
say,
such that
q(i) < p(i).
If
and
r(i) < t(i)
for all
such that
such that
Then we have
(p + r)(n) < (q + t)(n)
hence
i < k,
n
and
i < n
(p + r)(i) < (q + t)(i)
for
p + r <' q + t.
netting
r
then there exists
such that
.
^
there exists an
for all
and for all
Now to prove iv) above, we show
=
we are
However, if there
r(m) > t(m),
r(n) < t(n)
there is
i < k,
(p + r)(i) < (q + t)(i)
an
i < n;
or
p <' q,
and for all
m < k
all
Since
(p + r)(k) < (q + t)(k).
r(i) < t(i).
Then
q(j) > p(j)
q(k) < p(k)
exists an
n < m
so
q(j) > p(j).
finished since then
i < k
p + r <' q + t.
(q + t)(j) > (p + r)(j).
g ( j ) + t ( j ) > p(j) + r(j);
t(i) > r(i),
implies
such that
- \
* * * n'
(s, t) €
(s,r) 6
^
.
. \
^
4-2
and
(r,t) €
Ihus
&(s,r) € N*(e/3; i^,...,i^)
so there exist
6(s,r) <
g^
and
6( r , t ) € N*(e/5i
€ N(e/3; i2_j • • • j ij^)
and
&(r,t) <' g^.
G(s,t) <• &(s,r) + 6(r,t) <
9l + 92 «
.. . ,
such that
By Lemma $.1,
g^ + g^;
but
as
1(91 + S2)(i]^)l = 191(1%) + ggC^k)! 5
< e/5 + s/5 < Ei
therefore
+ 192(1%) I
5(s,t) € N* ( e ; i^^, ..., i^^)
and
(s,t) e
Hence we have shown i) - iv) are satisfied; therefore
is a base for a quasi-uniformity "K
We now show that
Let
such that
3 € P
P = T{^)
and
s 6 9-
s € U^^(s) c 6.
Ug,*[s] = [t I (s,t) €
s € U^*[s] c 8;
s F 8,
Since
so
U €
We have shown
Q = T(1^ ^)•
Then there exists
But
6 6 T(% .
U^^(s)
€ "U and
Now if
U € t(
there exists
Thus
S x S.
= ft i 6(s,t) e N*3 = U^*(s).
then there exists
Uj-^ c U.
and
on
0 € T(W)
such that
where
Similarly
N* € B
Q = T(î/
and
s € U[s] c 8.
s c U^*(s) = U^^[s] c U[s] c 8
P = T(t().
Thus
such that
and
8 € P.
.
So far in this chapter we have taken a generalized
quasi-metric space
uniformity
(s^i)
and we have exhibited a quasi-
such that the bitopological space induced by
the generalized quasi-metric
&
is the same as the
;
45
bitopological space induced by the quasi-xiniformity X .
Ihus we have proven the following:
Theorem 5.1:
Every generalized quasi-metric space is a
quasi-uniform space.
We now show the following:
Theorem 5.2:
Every quasi-uniform space
is separated
(nt( = A),
(X,*U)^
vAiere
is quasi-uniformly isomorphic to
a generalized quasi-metric space.
Before beginning the proof of this theorem, we prove a
generalization of a theorem from Kelley.
The next
three definitions are due to Reilly [16]:
Definition 5.2:
spaces.
Let
(X,W)
The function
f : (X,t()
continuous if for each
that
(x,y) € U
Definition 5.5:
spaces.
f
and
If
f ^
V 6
implies
Let
f:X - Y
and
(Y, V)
be quasi-uniform
(Y,y)
is quasi-uniformly
there exists
U € (/ such
(f(x),f(y)) € V.
(X,^)
and
,V)
be quasi-uniform
is a one-to-one function such that
are both quasi-uniformly continuous, then
f
is a quasi-uniform isomorphism.
Definition 5.4:
form space
(X,î^
A real valued function from the quasi-uni­
is quasi-uniformly upper semi-continuous
(q.u.u.s.c) if for each
e > 0
there is a
U 6 Î/
such that
44
(x,y) 6 U
implies
f(y) < f(x) -f e.
f
is quasi-uniformly
lower semi-continuous (q.u.l.s.c) if for each
V Ç
is a
such that
Theorem S.3:
(x,y) 6
implies
e > 0
there
f(y) > f(x) - e.
is quasi-uni-
Each quasi-uniform space
formly isomorphic to a suLspace of a product of quasipseudo-metric spaces.
Proof:
By Reilly [16, Theorem 2.13], ^
family
P
of all quasi-pseudo-metrics which are q.u.u. s.c
.-1 xt/).
(X X X, u~
on
f ; X -• Z
Xp
Let
be defined by
Z = 7rf(X,p) I p € P}
(f(x))p = x
for all
be assigned the quasi-uniformity
and
Z
We need to show
f
and
and
x € X.
Let
generated by
the product quasi-uniformity.
one-one.
is generated by the
Obviously,
f ^
f
P
is
are quasi-uni formly
continuous.
Let
W
be a subbasic element of the product quasi-
uniformity for
Z,
that is
W = {(u, v) | (u^, y^) €
the quasi-uniformity generated by
implies
U
is such that
f
€
since 1/ is induced by
(x,y) € U
implies
But
P.
U
€^
Thus
(f(x),f(y)) € W,
U
€^
and
is quasi-uniformly continuous.
Now to show
U
p}.
^
f ^
is quasi-uniformly continuous, let
be a subbasic element of^ , U
= [(x,y) | p(x,y) < e}
45
for some
p € P, e > 0.
Choose
V = {(u,v) e f (x) X f (x) I (u ,V ) e u } .
P P
PG
îhen clearly
(u,v) 6 V
and
implies
(f~^(u),f~^(v)) E U
quasi-uniformly continuous.
isomorphism from
X
Hius
f(x)
onto
f(x)
isomorphic to a subspace
f
f ^
is
is a quasi-uniform
and
X
is quasi-uniformly
of a product of quasi-
pseudo-metric spaces.
Proof of 'Theorem 5.2:
By the previous theorem we can let
the quasi-uniform space
to a subspace
(X,^
of a product
Z
be quasi-uniformly isomorphic
P =
TT P ,
where
P
(i€M ^
are
^
quasi-pseudo-metric spaces with quasi-pseudo-metries
which generate *U .
sets of
M.
For
m € M ,
where
component of
distance in
be the set of all finite sub­
x,y € Z,
r
x
P
|i
Let
p^
define
&(x,y) = (r^)
for
= sup(dist(x ,y )) where x
is the
1J.€M
^ ^
^
in P^ and dist(x^,y^) is the ordinary
between
x
and
n
y , and
n
(r_)
^ m'
is an
MQ
element of
G = R
& : Z X Z -* G
.
makes
It remains to show this definition of
(Z,6)
a generalized quasi-metric space
which is quasi-uniformly isomorphic to
Obviously
X
0 <
S(x,y)
= y, 6(x,y) =0. If
(x,y)
U
as
X
for all
Y3
is separated.
(xfU)•
x^y € Z,
there is
and if
U € %( such that
^ is generated by
46
so there is a finite number of
n
U =) n U
^ . As
i=l Pp^Gi
there is some
So
p
, e.
^i
^
(x,y) / U, (x,y) X
k, 1 ^ k < n,
(x,y) >
p
such that
such that
n
n U
i=l
(x,y)
and there is at least one
,
i
so
U
^ .
%^
m
such that
r r 0. Hence (r^) = ô(x,y) / 0. To verify that 5 is a
m'
^ m
^
'
generalized quasi-metric, we have yet to show that the
triangle inequality holds.
Let
ô(x,y) + 6(y,z) = (r^) + (r^) = (r^+
5(x,z) = (r^).
then
If
r;; > r^ + r^
o
o
o
for some
and
m^ €
sup (dist(x ,z )) > sup (dist(x ,y ))
)j€m^
^€m^
+ sup (dist(y ,z ));
nat.^
"
hence there is a
Uq ^ m^
such that
Pu (x ,2 ) > sup (dist(x ,y )) + sup(dist(y ,z )).
•"^o '"^o '^o
LiGno
""^o
o
ii€mo
'"^o
o
Hence
"
contradiction.
(Z,6)
Thus the triangle inequality holds and
is a generalized quasi-metric space.
(x/U)
Now it remains to show that
is quasi-uniformly
isomorphic to the quasi-uniform space induced by the gener­
alized quasi-metric space
uniformly isomorphic to
(Z,5).
Z
We have
quasi-
with the product quasi-uniformi-
ty, so it is sufficient to show
Z
with the product quasi-
uniformity isomorphic to the quasi-uniform space induced by
47
(Z,6).
We show the identity function is a quasi-uniform
isomorphism.
Clearly the identity function is one-one, so it is
sufficient to show
tinuous.
Let
induced on
V
Z
i
and
i~^
be an element of the quasi-uniformity
by
V„„/ ,
, \
N*(e;kj_,...,k^J
6.
Then
V
n
m = U k. and consider
° j=l ]
< e ,j = 1,2,...,n}.
Z.
since
If
(x,y) € U,
^
and
i
then
formity.
m^ =
subsets of
6(x,y) € N*(e;k^,—,k^)
r^ = sup {p^(x^,y|_^)}
3
P-€kj
therefore
(x,y) « V(e;lc^,..
is quasi-uniformly continuous.
Now let
by
U = {(x,y) € Z x Z 1 p (x,y) < e,
^
ô(x,y) = (r^) where
-
Let
Let
which is an element of the product quasi-uniformity
for
c V
contains a base element
as defined before where
N*(s;k^,= {(r^) 1 r^
^
are quasi-uniformly con­
U
Then
be an element of the product quasi-uniU
contains a base element
I i = 1,2,...,n}
m^;
m = {k.}f
'• 1/1=1
then
m
and let
^
^ = 1.2.•••.'>}•
m
be the set of all
is finite and can be represented
Consider then
. \
N*( e;k^,...,k^j
element of the quasi-uniformity induced on
(x.y) €
Z
by
a base
6.
If
5(x,y) f N*(e;ki,...,k,);
48
so
ô(x,y) = (r^),
Thus
where
sup {p (x ,y )} < e;
ia€k^ m ^
r^
< e
hence
for
i = 1,2,
sup {p (x ,y )} < s;
k^6m
hence
p
(x ,y ) < e for i = 1,2,...,n, which means
^i ^^i ^i
fx.y) € U„„/
\ c U. Thus i~^ is quasi-uniformi-
ly continuous.
Ihis completes the proof of Theorem 5-2-
49
VI.
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2
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51
VII.
ACKNOWLEDGEMENTS
The author would like to express his appreciation to
Dr. J. C. Mathews for his generous assistance and many help­
ful suggestions during the preparation of this thesis.