SOME RESULTS IN SEMI-METRIC SPACES
by
ROBERT EUGENE STUBBLEFIELD, B.S.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
Approved
Accepted
August, 1972
_•--' c3-'
"3
Z?f ? 1
(..•Cpt 3t,
PREFACE
This thesis was prepared under the direction of
Dr. Harold Bennett.
It has one major purpose.
That
purpose is a concise compiling, for the first time,
of major results in the study of semi-metric spaces.
It also includes several examples answering questions
raised concerning semi-metric spaces along with implications noticed during the research and writing period.
In Chapter I is an introductory history of semimetric spaces along with definitions and several inherent properties plus a most useful characterization
of semi-metric spaces.
Chapter II presents conditions under which semimetric spaces are developable spaces or even metric
spaces.
Also included are some results concerning
continuity of the semi-metric function.
Chapter III presents an example of a regular
Lindelof semi-metric space Y such that Y x y is not
normal.
The question concerning Y a normal semi-metric
space implying Y x Y normal has been an outstanding
problem (see Michael [26]).
I wish to express my thanks to Dr. Harold Bennett
for his direction during the creation of this paper.
Also
gratitude is expressed to the other members of my committee. Dr. Ali Amir-Moez, and Dr. Charles Kellogg.
11
TABLE OF CONTENTS
PREFACE
ii
LIST OF FIGURES
iv
I.
CHAPTER I
Introduction
1
Properties
4
II.
CHAPTER II
22
III.
CHAPTER III
42
BIBLIOGRAPHY
46
111
LIST OF FIGURES
Figure 1
Figure 2
8
Figure 3
17
Figure 4
44
IV
CHAPTER I
Introduction
Semi-metric spaces were given little attention prior
to a systematic study begun in 1950 by F. B. Jones.
How-
ever, work in semi-metric spaces was done by Frechet,
1928 [15]; Wilson, 1931 [31]; Chittenden, 1917 [12];
and Niemzytzski, 1927 [28].
Since a regular semi-metric need not be developable,
it is remarkable that many theorems concerning developable spaces have analogies in semi-metric spaces [16].
Now the definitions of a semi-metric space, a
developable space, and a metric space are presented.
For the purpose of this study all topological spaces will
be assumed to be Hausdorff.
All undefined terms and
notations will be as in [14], in particular Z
will
denote the set of natural numbers, and if A is a subset
of a topological space X, then cl^(A) denotes the closure
of the set A in X and Int(A) denotes the interior of the
set A in X.
Definition 1.1.
Let X be a nonempty set and d:X x X -^ R
be a distance function such that:
(1) d(x,y) = d(y,x) >_ 0 for all x, y e X.
(2) d(x,y) = 0 if and only if x = y.
(3) d(x,z) ^d(x,y) + d(y,z) for all x, y, z e X.
Then the function d determines a topology on the set X
and the pair (X,d) is called a metric
function d is called the metric
Definition 1.2.
space.
The
function.
Let (X,x) be a topological space.
If
there exists a sequence of open covers {G | n e Z } of
X, such that for each x E X, {st(x,G )| n e Z } forms a
neighborhood base at x, then (X,x) is called a
ahle
space
and {G | n e Z } a development
develo-p-
for X.
Developable spaces are of interest since a regular
developable space is a Moore space.
See [27] for a large
collection of work in Moore spaces.
Definition 1.3.
Let (X,T) be a topological space with
a distance function d:X x X ^ R
such that:
(1) d(x,y) = d(y,x) >^ 0 for all x, y e X.
(2) d(x,y) = 0 if and only if x = y.
(3) For A c X, X e cl.-(A) if and only if
d(x,A - {x}) = Inf{d(x,y)' y e A - {x}} = 0.
Then (X,T,d) is called a semi-metri^'
space,
and a function
d satisfying (1) and (2) is called a semi-metric
function.
Notice that (3) implies that the semi-metric function d
preserves the topology on X.
In [4] an example is given
showing that a set with a function d satisfying (1), (2)
and using (3) of (1.3) as a definition of a limit point
need not generate a topology.
Clearly the open covers of a metric space (X,d)
by 1/2
X.
spheres about each point form a development for
Thus each metric space is also a developable space.
If a space, (X,x) has a development (G | n e Z }
then define
d(x,y) = Inf{l/n| x e g, y e g for some g E G }.
Clearly d(x,y) = d(y,x).
If x = y then d(x,y) = 0 and
if d(x,y) = 0 then x, y e g e G
for all n e Z . Since
only Hausdorff spaces are considered, x = y.
Now, if X is a limit point of A c; X, then in every
G
there is a member g containing x and this g also
contains some y e A, x j^ y,
since g e G
is open in X.
Hence d(x,A -{x}) = Inf{d(x,y)| y e A - {x} } = 0.
Conversely if d(x,A - {x}) = 0, then there is a sequence
{y } in A such that lim^_^^d (x,y^) = 0.
open set containing x.
Let O be an
Then for every n e Z , there is
an m e z"^, i^ 1 ^' such that there is a member g in G^^
with X, y. e g c 0 for some i e Z . For suppose there
is no m e z"^ such that some g in G
for some i e Z^.
contains x and y^
Then d(x,{y.}) > 1/n, for i e Z .
Hence there is a y. e g c O for some i e Z+ , g e G
for
some m >_ n.
Thus x is a limit point of A.
Properties
The following are inherent properties, examples,
and characterizations of semi-metric spaces.
The first
property is given by the following lemma.
Lemma 1.4.
Let (X,x,d) be a semi-metric space.
If
A c X is a closed set, then d(A,x) > 0 for all x E X - A
Proof:
If x E X - A, then x is not a limit point of A.
Thus by (1.3) d(A,x) -^ 0.
Hence d(A,x) = E for some
E > 0.
Immediate questions arise concerning the relationship between E-spheres determined by the function d, i.e,
S,(x,e) = {y E X| d(x,y) < E } ,
and topologically open sets.
In light of this question
the following results are given.
Theorem 1.5.
Let (X,T,d) be a semi-metric space.
Then
Int(S.(x,£)) is non-void for all x E X and E > 0.
Proof:
Consider the set cl^(X -
d(x,X -
S^(X,E))
X E Int (S. (x,
S^(X,E)).
>^ e > 0, X / cl^(X -
G) )
.
Since
S^(X,E))
and thus
A consequence of the preceding result is the
following.
Theorem 1.6.
Let (X,x,d) be a semi-metrie space, then
(X,x,d) satisfies the first axiom of countability.
Proof:
Let x E X and U a x-open set containing x.
Thus X - U is closed and d(x,X - U) = E > 0.
S,(X,G/2) C
U.
Hence
Choose GQ < E/2 such that E^ is rational
Then Int(S, (x,EQ)) c S , ( X , E Q ) C S , ( X , E / 2 ) C U , and
{Int(Sj(x,E))I E > 0, E rational} is a countable base
for the topology at x.
It should be noted that E-spheres determined by
d need not be open.
An example will be given later
having this property.
But first consider the following
example of a separable, paracompact, non-developable,
semi-metrizable space.
Example 1.7.
plane.
(McAuley [23]):
Let X be the Euclidean
Let the base for the topology on X be:
(1) For X = {x^,x^),
x^ ^ 0, basic open sets are
S (x,£), where p(x,y) = |x - y|, such that
P
S (X,G) does not intersect the x-axis.
P
(2) Figure-eight bowtie regions about x = (x^,0),
i.e. B ( X , E ) = {y| p(x,y) + a(x,y) < E } .
where a(x,y) is the acute angle formed by the
line connecting x and y and the x-axis (see
figure 1 ) .
^
^
a)
^
^ S (X,G)
\ ^
/
\
f
x
-^ ^1
/
\
s
/
Figure 1
"~
Clearly this space is Hausdorff.
Noticing that
the closure of a sphere or a bowtie will add only the
Euclidean boundary, X is seen to be regular.
The upper
and lower half planes considered as subspaces have the
"usual" topology of the real plane.
lower planes are Lindelof.
X-axis as a subspace.
paracompact.
Thus the upper and
The same is true for the
Hence X is Lindelof and, thus,
Clearly the points with rational coordinates
are dense in X and, thus, X is separable.
Notice that any base for the topology of X must
contain an uncountable number of bowtie regions about
points on the x-axis, and thus X is not a second countable
space.
Since a paracompact, developable, Hausdorff space
is metrizable and a separable metric space is a second
countable space, X is not developable.
Consider the following distance function on X,
d:X X X -> R
such that:
(1) d(x,y) = d(y,x) for all x, y E X.
(2) d(x,y) = |x - y| for all x, y both on the
X-axis or both off the x-axis.
(3) d(x,y) = |x - y| + a(x,y) for one point on
the X-axis and one point off the x-axis,
where a is given by (2) of (1.7).
An £-sphere about x = (x^,X2), x^ 7^ 0, and e
less that \x^ - 0 |, is a basic open sphere in X (see
figure 1, a)). An e-sphere, where E is greater than
|x^ - 0 I will be "dumbell-shaped" (see figure 2, a ) ,
b ) , c)). But these spheres still contain basic open
sets about each of their points and thus are open in X.
An E-sphere about a point x = (x^,0) is figure-eight
bowtie-shaped and is open in X (see figure 1 ) .
Since s-spheres with respect to d are open topologically, d must be consistent with the topology of X,
and hence (X,x,d) is a semi-metric space.
The original space built by McAuley [23] , had
8
x^r^v^)
b)
d)
1
->
<;
e)
1—I
I—I
Cf6 oad
ado
c»'c:-
Figure 2
^
c)
f)
(Ii
tf^
1^
E^
Mr-
strict bowtie-shaped neighborhoods about points on the
x-axis (see figure 2, d)). However, it is easy to see
that the looser "figure-eight" bowties are equivalent
to the strict bowties used by McAuley (see figure 2,
e) , f)).
The preceding example has been slightly modified
to generate a topology on X in which no semi-metric
can have all spheres (with respect to the semi-metric)
open.
Example 1.8. (Heath [18]):
Let X be the Euclidean plane
Let the base for the topology on X be:
(1) For X = (x, ,X2) , x^ 7^ 0, basic open sets are
S (x,£), where p(x,y) = | x - y | , such that
P
S (X,E) does not intersect the x-axis.
P
(2) For X = (x,,0), x^ rational, basic open sets
are S (x,E).
P
(3) For X = (x,,0), x^ irrational, basic open
sets are B(x,£) = {y| P(x,y) + a(x,y) < z) .
Let d be defined on X as follows:
(1) d(x,y) = p(x,y), for x = (x^,X2), y = (yi'y2^
x , y^ 7^ 0 or X2 = 0, x^ rational.
(2) d(x,y) = p(x,y) + a(x,y) for x^ = 0, x^ irrational.
10
Then d serves as a serai-metric for this space.
let 6 be any semi-metric on X.
However,
Define
A(n,l) = {(x,0)| X E R - Q and Sg(x,l/n) c B(x,l)}.
Then I = U{A(n,l)| n G z"*"} = { (x,0) | x irrational}.
The topology that the x-axis inherits from X is
the usual metric.
Since it is well known that this
space is complete and the set of rational points on the
X-axis are first category, I is not of the first category.
Thus some A(m,l) is somewhere dense, say A(m,l) = M.
Since M is somewhere dense there exists a point p = (x,,0),
x^ rational, such that p is a limit point of M.
Thus
there exists a sequence (x } in M such that {x } converges
to p.
Choose a point R =
{Y-^,Y2^
' yi "= ^i' Yo ^ ^'
such that p E S^(R,l/2m) and R e S.(p,l/2m).
But for
sufficiently large n, notice that R / B(x ,1). However,
S.(x ,1/m) ^ B(x ,1) which implies R / S^(x ,1/m) and
thus 6 (x ,R) >^ 1/m for sufficiently large n.
sufficiently large n, x
Thus, for
/ S (R,l/m), but recall that
S.(R,l/2m) c Sg(R,l/m) and hence S^(R,l/2m) contains at
most a finite number of the points of the sequence {x^}.
Hence S^(R,l/2m) is not open in X.
6
The following result concerns closed sets in a
11
semi-metric space.
Theorem 1.9.
Let (X,T,d) be a semi-metric space.
If
A c X is closed, then A is a G. set in X.
Proof:
Let U. = |j{lnt (S, (x,l/i) ) | x G A}. Then U.
is open.
Since x / A implies d(x,A) = G > 0, x / U.
where 1/i < G. Thus, A =p{u.| i G Z"*"} by (1.3).
In order to more clearly represent the structure of
a semi-metric, the following characterization of semimetric spaces is given.
Theorem 1.10. (Heath [17]):
A topological space (X,x)
is semi-metric if and only if there exists a function
g:Z
X X -> T such that:
^.
(1) For each x e X, {g(n,x) I n e Z } is a nonincreasing sequence of open sets forming a
local base for the topology at x.
(2) If y is a point in X and {x } a sequence
such that for each m £ Z , y e g(m,x ) ,
m
then {X } converges to y
n
Proof:
If X is a semi-metric space, let
g(n,x) = p{Int(S, (x,l/i)) I i ^ n} •
12
Then g clearly satisfies all of the given conditions.
Suppose X is a topological space and there is a
function g satisfying the given conditions.
Let
m(x,y) be the smallest integer k such that y i
g(k,x).
If X 7^ y, then ]c exists since X is Hausdorff.
Now
define a distance function d on X such that:
(1) d(x,x) = 0.
(2) d(x,y) = d(y,x) = minil/m(x,y),1/m(y,x)>.
Thus d(x,y) >_ 0 for all x, y in X and x = y implies
d(x,y) = 0 .
If d(x,y) = 0, then for any n E z
X G g(n,y) or y G g(n,x).
either
In either case, since X is
Hausdorff, this must imply that x = y.
Thus d need
only be consistent with the topology on X to be semimetric.
Let A c X and x be a limit point of A.
Then a
sequence {x } in A can be found satisfying condition
(2) of the function g.
Thus
d(x,x ) = min{l/m(x,x^),l/m(x ,x)}
' n
n
n
approaches zero as n approaches infinity since x E g(n,x^)
for all n.
Thus x is a "distance" limit point of A,
i.e. d(x,A - {x}) = 0.
Conversely if d(x,A - {x}) = 0,
then there exists a sequence {x^} in A such that
lim ^^d(x,x ) = 0 = lim^^^[min{l/m(x,x^),l/m(x^,x)}].
13
Assuming ^ ?^ x^, for all n G z"*", then either (1)
l/m(x,x^) approaches zero or (2) 1/m(x ,x) approaches
zero.
If (1) occurs, then there exists N such that
if n > N, x^ £ g(n,x).
If (2) occurs, then there
exists N such that if n > N, r > N, x
E g(r,x).
In
either case, x is a topological limit point of A.
Hence,
d is a semi-metric for X.
The following characterization for developable
spaces is presented for future reference.
Theorem 1.11. (Alexandrov and Niemzytzski [2J): A
regular topological space (X,x) is developable if and
only if there is a semi-metric function d for X such
that:
(1)
lim^^^d(x^,p) = lim^^^d(y^,p) = 0 implies
(2)
lim^^^d(x^,y^) = 0.
Using the characterizations of semi-metric spaces and
developable spaces given by (1.10) and (1.11) respectively,
the examples previously given can easily be seen to be
semi-metric spaces but not developable spaces.
The following questions concerning semi-metric
spaces naturally arise.
Are products of semi-metric
spaces semi-metrizable?
Are quotients of semi-metric
14
spaces semi-metrizable?
Is the range of a semi-metric
space under a perfect map a semi-metric space?
In
light of these questions the following results and
examples are given.
Theorem 1.12.
A countable product of semi-metric spaces
is a semi-metric space.
Proof:
Let {(X ,d ) n E Z } be a countable collection
n n '
A.
of semi-metric spaces.
Let n{X | n E Z } have the
Tychonoff product topology.
functions on X
Place bounded distance
as follows:
n
d*(x,y) = min{d^(x,y),1/2^} for x, y E X .
n
n
n
Clearly d* is still a semi-metric for X^.
Define d on
n{X I n G z"**} by the following:
d(x,y) = -li^i^^i'^i^' where x, y £ n{X^| n e Z } and
X., V. are the ith coordinates of x, y respectively.
1
-^ 1
Now -Ii^^i^^i'yi) converges for all x, y E n{X^| n e Z }
since it is bounded by ^|^l/2^.
Clearly
(1) d(x,y) = d(y,x) >^ 0,
(2) d(x,y) = 0 if and only if x = y.
15
Thus d need only be consistent with the topology on
n{x^| n G Z } to be a semi-metric space.
I
+
-I-
Let {x^l a G Z } be a sequence in n{x | n G Z }
which converges to some p G n{x | n G Z"*"} . Then
i'^v,(x„)| a E: Z } converges to TT (p) in X , where TT is
" "
n "^
n
n
the nth projection map.
Thus d* (TT (X ) ,7r (p) ) approaches
n n a
n
*^^
zero in X^ as a approaches infinity. Hence d(x ,p)
approaches zero in n{X^| n G z"*"} as a approaches infinity.
If d(x a ,p) approaches zero in n{X n I n
G Z"*"},
then
d* (TT^ (x^) ,7T^(p)) approaches zero in X . Thus
{TT11 (x oo) I a G Z } c o n v e r g e s i n Xn t o TTn (p) .
(x cx II a G Z+ } c o n v e r g e s t o p i n n { xn I| n G Z }+
Thus
and t h e
theorem is established.
Since an uncountable product of the unit interval
with the usual metric topology is not a first countable
space, the uncountable product of semi-metric spaces
need not be a semi-metric space.
Quotients are not as cooperative as products.
Even
if the quotient space elements are compact subsets of
the original space, the quotient need not be a semi-metric
space.
spaces.
Hence, perfect maps need not perserve semi-metric
16
Consider the bowtie space of (1.7) with the
following equivalence relation:
(1) xRx for all x ^ 10,1] .
(2) xRy for all x, y e [0,1].
The induced quotient space must be a first countable
space to be a semi-metric space.
Suppose (3 | n G z } is
a countable base for the point [0,1] in the quotient.
Each 6
is the union of basic bowtie open sets.
6n = U^B(x
,Ga')I
^ a'
' a G An},
where A is some index set.
n
Since [0,1] is compact, a
finite collection iB(x^ ,G^ ) | i = l,...,n} is sufficient
i
to cover [0,1].
[0,1] CU{B(X^
i
Thus
,G^ ) | i = l , . . . , n } c U ^ B ( x ^ , e ^ ) |
i
i
a e A^}
Let
3' = UlB(x^
,G
1
)I i = l , . . . , n }
1
and thus (3' I n E Z"^ } is a base for [0,1].
n
Let K be
^^
the collection of Icnot points of 3^1^ (see figure 3 ) ,
that is K c: {X I i = l,...,n}.
n
a•
Hence, there is only a
17
i
countable number of Icnot points in all 3', n e Z .
Choose X £ [0,1] such that x is not a Icnot point of
any 3^, for any n E z . Then B(x,2) is an open set
containing [0,1], but no 3' is contained in B(x,2).
n
Thus the quotient cannot be a first countable space
and, thus, not a semi-metric space.
Since inverse
images under this quotient are compact, this example
shows that the image of a semi-metric space under a
perfect map need not be a semi-metric space.
Knot Point
V
.--
\ .
Figure 3
/
w^
18
The following discussion involves a result only
recently resolved.
Definition 1.13.
A topological space (X,x) is said to
be suhparacompact
if for every open covering U of X
there is a sequence of open covers (U | n E z"*"} such
that for any x G X there is a positive ihteger n(x)
and u G U with st(x,U , v) c u.
n (x)
Definition 1.14.
A collection, D, of subsets of a
topological space (X,x) is called discrete
if each
X £ X has a neighborhood that intersects at most one
element of D.
Definition 1.15.
F -screenahle
A topological space (X,x) is called
if and only if for each open cover H of X,
there exists a sequence {X.} such that:
(1) For each i, X. is a discrete collection of
closed sets, each of which lies in an element
of H.
(2) \J{X, I i G Z"^} = X.
Recently, Burlc [9] has shown the equivalence of
F -screenable and suhparacompact.
a
Theorem 1.16. (McAuley [22]):
(X,x,d) is suhparacompact.
A semi-metric space
19
Proof:
of X.
Let a denote a well-ordering of an open cover H
For each h G H, let
p(h,i) = {y G h| y ji h' for all h' preceding
h in a and S^(y,l/i) c h}.
Let X. = {p(h,i)| h G H}. Each X. is discrete, for let
X E X and ho be the first element in H such that x G h_
Consider Sj(x,l/i), and for some A < 3, suppose
S^(x,l/i) n P(h^,i) ?^ <}..
Then let y be an element of this intersection, which
implies that S, (y,l/i) c h^ which implies x G h^ since
X G S-,(y,l/i).
Thus h„ is not the first element of H
d
p
containing x and hence
S^(x,l/i) n P(hj^,i) = <^ for all A < 3
Suppose
S. (:x,l/i)p p(h ,i) 5^ 4, for Y > 3
a
Y
Then there exists y G h^ such that x z S^(y,l/i) c h
But by definition x G h
implies x i h . Hence
p
Y
S^(x,l/i) p p(h ,i) = <^ for all Y ?^ 3-
20
Thus X^ is discrete.
For each x r x there is a first element of H with
respect to a containing x.
Since this h G H is an open
set it must contain some c-sphere about x.
X G p(h,i) for some i G Z
X.
Hence
and IJ^X. | i G Z"*"} contains
Thus consider {cl^(X.)| i G Z"^}, where
A
1
cl^(X^) = {cl^(p(h,i)) I h G H}
which is still discrete and thus X is F -screenable or
a
suhparacompact.
The question arises whether a topological space
satisfying the first axiom of countability is a semimetric space.
The following example answers this question
negatively.
Consider the ordinal space [0,f^[ from [14], p. 66,
where Q. is the first uncountable ordinal.
first countable space.
space.
This space is a
Assume [0,^[ is a semi-metric
Then [0,f2[ x [0,^[ is a Hausdorff semi-metric
space by (1.12).
Because [0,Q[ is Hausdorff, its diagonal
is a closed set and, thus a G. set by (1.9).
Note that
0
a topological space (X,x) is linearly orderable if there
is a linear order on X whose induced topology agrees with
the topology on X.
It is shown by Lutzer 121] that a
linearly orderable space with a G -diagonal is metrizable.
w
21
Since [0,J^[ is linearly orderable, if [0,fi[ is a
semi-metric space, then 10,fi[ is a metric space.
However, [0,f2[ is Icnown to not be a metric space and,
thus, not a semi-metric space.
CHAPTER II
In this chapter conditions are given under which a
semi-metric space is a developable space and even a metric
space.
Some results concerning continuity of the semi-
metric function d are also given.
Definition 2.1. (Arhangel' slcii [3]): A completely
regular space (X,x) is called a p-space
if in its
Stone-Cech compactification 3(X) there is a sequence
of families (C I n G Z }, where C
n'
is a collection of
n
open sets in 3(X), which cover X and satisfy the
condition that for each x G X, n{st(x,C^)| n G Z } c x.
The sequence of families (C | n G Z } is called a
pluming
for X in 3(X).
Definition 2.2.
A collection, P, of subsets of a space
(X,x) is called a network
if for x G X and O open in X
with X G O, there is a p G P such that x G p c O.
The following lemmas and theorems (due to Burlce and
Stoltenberg [10]), are needed to prove the major result
of this chapter.
Lemma 2.3.
If (X,x) is a topological space with the
property that every open cover of X has a a-discrete
refinement, then X is suhparacompact.
22
23
Proof:
Let U be any open cover for X and suppose
P = U^P^I n G Z } is a a-discrete refinement of U with
n
P
a discrete collection of closed sets.
For each p G P,
piclc U(p) G U such that p c U(p). For each n G Z
an open cover U
define
as follows:
(1) If X G X and X £ p for some p G P
U^(x) = U(p) r\ [X - U{p"| p' G P^, X / p'}].
(2) If X /
Thus U
n
n G z"^.
U P I P G P^},
U^(x) = X - U ^ P I P G P ^ } .
= {U (x)I X G X} is an open cover of X for each
n
'
^
For each x G p G P , st(x,U^) c st(p,U^).
z £ st (p,U ) .
^
n
Let
Then
z £ Uj^(y) = u(p^) n [X - U p ' I P' e Pj^. y / P'>K
for some y G X, such that y G p^ G P^, or
z G U^(y) = X -
UPI
P
e P^^
for some y e X, such that y / U p l P ^ ^n^ *
case, U (y) contains some b G p G P^^.
"^^ ^^® first
Thus, since P^ is
discrete p^ = P and hence z G U^(y) c U(p).
case cannot occur since U^(y) H P = "l^ • '^^^^
st(x,Uj^) c st(p,U^) e U(p),
The second
24
if x I p u P^.
Since every x G X is contained in some
p e P^ for some i G Z , X is suhparacompact by (1.13).
Lemma 2.4.
Any topological space (X,x) with a a-discrete
networlc is suhparacompact.
Proof:
Since X has a a-discrete networlc any open cover
has a a-discrete refinement by (2.2).
Hence by (2.3)
X is suhparacompact.
Lemma 2.5.
If (X,x) is a p-space and x G X is such that
{x} is a G. set in X, then x has a countable neighborhood
base.
Proof:
Let (C | n G Z } be a pluming for X in 3(X) and
suppose { x } = n ^ o | n G Z } , 0
open in X.
For each n,
there is an O', open in 3(X) such that 0^ = X p 0'. Thus
n
n
n
{x} = [P{0'| n G Z"^}] n [n{st(x,C^)| n G Z"^}] by (2.1).
n
n
Hence {x} is a G^ set in 3 (X) and must have a countable
0
neighborhood base since
3(X) is compact.
For suppose
(x) has no countable local base in 3(X) and
{x} = f]{\J I n £ z"^}, Uj^ open in 3 (X) .
Since 3(X) is a regular space, the U^s may be chosen
such that cl^^^j ^^n+1^ ^ ^n'
-^^ ^ ^^^
^° countable base
25
in B(X), there is an open set U containing x such that
U^ 9! U for all n e z"^. Now
n{cl^^^j (U^) I n G Z"^} = (x),
thus consider the open cover of 3(X) given by
{U,{X - cl^^^j (U^) I n G Z"*"}}.
Clearly no finite subcover will cover 3 (X), thus x has
a countable local base.
The relative local base system
in X will be countable and the proof is finished.
Theorem 2.6. (Arhangel•skii [3]). A topological space
(X, x) with a a-discrete network, P, is semi-metrizable if
and only if it is first countable.
Proof:
Since a semi-metric space is first countable,
one direction of the proof is obvious.
other, let U
To prove the
= {U(i,x)| i G Z } be a countable base at
X and l e t P = { p | n G Z } , b e a
a - d i s c r e t e network for X
n
Define
M(n,x) = U^P e Pjl J 1 n, X / p)
g(n,x) = U(n,x) - M(n,x) for each n G Z
Notice that M(n,x) is closed since each P., for j <^ n.
26
is a discrete collection.
Without loss of generality,
assume U(i+l,x) c U(i,x) and hence g(i+l,x) c g(i,x)
and {g(i,x)| i G Z } is a nonincreasing local base at x.
Thus g is a function satisfying the first condition of
(1.10).
Suppose X G X and y = {y } is a point sequence
such that X E g(n,y ) for each n G Z . Note that if
X E p E P., for j <^ n, then y
of X containing x.
£ p.
Let 0 be an element
Then there is a p E P., for some
i E Z , such that x G p C O. Because x G p, y. must be
contained in p and hence x is a limit point of ^y„^'
Thus, the second condition of (1.10) is satisfied and
(X,x) is semi-metrizable.
Lemma 2.7.
A p-space (X,x) with a a-discrete network,
P, is semi-metrizable.
Proof:
Let P = {P I n G Z } be the a-discrete network
n'
for X.
If X G X, the let
F(n,x) = X - U{P e PjI ^ e P^ 3 i n } .
It follows that {x} = n[F(i,x)| i G Z"^}. For suppose that
there exists y ?^ x and y G f^F (i,x) | i G Z }.
each n G z"*", if y G p G P^, then x G p.
Then for
But since (X,T)
is a Hausdorff space, there must be an m E Z
and P e Pj^
27
such that y £ p and x / p since P is a network.
Hence
a contradiction exists and {x} = rKF(i,x)| i E Z"^}.
Thus by (2.5), X is a first countable space with a
a-discrete network and is semi-metrizable by (2.6).
Lemma 2.8.
A topological space (x,x) with a development
^G^l n £ Z } has a a-discrete network.
Proof:
A developable space is a semi-metrizable space
and thus each G^ has a a-discrete closed refinement
n
P(n,i) by (1.15).
Thus
{P(n,i)|
(n,i) E Z"*" X Z"^} is
clearly a a-discrete network for X.
The following theorem, due to Burke and Stoltenberg
[10] gives the necessary framework for the upcoming theorem
concerning metrizability of semi-metric spaces.
Theorem 2.9.
For a completely regular space (X,x) the
following are equivalent:
(1) (X,x) has a development;
(2) (X,x) is a p-space with a a-discrete network;
(3) (X,x,d) is a semi-metrizable p-space.
Proof:
It is well known that a completely regular
developable space is a p-space.
Thus (1) implies (2)
and (2) implies (3) by (2.8) and (2.7).
(X,x,d) is suhparacompact by (1.16).
If (3) holds,
Let {C^| n E Z }
28
be a pluming of X in 3(X). For each x E X , let
S^(x,l/n) be an open set in 3(X) such that
S^(x,l/n) n X = Int(S^(x,l/n)).
Recall by (1.6), {S^(x,l/n)| n E Z+} is a local
neighborhood base at x in X (not necessarily open).
For X E X, let U^(x) be an open set containing x in
3(X) such that
^^3(X)^Un^^^^ <^ S^(x,l/n)
and such that the family ^ol^^^^ (U^(x))| x E X) refines
C^.
Let U(n) = {U^(x) p X| x E X}. Since X is
suhparacompact, for each n E Z , there is a sequence
{U^(n)I m £ Z } of open covers of X such that for each
X £ X, there is a positive integer m(x) and U E U
that st(x,Um /^N
(n)) c U.
\X}
such
Assume, without loss of
generality, that U (n) is refined by U .n (n).
Finally,
for n £ Z , let G be an open cover of X such that G
n
*^
n
refines U„(t) for s < n, t < n and G ,, refines G .
Consider the sequence of covers { G | n G Z } .
Ifx£X
is fixed and k is any positive integer, then there is some
x^
£ X and n^^ E Z
with nj^ >_ k such that
29
st(x,U
(k)) c Uj^(Xj^) c X,
k
since X is suhparacompact.
So
st(x,G^ ) c st(x,U^ (k)) c U, (x ) p X,
^k
""k
^ ^
since G
refines U (k) . Assume n, ., > n, due to the
^k
""k
^"^^
^
direction of refinement of the covers involved.
Thus
st(x G
) c st(x,G ). Suppose O is any neighborhood
' ^k+1
""k
of X, open in X, and that O', open in 3 (x) r is such that
0 = X n O'.
If
r\icl^^^^(\U^))\
k = l,...m}<^ O',
+
for any m G Z , then
[n{clg(x)<Uk<^k>'l k = l,...,in} - O'l m e Z*]
is a decreasing sequence of non-empty closed sets in
6(X) and hence
Now,
^^^H(x)'\(V)l "^ ^ 2*} <=
30
lp(st(x,Cj^)| k G Z"^}] p ip{S^(Xj^,l/k)| k G Z'^}]c
X n [n[s^(Xj^,i/k)| k G z"^}] = n(s^(xj^,i/k)| k ^ z"^),
since cl^
p /vx
vX) (Un (x)) refines both Cn and S'(x,l/n).
u
Hence, if
y G irtcl^(x)^^k^''k^^' k G Z"^} - O'],
then
y G n{S^(Xj^,l/k)| k G z"^}
and thus ix^}
-> y.
But (x^^) -> x, and hence x = y,
which is a contradiction.
Therefore a positive integer
m exists such that
n{cl3(x)(\^^k^^l ^ = 1
m) c 0'.
Thus
st(x,G^ ) c p{st(x,G
m
)| k = l,..wm} c
^
[p{Uj^(x^)l k = l,...,m} p X] c 0- p X = 0.
e X £ st(x,G^ ) e 0 and {G^l n G z"^} is a developm
ment for X
31
Now the following theorem concerning the metrizability of a serai-metric space can be presented.
Theorem 2.10.
A compact semi-raetric space is a separa-
ble metric space.
Proof;
A compact semi-metric space is a completely
regular p-space and by (2.9) is a developable space.
A compact developable space is second countable.
Thus
a compact semi-metric space is regular and second
countable.
Hence a compact semi-metric space is a
separable metric space.
It is easily seen that a regular Lindelof, developable space is a separable metric space.
Example (1.7)
shows that a regular Lindelof semi-metric space need
not be a developable space.
The attention of this chapter is now turned to
other conditions under which semi-metric spaces are
developable spaces.
Definition 2.11. (Alexandrov [1]): A topological
space (X,x) is said to have a point
countable
base B
if every point x G X is contained in only countably
many elements of B.
The property of having a point countable base in
a semi-metric space is one of the topological properties
32
that yield a developable space.
Theorem 2.12. (Heath [20]):
A serai-metric space (X,T,d)
with a point countable base B is developable.
Proof;
For each x G X, let {R(i,x)| i G Z"*"} be a simple
ordering of the set of elements of B containing x.
+
each n G z , let
For
N(n,x) = Int(S,(x,l/n)) and.
h(n,x) = N(n,x) p R(n,x) .
Clearly R(i,x), N(i,x) and h(i,x) are open and contain
X for all i E Z . Let ot be a well-ordering of X, and
+
for each x E X and n G Z , let y[n,x] be the first
element z of X such that x G h(n,z).
X £ h(n,x).
Now z exists since
For each x E X and n G z , define
g(n,x) = N(n,x) p [p{h (i,y [i ,x]) | i <^ n}] p
in{R(j/yIifXj) I X G R(j,y[i,x]) , j 1 n, i £ n}] .
Since x E h(i,y[i,x]) for all i 1 n and
X E n^R(J^y[i'Xl) I ^ E: R(j,y[i,x]), j <_ n, i £ n}
by definition, then g(n,x) contains x and is clearly
open.
Define G
= {g(n,x)| x G X}.
Suppose X E X, Q is an open set containing x and
33
p = {p^} is a point sequence in X such that for each n,
X E g(n,p^).
Let M and L be natural numbers such that
R(M,x) c Q and g(L,x) c
S^(x,l/L) c
R(M,x).
R(M,X)).
R(M,X)
(pick L such that
Notice that for some K, R(K,y[L,x])
Since x E R(K,ylL,x]) fl h (L,ylL,x] ) and p
converges to x by (1.6), there is a natural number H
such that for i >_ H, p. E h(L,y[L,x]) and
ylL,p^] <^ ylL,x]
in a.
Now for all i >_ L, x s g(i,p.) c h (L,y [L,p. ] )
by the definition of g(i,p.).
Thus
y[L,x] >_ y[L,p^]
in a.
Hence, for i > H + L,
y[L,xJ = ylL,p^J.
Therefore for i ^ L + H + K,
g(i,p.) c: R(K,ylL,p^]) = R(K,y[L,xJ) <= Q.
Hence for i ^ L + H + K, g(i,Pj^) ^ QSuppose that {G | n e z"^} is not a development for
X.
Then, there exists an x and an open set 0 such that
X E 0 and st(x,G^) 9^ 0, for all n E Z"^. Thus, for all
34
n, there exists a g(n,q^) E G
n
and g(n,q^) £ 0,
such that x E q(n,a )
n
3
' ^j^
However, the above discussion states
that for some N, g(N,q.J c O.
N
is a development for (X,x,d).
Hence {G I n E Z"*"}
n'
The following results are concerned with continuity
of the semi-metric function d.
Lemraa 2.13.
Let (X,x,d) be a semi-metric space. If
d is continuous in one variable, then X is regular.
Proof:
If X £ X and 0 is an open set containing x,
then there is an n such that x E S, (x,l/n) ^ 0.
Sup-
pose d(x,y) is continuous in one variable and consider
the set cl (S, (x,l/2n)).
If y is a limit point of
S^(x,l/2n), then d(y,S^(x,l/2n) - (y}) = 0.
Thus
there is a sequence {y.} in S^(x,l/2n) such that
lim
d(v,v.) = 0.
1
n-^°°
Since d is continuous in one variable,
lim]^->-ood(x,y.)
= d(x,y).
i
Thus lim^_^
d (x,y.1 ) = d(x,y) i l/2n,
n"^°^
and
cl^(S. (x,l/2n)) c {y G X| d(x,y) ^ l/2n}.
X
ci
Now it is clear that
X G Int(S^(x,l/2n)) c S^(x,l/2n) c
35
cl^(S^(x,l/2n)) c S^(x,l/n) c 0.
Therefore X is regular
Theorem 2.14. (Cook [13]):
Let (X,x,d) be a semi-metric
space.
If d is continuous, then X is developable.
Proof;
Since the serai-metric d is continuous in one
variable, (2.13) implies that X is regular.
two sequences { x | n G Z } , { y | n G Z }
to the same point p.
Consider
that converge
Hence
li"'n.„'i<Xn'P) = limn-^-ityn'P) = 0
and by the continuity of d
lim^^^d(x^,y^) = d(p,p) = 0
Thus by (1.11), X is developable.
The following results concern Cauchy sequences in a
semi-metric space.
Definition 2.15.
Let (X,x,d) be a semi-metric space.
A sequence {x | n G Z"^} in X is said to be Cauchy if,
for every G > 0, there is some k G Z
d(x ,x ) > G for all n, m > k.
n m
—
such that
36
Theorem 2.16. (Burke [8]): if (x,T,d) is a semimetric space, the following conditions are equivalent:
(1) Every convergent sequence in X has a Cauchy
subsequence;
(2) If {x^l n G Z } is a convergent sequence in
X and G is a positive number, then there is
a subsequence (z | n G z"*"} of (x I n G z"^}
n
n'
such that d(z^,z^) < G for all n, m G Z"**;
(3) If F c X and there is a positive £ such that
d(x,y) >_ £ for all distinct x, y E F, then F
is closed.
Proof:
Clearly (2) implies (1). Assume (1) holds.
Let
F c x such that d(x,y) >_ E > 0 for all distinct x, y E F.
Suppose that F is not closed.
Then there is a point y,
a limit point of F, and y / F.
Therefore there is a
sequence in F that converges to y.
By (1) this sequence
must have a Cauchy subsequence, however, since this
sequence cannot have a Cauchy subsequence, F must be closed
Assume (3) holds.
Let {x | n e Z } be a convergent
sequence in X, say {x } -> x.
n E Z
and let E ^ 0.
Suppose that for every subsequence
{ y | n e Z } o f { x | n £ Z }
•'n I
{z
n
•'
n E Z } of
for a l l n > 1.
n '
(y
n
Assume that x 7^ x^ for all
t h e r e i s a subsequence
^
n E Z } s u c h t h a t d ( z , , z ) > E,
1 n —
Then a s u b s e q u e n c e
( z ' | n E Z } of
37
of {z I n E Z } can be constructed such that d(z',z') > E,
"
n m — '
for all distinct n, m E Z . By hypothesis {z'| n E z"*"}
is closed.
However, {z'| n E Z }is a subsequence of
(x I n G Z }and must converge to x.
Since x
?^ z' ,
for all n G Z , a contradiction exists. Hence there is
a subsequence { y | n G Z } o f { x | n G Z }
subsequence { z | n G Z } o f { y | n G Z }
^
n'
•'n'
such that every
c o n t a i n s a z_,
m
for some m > 1, and d(z^,z ) < G. Thus a subsequence
{ z ' | n G Z } o f { z n G Z }
n'
n
c a n be found s u c h
that
d(z',z') < G for n, m G z"^, and (1) is satisfied,
n m
Theorem 2.17.
Let (X,x,d) be a semi-metric space.
If S.(X,G) is open in X for all x
G
X and
G
> 0,
then convergent sequences have Cauchy subsequences.
Proof:
Let F c X such that d(x,y) >_ G for distinct
X, y G F.
e > 6 > 0.
Consider X - F, and let x G X - F.
Let
If S^ (x,6) p F = (f) then x is an interior
^d
point of X - F.
intersection.
If F p S^(x,6) ?^ cj), choose z in the
Then
{S^(x,6) n
is a neighborhood of x and
S^(Z,G)
} - {z}
38
[{S^(x,6) p S^(z,G)} - {z}] p F = (I).
Thus F is closed, and by (2.16) convergent sequences
have Cauchy subsequences.
'^^^o^^^
2.18. (Sims 129]);
space.
If d is continuous in one variable, then S.(X,G)
Let (X,x,d) be a semi-metric
is open for all x G X and G > 0.
Proof:
Let y be a limit point of X - S, ( X , G ) .
For
each n G Z , there is a y^ G X - S ^ ( X , G ) such that
0 < d(y,yn ) < 1/n and thus {y^}
n -^ y.
d(x,y ) >_ G.
For each n G Z"*",
Since d is continuous in one variable,
liinj^^«,<i(x,y^) = d(x,y) >_ G.
Hence y G X - S, (X,G) and S ^ ( X , G ) is open.
Lemma 2.19.
Let (X,x,d) be a semi-metric space.
If
d is continuous in one variable then convergent sequences
have Cauchy subsequences.
Proof:
The proof of the lemma is by (2.18) and (2.17).
In the following it is shown that a semi-metric
space with a continuous semi-metric function need not be
a metric space.
This is a contradiction to a mistake by
39
Brown in [7].
Definition 2.20.
Let r be the closed upper half plane,
{(x,y)I y ^ 0}, in R X R.
Let the base for the topology
on r be:
(1) For X = (x^,X2), x^ 5^ 0, a basic open set will
be S ( X , G ) , where G < x
and p is defined in
(1.7) .
(2) For X = (x,,0) a basic open set will be the
interior of a tangent sphere T ( X , G ) plus the
point X, i.e.
T(X,G)
= {y G r| p((x^,G),y) < G} IJ^^^-
The set r with the induced topology is called the Moore
plane.
Notice that r is a regular, separable, developable
space.
Theorem 2.21.
There exists a semi-metric space with a
continuous semi-metric function, d, that is not a metric
space.
Proof:
Let r be the Moore plane.
Then
G = {S ( X , G ) I X = ( x , , x ^ ) , X 7^ 0 ,
n
p
J- "^
-^
{T(x,l/n)I
X =
(x^,0)}
£ = min{l/n,X2}}U
40
is a development for r.
Define d on r x r as follows:
d(x,y) = d(y,x) = min{l,Inf{l/n| x G st(y,G )}}.
Then recalling the discussion following (1.3), d is
a semi-metric for r.
To see the d is continuous, let x, y both be off
the X-axis and { x } - ^ x , { y } - ^ y .
n
-'n
-^
If there is some
z = (z^,Z2) with z^ -^ 0 and some E > 0 such that
X, y E S
(Z,E),
then clearly lii^n->~^ ^^n'^n^ " d(x,y).
Similarly, if x, y are both on the x-axis and
{ x } - > x , { y } - ^ y , E > 0 , there exists N E Z such
n
n
that y
£ T(y,, E) and x^
G T(X,G)
for all n ^ N.
Thus, for sufficiently small G, y^ and x^ cannot both
be in T(z,6) for z = {z^,0)
and all 6 > 0.
Similarly,
V and X cannot both be in S (a,Y) for a = (a-^^a ) with
-^n
n
P
a^ ?^ 0, a. > Y.
2 ^ ' 2 —
Hence liin„_«,d(x ,y ) = d(x,y) = 1.
"v-x.x^v.
j^_^^ n n
If X is on the x-axis and y is off the x-axis,
then no basic open set about some z = (z^,Z2), -z^ ^ 0'
contains both points.
However, there is an a = (a^,0)
such that X, y G T(a,£) for some G > 0.
Again this
implies that li%_«,d (^n'^n^ " d(x,y).
Thus d is continuous in both variables together.
41
However, this space is not a metrizable space.
But
r is a semi-metrizable space with a continuous semimetric function.
CHAPTER III
In this chapter an example of a regular, Lindelof
semi-raetric space Y such that Y x y ig ^ot normal is
presented.
The question of what conditions force the
product of two normal spaces to be a normal space has
been of major interest for some time. This shows that
semi-raetrizability is not sufficient.
This example is
due to Michael [25] with inspiration from Berney [5].
Example 3.1.
Let X be the Euclidean plane R x R
topologized with strict bowtie neighborhoods everywhere
(see (1.7)).
Let
X ^ A = Q X Q ,
Q
the rationals, and
suppose that U is a neighborhood of A in X.
If V is the
Euclidean interior of U, then V is dense in R x R, and
hence V~
U n U
= {(y,x)| (x,y) E V} is dense in R x R. Thus
contains a dense open subset of R x R, with
respect to the Euclidean topology.
Let 3 be a base for X, with the bowtie topology,
of cardinality b<, . With no loss of generality, suppose
3 is closed under countable unions.
Let
W = (B £ 3| A C B } and write
W = {Ba ' a < f^}
where ^ denotes the first uncountable ordinal. By
transfinite induction, choose {y(a) | a < f^} c X
42
43
such that
YM
C p{B^ pB-^l 3 1 a} - lA U {y(3)| 3 < a}]
for all a < f2. This is possible since p{B, 0 B^"^ | 3 < a}
p
3
contains a dense G^ subset of R x R and thus is
uncountable, while A y {y(3)| 3 < a} is countable.
Let
Y = A U {y(a) I a <fi}y {y"-'-(a) | a < n},
y
^'^^ ^ ^^a' ^a^'
open cover of Y.
^ ^'^oi' ^a^ '
Suppose that M is an
No generality is lost if it is assumed
that elements of M are also elements of 3.
Let M^ be a
countable subcollection of M covering A, and let
N = [J{m\ m £ M, } . Then N = B , for some a < U, since 3
is closed under countable unions.
Then, by construction,
y(3) and y(3)'"''" are in B^ for all 3 ^ ot.
So Y - B^
is countable and hence can be covered by a countable
subcollection M^ of M.
Thus, M^ IJ M^ is a countable
subcover of W, and Y is Lindelof.
To see that Y x y is not normal, it will be shown
that Y is separable (A x A is a countable dense subset)
and that {(y,y~ )| y E Y} is an uncountable, closed,
discrete subset of Y x y.
uncountable.
Clearly {(y,y
)| y E Y} is
Choose a basic open set U containing y
44
such that the angle formed by the horizontal line through
y and the line through y and any z E u is less that v/4
radians.
y
Similarly, choose a basic open set V containing
. Hence V x u""^ = {y""^}.
Thus
U X V = {(y,y"^) } c {(y,y~-^) | y E Y}
and {(y,y
-1
) | y e Y} is discrete (see figure 4)
U
-1
>\7
V
<r
u
•
/
I /<g_
<7r/4
fw^
Figure 4
Let (x,z) £ Y X Y such that x ?^ z"" . Thus
neighborhoods U and V of x and z-1 respectively, can be
45
found such that u p V = (j) . if there is an element of
the form (y,y
) in U x v""*", then y £ U and y"""" E V""''.
This iraplies that y s V which is a contradiction.
(x,z) G U X v~
Hence
and
(U X v"-^) p { (y,y''-^) I y E y} = c|).
Hence {(y,y
)1 y G y} is closed.
Thus Y X Y is not normal, since no space containing
an uncountable, discrete, closed set can be normal.
This result is due to F. B. Jones (see [14], p. 144).
This exaraple shows that the product of normal
semi-metric spaces need not be a normal space.
BIBLIOGRAPHY
[1].
Alexandrov, P. S., "Some Results in the Theory
of Topological Spaces, Obtained Within the
Last Twenty-Five Years", Russian Math.
Surveys, 15 (1960), pp. 23-83.
[2].
Alexandrov, P. S. and Niemzytzski, V. v., "Der
Allegemie Metrisatienssatz Und Das
Syraraetricaxiom", (In Russian, German Summary),
Mat. Sbornik, 3 (1938), pp. 663-672.
[3].
Arhangel'skii, A. V., "Mappings and Spaces",
Russian Math. Surveys, 15 (1960, pp. 87-114.
[4],
Bennett, H. R. and Hall, M. H., "Semi-Metric
Functions and Semi-Metric Spaces",
Proceedings of the Washington State University
Conference on General Topology, March (1970),
pp. 34-38.
[5].
Berney, E. S., "A Regular Lindelof Semi-Metric
Space Which has no Countable Network",
Proceedings of the American Mathematical
Society, 26 (1970), pp. 361-364.
[6].
Bing, R. H. , "Metrization of Topological Spaces",
Canadian Journal of Mathematics, 3 (1955),
pp. 175-186.
[7].
Brown, Morton, "Semi-Metric Spaces", Summer
Institute on Set Theoretic Topology, Madison,
Wisconsin, American Mathematical Society (1955),
pp. 64-66.
[8].
Burke, D. K., "Cauchy Sequences in Semi-Metrics",
Proceedings of the American Mathematical
Society, 33 (1972), pp. 161-164.
[9].
Burke, D. K. "On Suhparacompact", Proceedings
of the American Mathematical Society, 23
(1969), pp. 655-663.
HO].
Burke, D. K. and Stoltenberg, R. A., "A Note
on p-Spaces, and Moore Spaces", Pacific
Journal of Mathematics, 30 (1969), pp. 601-608.
[11].
Ceder, S. G., "Some Generalizations of Metric
Spaces", Pacific Journal of Mathematics,
11 (1961), pp. 105-125.
46
47
[12].
Chittenden, E. W., "On The Equivalence of Ecart
and Voismage", American Mathematical
Society Transactions, 18 (]Q17), pp i^i-i^c;,
[13].
Cook, H., "Cartesian Products and Continuous
Semi-Metrics", Topology Conference, ASU,
(1967), pp. 5S-rr,
[14].
Dugundji, J., Topology, (1968), Allyn and Bacon
Inc., Boston.
[15].
Frechet, M., Espaces Abstracts, (1928), Paris.
[16].
Heath, R. W. , Arc-Wise Connectedness in SemiMetric Spaces, Ph.D. dissertation.
University of North Carolina at Chapel Hill,
(1959).
[17].
Heath, R. W. , "Arc-Wise Connectedness in SemiMetric Spaces", Pacific Journal of Mathematics,
12 (1962), pp. 1301-1319.
[18].
Heath, R. W., "A Regular Semi-Metric Space for
which there is no Semi-Metric Under Which
all Spheres are Open", Proceedings of the
American Mathematical Society, 12 (1961),
pp. 810-811.
[19].
Heath, R. W., "On Certain First Countable Spaces",
Topology Seminar, Wisconsin, (1965) , pp. 103113.
[20].
Heath, R. W-, "On Spaces With Point Countable Bases",
Bulletin Des L'academie Polonaise Des Sciences,
13 (1965) , pp. 393-395.
[21].
Lutzer, D. J., "A Metrization Theorem for Linearly
Orderable Spaces", Proceedings of the American
Mathematical Society, 22 (1969), pp. 557-558.
[22].
McAuley, L. F., "A Note
wise Normality and
Proceedings of the
Society, 9 (1955),
[23].
McAuley, L. F., "Relation Between Perfect
Separability, Completeness and Normality
in Semi-Metric Spaces", Pacific Journal
of Mathematics, 6 (1956), pp. 315-326.
on Complete CollectionParacompactness",
American Mathematical
pp. 64-66.
48
[24].
McAuley, L. F., "On Semi-Metric Spaces",
Suraraer Institute on Set Theoretic Topology,
Madison, Wisconsin, American Mathematical
Society (1955), pp. 60-63.
[25].
Michael, E. , "A Regular Lindel'df Semi-Metric
Y such that Y x y is not Normal", Unpublished
Manuscript.
[26].
Michael, E., "The Product of a Normal Space and
a Metric Space Need Not be Normal", Bulletin,
American Mathematical Society, 69 (1963),
pp. 375-376.
127].
Moore, R. L. , Foundations of Point Set Theory,
Vol, 13, (1932) , American Mathematical
Society Colloquim Publications, New York.
128J . Niemzytzski, V. V., American Mathematical
Society Transactions, 29 (1927), pp. 507,513.
129].
Sims, B. T., "Note on an Example of Heath",
Mathematical Notes, October (1963),
pp. 854-855.
[30].
Willard, S., General Topology, (1970),
Addison-Wesley Publishing Company, Inc.,
Reading, Massachusetts.
131].
Wilson, W. A., "On Semi-Metric Spaces",
American Journal of Mathematics, 53 (1931),
pp. 361-373.