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SPACES WITH THE UHOS PROPERTY

A Dissariohon
I'm- IIne Degree oI pk. D.

MICHIGAN STATE UNIVERSITY
Leon Brewster Hardy
I976

 

THESIS

Date

0—7 639

This is to certify that the

thesis entitled

.. 9':
'

SPACES WITH THE UHOS PROPERTY
presented by
Leon Brewster Hardy
has been accepted towards fulfillment

of the requirements for

Ph . D . degree in Mathematics

 
   

P. H. Deyle

 

 

Major professor

April 30, 1976

 

 

ABSTRACT

SPACES WITH THE UHOS PROPERTY

BY

Leon Brewster Hardy

A topological space has uniformly homeomorphic open
sets (has the UHOS property) if all nonempty open sets
are homeomorphic. We prove that the category of spaces
with this pr0perty is large, and that the rational and

1 are in this category.

irrational numbers in E

A topological Space, X, is said to be invertible if
for every open set U in X, U # ¢, there is a homeomor—
phism hCZ}{satisfying h(X-—U) C U. We prove, in Chapter
II, that every topological space embeds in an invertible,
UHOS—space.

Characterization theorems for the rational and irra-
tional numbers with respect to the UHOS property are pre—
sented in Chapter III.

we prove, in Chapter IV, that compact, UHOS-spaces are
connected.

In Chapter V, we prove that a topological space, X,

has the UHOS property iff any nondense set D c X may be

taken into X-—D by a homemorphism h:X 4 X-—5.

SPACES WITH THE
UHOS PROPERTY

BY

Leon Brewster Hardy

A DISSERTATION

Submitted to

Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1976

To my wife Nellie, and my family

ii

ACKNOWLEDGMENTS

I would like to thank Professor Patrick Doyle for
suggesting my research problem, and for his patience and
encouragement during the entire course of my dissertation
work. I would like to thank Professor H. Davis for his
suggestions and discussions. Finally, I thank A. wardlaw,

R. Parson, B. Smith and N. Sheth for pointing the way.

iii

TABLE OF CONTENTS

Page
INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1
CHAPTER
I. NONTRIVIAL EXISTENCE . . . . . . . . . . . 3
l. The Definition and Existence of
Spaces with Uniformly Homeomorphic
Open Sets . . . . . . . . . . . . . . . 3
2. Examples . . . . . . . . . . . . . . . 7
3. Spaces that are Groups . . . . . . . . 13
II. THE EMBEDDING THEOREM . . . . . . . . . . . 18
III. THE METRIC UHOS-SPACES . . . . . . . . . . 23
IV. CONNECTED UHOS-SPACES . . . . . . . . . . . 34
1. Closed and Open Sets in UHOS—Spaces;
Components . . . . . . . . . . . . . . 34
2. Homeomorphic Closed Sets . . . . . . . 36
3. Disconnected UHOS-Spaces with
Clopen Sets . . . . . . . . . . . . . . 36
4. Miscellaneous Results . . . . . . . . . 38
5. The Main Theorem . . . . . . . . . . . 39
V. THE INVERTIBLE CASE . . . . . . . . . . . . 42
BIBLOIGRAPHY . . . . . . . o . . . . . . . 46

iv

INTRODUCTION

This thesis deals with topological spaces with the
property that all non-void open sets in them are homeo-
morphic (UHOS—spaces). Questions about the nature and
existence of nontrivial spaces of this type arose from dis—
cussions of general topology.

Chapter I explores the category C of such spaces,
and establishes familiar spaces such as the rational numbers
and irrational numbers as members of C. The morphisms of
the category are quite unimportant as a rule and for con-
venience may be taken as maps (continuous functions). No
separation axioms are assumed for topological groups in
this chapter.

Chapter II establishes C as a universal embedding
class for all topological spaces. The argument leads to a
corollary showing that every topological space embeds in an
invertible space [1]. A topological space X is invertible
if each proper closed set ‘W in X is carried to its com-
plement in X by some homeomorphism h of X onto X;
i.e. h<3 X and h(W) c X-W’ [1]. Early examples suggested
a strong relationship between invertible spaces and the ob—

jects in C. It is certain that in general invertible spaces

are not UHOS spaces since the n—sphere is invertible,

but not a UHOS-space. A simple example of a UHOS—space
that is not invertible appears in Example 5.1. In addition,
theorem 5.4 shows the weaker relation in a UHOS—space be-
tween a closed set and its complement.

Chapter III deals with metric UHOS—spaces. Our in—
terest is largely confined to the separable case. The
Menger-Urysohn definition of dimension is used. The most
general result here is the existence of an infinite parti—'
tion into open and closed sets of every infinite UHOS—
space. Finally the rational and irrational numbers are
characterized among UHOS~sPaces.

Chapter IV studies properties of UHOS-spaces related
to connectedness. Theorem 4.5.1 establishes the surprising
result that compact UHOS-spaces are connected.

Finally Chapter V deals with the rather weak connec—

tions between UHOS-spaces and invertible ones.

CHAPTER I

NONTRIVIAL EXISTENCE

l. The Definition and Existence of Spaces with
Uniformly HOmeomorphic Open Sets.

In this section of Chapter 1, we prove that a large
number of topological spaces have the UHOS-property. In
particular, we prove that the familiar, but seemingly un-
likely spaces R (the space of rational numbers in E1
with the relative topology), and I (the space or irra—

l

tional numbers in E with the relative topology) have

this property.

Definition 1.1.1. Let X be a non—empty topological

 

space. X has uniformly homeomorphic open sets (UHOS), or
is a UHOS-space or has the UHOS-property if all nonempty
open sets in X are homeomorphic.

Observe that the class of UHOS-spaces along with the

maps between them form a category C.

Lemma 1.1.2. C is a large category.

Proof: Any non-empty set with the indiscrete topology

(at most two open sets) is in C.

 

Lemma 1.1.2. The only finite spaces in C have the

 

indiscrete topology.

Proof: There exists no homeomorphism between two sets

of different cardinality.

Theorem 1.1.3: The irrational numbers belong to C.

 

Proof: Let I c E1 (En is euclidean n—space) be the

irrationals. By a theorem of Hurewicz-Wallman, we can con—

sider the irrationals on the real line [3], page 60. If

U c I is open, then U = I n V, where V is open in El.

n
NOW V = LJVi, where the Vi are disjoint open intervals,
1

and n = l, l < n < m for some positive integer m or

11:00.

Case 1: V = V1, a single open interval.
V-(V-I) is the set of rational numbers in V; a
countable dense set in V. There exists a homeomorphism

g:V 4 El, Which is generally the composition of two homeo-

morphisms; the standard homeomorphism from (-l,+l) to El

[2], and some linear map in E1. This g takes the count—

able dense set {V-—(V-I)} to a countable dense set in

E1. By a theorem of Hurewicz-Wallman [3], page 44, there

exists an onto homeomorphism g’:E1 4 E1 which takes the

countable dense set g{V-(V-I)} to the rationals in E1.

1 is a homeomorphism and

takes the rationals in V to the rationals in El. Hence

NOW the composition g’<>g:V a E

the irrational numbers (V n I) in V are preserved under

g’<3g and this completes the proof of case I.

Case 2: V =lJ V..

Again {V-—(V n 1)} is the set of rational numbers

in V, a countable dense set in V. There exists a homeo-

morphism gi:Vi 4 El, for each positive integer i = 1,2,...,n,

n
as noted in case 1. Then G:CJVi 4.LIE: is a homeomorphism,
l 1

where G(Vi) = gi(Vi), and each E: is a copy of El. G

takes the countable dense set {V-(V n 1)} to a countable

1

i' Again using the theorem of Hurewicz-

n
dense set in LIE
1

wallman employed in case 1, we can construct a homeomorphism
n n
G’:LJE% alJJEi taking the countable dense set G{V-—(V[WI)}
l l
. n 1 . . n
in UEi onto the rational numbers in LJE
1 l

on each Vi n I after using the Hurewicz-Wallman result

1
i

(G is defined
mentioned in case 1 on each ViflI)-

NOW, as a result of case 1, a homeomorphism 9; can
be constructed (preserving rationals) from each Ei, i =
l,...,n, to any open interval. HOwever, we choose our in—

l l

tervals so that gi:E1 4 U1 = (-m,0), 92':E2 4 U2 = (0,1),
Observe that I is contained in the union of these n-intervals,
and inherits the usual subspace topology from this union of

open intervals. Now

where G”(Ei) = g{(Ei) = Ui is a homeomorphism and pre—
serves rational numbers.

The composition (G”<>G’<:G) on {V-(VrWI)} is a
homeomorphism and preserves rational numbers. Hence
(G”<>G’<>G)(V{11) = I, and this completes the proof of
case 2.

Case 3: V=UV..
———-—- 1 1

Again {V-—(erI)} is a countable dense set in V.
There exists a homeomorphism g:Vi 4 E: for each positive

. .. .. 1 _ .
integer. Then G.LlVi 4ILIEi, where G(Vi) — gi(Vi) is a

l 1
co
homeomorphism and LIE: is a countable union of copies of
1
El. G takes the countable dense set {V—-(Vr11)} to a

1

Q
countable dense set in LlEi.

1
each E: is countable and dense in E1. Again using the

The image of {V-—(Vr11)} in

theorem of Hurewicz-Wallman in case 1, there is a homeomor-

phism G’:CIE: 4(3 E: taking countable dense G{V-—(VrWI}
1 1
co co
in (J E: to the rationals in (J Ei. NOW again by case 1,
1 1

a homeomorphism can be constructed (preserving rationals)

from each E: to some open interval in Ll(k,k-+l), k =
k

0, :51, j;2,..., which contains 1. The rational preserving

1

G”:UEi 4 E1 is then evident. The composition, (G”<>G’<>G)

is then a homeomorphism from (erI) to the irrationals in

El, and this completes the proof of case 3.

Theorem 1.1.4. The rational numbers belong to C.

 

Proof: Using the same theorems and similar procedures

as in Theorem 1.1.3, this result follows.

The above results may lead to faulty intuition and

we note: The Cantor ternary set C* does not belong to C.

Proof: The Cantor ternary set C* is compact and per—
fect. C* - {O} is open and not compact in C* and hence

not homeomorphic to C*.

2. Examples of UHOS-Spaces.

1.2.1. Let X # ¢"be any set with indiscrete topology.

 

X is a UHOS-space.

1.2.2. Let X be a one point space. X is a UHOS-space.

1.2.3. Let X = E1 and the topology consists of ¢' and

any open interval about zero (0). This space is

UHOS. X is also Tb but not Tl.

1.2.4. Let Xi be countably infinite, i = l,...,m; Xithj

o’ for i7=’j. Consider Y=Gxi.
1

U in Y is open if H = (Y-—U) is a union of

finitely many Xi'

Y is then a UHOS—space. Y is not TO'

1.2.5. Let X be a countably infinite set, and let U c X

be open if H is finite. X is then a UHOS-space.

1.2.10.

X is T T but not T

0’ 1
finite topology) [5].

2. (X has the co—

Let X be uncountable, and let Uc:X be open if
'6 is (i) finite or, (ii) countable. x is then a

UHOS-space in both cases. X is T Tl’ but not tr

0’ 2'

The rationals form a UHOS-space.

The irrationals form a UHOS-space.

Let X = El. U'c X is open if U = [x > a]a

any real number}. X is a UHOS-space. X is
TO but not Tl!
Let (X,T) be a topological space which has the
UHOS-property. Consider the collection, T, of
open sets in X. We order the open sets in T
by set inclusion, and the pair (T,"c") ‘becomes
a partial ordering. However, the pair (T,"c")
need not be a total ordering.
Define a new topological space (X,T’) as
follows:
1. While (T,"c") is not necessarily totally
ordered, there are chains in T. Consider
a maximal chain in (T,"c").
2. The open sets comprising this maximal chain

will be the open sets in our new topology,

T’.

1.2.11.

Is (X,T’) a topological space? A straight—
forward proof using the definition of chain with
respect to set inclusion shows that our collection
of open sets, T’, is closed under arbitrary unions
and finite intersections. The definition also
shows that the sets ¢ and X are both in T’.
(X,T’) is therefore a topological space.

Is (X,T’) a UHOS—space? Let U,V be open
in T’. U,V are open in T, by definition of
T’, hence there is a homeomorphism h:U 4 V. This
same h applies in (X,T’). Therefore (X,T’)
is a UHOS-space.

This result yields a method of generating

new UHOS-spaces from known UHOS-spaces.

A counterexample to the conjecture that the product
of UHOS-spaces is a UHOS-space.

Z+, with cofinite topology, for each

Let X.
1
i = 1,2,... . Observe that X1 is compact and

a UHOS-Space for each i; hence
m 0
Y = H X. 18 compact.
1 1

But we check the UHOS property in Y.

Let Ci = [li,Zi], i = 1,2,... , and consider

C = n {11,21}.
1

Claim: Y-C is Open.

 

1.2.12.

10

Proof: Let p = [pi] E Y-C. Then there is an

integer j with pj 2-3j in Xj. Let Uj =
[3j,4j,5j,...].l Then (Pl’P2”'"pj-l’3j’pj+l’°°')
3—
belongs to (1'1 X.) xU. x( 1.91 X.), an open set in
1 j . 1
1 3+1
Y-C. NOtice that Uj’ j = 1,2,... generates
an infinite open covering of Y-C, namely,

3-1 m
A={(IIX.)XU.X(I1X.)} .
1 l 3 j+1 1 J=1

But points of the form,

(1 1 3,1 ).j=JIL.H,jn Y-q

1’ 2"°"lj-l’ 3 j+1”°°

are found in only one such covering element. Hence
a finite subcover of Y-C from A is impossible.
Y-—C is therefore not compact. we conclude then
that Y is not a UHOS—space, and that the product

of UHOS-spaces is not necessarily a UHOS-space.

Counterexample to the reasonable conjecture that a
G6 set in a UHOS-space is void or UHOS. (Recall
that a G5 set in a topological space is the in—
tersection of countably many open sets [5]).
Consider the rationals R in E1, which is

as we showed earlier, a UHOS-space. Consider the

open sets in R determined by [Sl(a)[JSl(b)},
'fi '3
a < b rational, n = 1,2,... . NOw,

1.2.13.

11

5 {31(3) U81(b)} = [a,b} , which
“=1 E '5'

has the discrete relative topology as a subspace

of R. But since finite discrete topological spaces
are not UHOS-spaces (see lemma 1.1.2), we conclude
that 66 sets in UHOS—spaces are not generally

UHOS—spaces or void.

N922: Consider the result of requiring X to be
a metric UHOS-space and each G6 in X a UHOS—
space (as a subspace). Using an argument like the
one above, we establish that the two-point subspace,
[p,q], is a G6 set, hence a UHOS—space. But
again, [p,q} has the discrete topology in X. It
follows that X contains at most one point.

The same applies to Hausdorff first countable

spaces.

It will appear in Chapter III that the 66
property is useful in conjunction with the UHOS
property to characterize the irrational numbers.

Another reasonable question is considered in

the following example.

Do onto, open maps preserve the UHOS property?
Consider the rationals, R, which we showed

earlier has the UHOS-property, and the subspace

[0,1] of the rationals. Here {0,1} has the

discrete subspace topology, and hence is not a

1.2.14.

12

UHOS-space. But the function F:R 4 {0,1} de-
fined so that F[x|x rational; x < n] = O and
F[x]x rational; x > n} = l, is open and continuous.
Hence onto, open maps do not preserve the UHOS-

property.

Examples of UHOS—spaces with special closed sets,
i.e., UHOS—spaces in which all nonvoid closed sets
are homeomorphic.
(1) One point spaces have this property.
(ii) Consider example 1.2.4 in which Xi is
countably infinite, i = 1,2,...; Xirwx. =

3
d, i #’j. We defined

and defined U c Y to be open iff 'ii =

n
lei. Y is a UHOS-space, and since non-
l

void closed sets are all finite unions of
Xi's, each Xi countable, we conclude that

nonvoid closed sets are homeomorphic also.

Construction Cti)suggests a scheme for generating
UHOS-spaces with the property that nonvoid closed
sets are homeomorphic using the technique in ex-
ample l.2.4. We only need require that each Xi
be "large enough", e.g. have cardinality c,2c,...,

etc.

l3

3. Spaces that are Groups.

Several of the examples introduced above carry a group
structure. In this section we prove that topological groups
([4], [5], [6]), G, which have the UHOS—property retain this

property in the quotient, G/A, for suitable A.

Definition 1.3.1. Let G be a topological group.
Define a subset A of G as follows: a g A if every
neighborhood of a contains e, and every neighborhood of

e contains a, and A is maximal with respect to this

 

property.

Lemma 1.3.2. A is a subgroup of G.

Proof: Let a,b E A. Consider any neighborhood W

of the product ab—l. The continuity of the mapping

T:(x,y) 4 xy_1 of G xG onto G guarantees the existence
of a neighborhood V, of (a,b) a T(V) c W. But (e,e) e V
(by definition of A and the fact that V = M)<N, M,N
open sets containing e), hence T(e,e) = ee"l = e E W.
Therefore, any neighborhood of ab"1 contains e.
Consider any neighborhood W’ of e. Again the con-

tinuity of T at (e,e) guarantees the existence of a
neighborhood V’, of (e,e) 3 T(V’) c W’. But (a,b) e V’,

hence T(a,b) = ab-1 6 W’. Therefore, any neighborhood of

e contains ab—l. We conclude then that A is a subgroup

of G.

 

14

Lemma 1.3.3. A is an invariant subgroup of G.

Ppppf: The map f 9 f(x) = g xg—l is a homeomorphism
and isomorphism of G onto G. Thus f must preserve A,
the set of all points not meeting any separation conditions
relative to e. Since f(A) = gAg—l, and A is maximal,

gAg = A. We conclude then that A is an invariant sub—

group of G.
Theorem 1.3.4. A is a closed subgroup of G.

Proof: Let x be a limit point of A.
1. Consider an arbitrary neighborhood UX of x. x is a
limit point of A, hence there is an a e A with a e U .

But a 6 UX implies that e 6 Uk.

2. Consider an arbitrary neighborhood, We’ of e. Suppose
x E We'
Preliminaries: a) x-1 is a limit point of A.
Proof: Clear (apply the homeomor—
phism T:x 4 x_1 of G

onto G).

b) Every neighborhood of x-1 contains

e.
Proof: Use a).

NOw, there is a neighborhood, Wé, of e since other—
wise x—1 K A and we are done. Then x,x_l f Welwwé.

However,

15

I
X E X(We Owe) 3
an open set containing x which does not contain e.

(e e x(Wé(1Wé) = e = xy 4 x_1 = y e WérTWé).

The preceding lemmas and theorem allow us to conclude
that G/A satisfies the separation axioms Ti’ i = O,1,2,3,
and is also a topological group (Husain [4], Chapter 3).

we are interested in whether or not the UHOS-property
is retained in the quotient space G/A. The following ex—
ample Shows that the UHOS-property is not generally retained

When passing to quotient spaces.

Example 1.3.5. Consider the rationals, Q, under mul—

 

tiplication, without zero. Let R be the equivalence re—
lation that partitions the positive and negative rationals

into two disjoint sets by means of the open sets,
S = (-oo,...,0) and T = (O,...,+oo), i.e.,

x ~ y e x and y are both in S pp in T. Now, Q/R is
a discrete two point space, hence Q/R is not a UHOS—space.
Note that we might as well consider Q to be a topo-
logical group, in whidh case Q/R a Z2, with the discrete
topology. Consequently the result holds for topological
groups as well.
The following result will be of great importance in

dealing with the quotient space G/A in the upcoming theorem.

 

 

16

Lemma 1.3.6. A is the smallest closed set that con—

tains e.

ngpf: Suppose e E B, a closed set and B g A. Then,
there is an element a e A with a g B. Now, B is an
open set and a E B, and Br1[e] = ¢. This however con-
tradicts our definition of A, so we conclude that A is

the smallest closed set containing e.

Observe that the equivalence classes, {gAzg e G], in
G/A are closed and homeomorphic to A, since left trans-
lation determines a homeomorphism. As in the previous lemma,
it is easy to show that an arbitrary equivalence class, say
gA, is the smallest closed set containing a representative
g. It is clear that under a homeomorphism, these closed
sets are permuted, an observation that is used in the theorem

that follows.

Theorem 1.3.7. If G is a UHOS-space, then G/A is

a UHOS—space. Any set S = {sAzs 6 S], and m = mA.

Proof: we show that every non—empty open set U c G/A

is homeomorphic to G/A. Consider the diagram,

 

 

G/A G
U f1 U
6 flu?) h 4 e ‘1 4. G/A,
u n
[uA:uEU] U {uA}

uEU

17

Where ¢’ is the natural map, and h a homeomorphism from
¢F1(U) to G. (This h is guaranteed since G is a UHOS—

space and ¢P1(U) is open).

(i) ¢h¢rl is onto: Since h is onto of G, and d’ is

-1 .
onto all of G/A, ¢h¢' 15 onto.

(ii) ¢h¢-l is 14. Let 5: a! 5; 6 i1. Then {1(2) 4 {1(9)
(because equivalent elements in G are identified in G/A
under ¢), and h(¢"l(x) ¢'h(¢’1(y)) (h permutes the closed
partition, {gA:g E G], of G). Finally, under d, different

equivalence classes in G are identified with different

points in 6/21. so ¢[h(¢"1(:’c))] # ¢[h<¢'1<9))1.

(iii) ¢h¢rl is open: V c U c G/A open 4 ¢"1(V) c ¢P1(U)
open in G 4 h[¢Pl(V)] open in G. But h[¢rl(V)] is the
complete inverse image of an open set in G/A, hence

¢Th[¢rl(V)]} is open by definition of open sets in G/A.

(iv) ¢h¢fil is continuous: Let V c U c G/A be open. Then
consider (¢h¢rl)—1(V) = ¢h-1¢F1(V). Now h-1¢P1(V) is
clearly open and is the complete inverse image of an open

set in G/A. By definition of d' and open sets in G/A,

¢[(h-l¢"1(V)] is open. Hence ¢h¢rl is continuous.

CHAPTER II

THE EMBEDDING THEOREM

This chapter considers the universal embedding char—
acter of UHOS—spaces, that is, the main result is that
every topological space embeds in a UHOS—space. The final
corollary proves that in fact, every topological space em—

beds as a closed subset in an invertible, UHOS—space.

Theorem 2.1.1. Each topological space embeds in a

UHOS—space.

Proof: Let X be an arbitrary topological space,
and let A = {Ua}aeQ be the collection of open sets in X.
Consider xo-copies of each of the Ua’ a E 0, giving rise

to the collection,

Now, we study the set,

Y = \\// U

0'69 an
nEZ

18

 

19

In our study, it would be helpful to think of each of the
Uan's in B as defining or representing a position in Y;
for example we can speak of the ys position, Y 6 Q,
s€Z+.

Define a basis element in Y as Y, excluding at most
a finite number of positions, those positions being filled
or replaced with d, or some nonempty open subset of the
representative of that position.

(a) The basis so defined is a basis for a topology

in Y. For, let M and N be basis elements in Y,

where

M

(Y‘IU } )VIU'} ,
Bm BEA 6m BEA
mew mew

N = (Y-[U ] )\/[U’ ] , B,m,Y,k indexed
Yk Yee Yk Yee
kes kes

over finite subsets of Q and Z+, and U5 some open

t
subset of UOt’ ‘Where 0 E ALJO, t e WLJS.

Now,

Y -— \/ ({UBm} u {UYk} ) c M M.
AUG {36A Yee
WUS mew keS

and is by definition a basis element in Y. We con—
clude that we have defined a basis for a topology on
Y. (Note that open sets in Y are of the same fOrm

as basis elements).

20

(b) X embeds in Y. Choose any one of the xO—copies

of X in [Uan] , say UHt° The embedding we want
060
nez+

is of course defined by, x._4 x and we only need

ut’
check that with the relative topology on UHt’
X a Upt‘ But by definition of basic open set, open

sets induced by the relative topology do not alter the

structure of Up (The relative topology in U is

11’ [1t
generated by copies of Y with the utEh position

occupied by open subsets of U We conclude that

pt)'
X embeds in Y.

(c) Y is a UHOS—space. Again, let M and N be

two nonempty open sets in Y, where

M

(Y ‘IU l )\/{U’ l ,
5“ BEA 5m BEA
mew mew
N = (Y"{U } )\/{U’ l ,
Yk YEG Yk YEG
kES kES
A,e,W,S indexed over finite subsets of Q and Z+, and
some open subset of Uot’

In accordance with our preliminary discussion, we can

I

U0t oeAue,teWuS.
consider the elements of the indexing set Q as deter—
mining an "open set type". With this notion the de—
sired homeomorphism F:M 4 N is easy to construct.

Each U’ in M and N is homeomorphic to each of

the elements of some "open set type", while it may not

 

21

be homeomorphic to the element of B = [U whose

}
an (169

nEZ
position it occupies. Since the U’ in M and N
are finite in number, they can easily be associated

with elements of the same "open set type", and since

each "open set type" is represented by members,

Yo
a homeomorphism defined piecewise on open set types

yieLdsthehomeomorphism F we desire.

For example, if we have

M=""VU V....VUélvU V....V....,and

nb 52

N = o...V... V .e..VU61\/U62V .........-o :

where Uéb a Uén’ n E Z+, F on the "5 Open set types"

might look like

M: VU \/ Uélv U62V

r
'nb
l l l

N=.....VU61VU62\/....... .

' I _
We Simply have Unb 4 Uél’ then U6c 4 U5(c+l)’ c —
1,2,... . We conclude that Y is a UHOS—space.

Corollary 2.1.2. X is embedded as a closed set in Y.

Proof: Let X 4 Xa be the embedding. By definition,

n

Xan = Y-—(Xan) is a baSic open set in Y, and X is em~

bedded as a closed set in Y.

22

Definition 2.1.3. A topological space X is said to
be invertible if for every non-empty open set U c:X, there
is a homeomorphism h d X (i.e. h:X 4 X) satisfying

h(X-—U) c‘U [1].
Corollary 2.1.3. Y is an invertible space.

Proof: Let S g Y be an open set. We show that there
is a homeomorphism F:Y 4 Y such that F(Y-—S) c S. Rep—

resent S as

S: (Y—[U] )v{u'} ,A,W
6’“ BEA 6‘“ 66A

mew mew

finite subsets of Q and Z+, and U’ c UBm some open

Bm

subset for each B,m. For each UBm’ choose some UBr z

{UBm}BEA’ giving rise to pairs [(UBm,UBr)]. Now, alter
mEW
the identity map I:Y 4 Y as follows. Instead of UBm 4 UBm

as required by I, map UBm 4 UBr and UBr 4 UBm for every
pair in {(UBm’USr)}‘ Call this altered map I’:Y 4 Y. I’
is clearly a homeomorphism and I’(Y-—S) c S by construc—

tion. We conclude that Y is invertible.

Corollary 2.1.4. Each topological space embeds in an

invertible UHOS—space as a closed subset.

Proof: Apply theorem 2.1.1 and corollaries 2.1.2 and

2.1.3.

CHAPTER III

THE METRIC UHOS-SPACES

In this chapter we study the extent to which compact
subsets of UHOS metric spaces, X, are moved by homeomorphisms
h c:X, and the incompatibility of compactness and metriz—
ability. ‘We also characterize the familiar metric spaces
R (rationals) and I (irrationals) in terms of the UHOS—

property.

Lemma 3.1.1. Let X be a metric space and p a point

 

in X. Then for any real numbers a,B, a < B, the set

A = [x]a < d(x,p) < B], is open in X.

Proof: A = [x\d(x,P) < B} n [x|d(x,p) > a}, is the

intersection of open sets; hence A is open.

Lemma 3.1.2. Let X be an infinite metric space or

 

a metric space with a limit point. Then from X, a dis-
connected subset, which is the union of infinitely many

disjoint open sets, can be constructed.

Proof: Let p be a limit point of X. Let pl,p2,...

be a sequence of points in X converging to p, with

 

d(pl,p) > d(p2,p) > d(p3,p) > ..., n = 1,2,3,... . Let
d(p .p)+d(p ,9)
an = n 2 n+1 , n = 1,2,3,... . Now,

23

24

pn E An = [x]an < d(x,p) < an_l], n = 2,3,..., an open
set by the lemma above. If we let Bn = [x[d(x,p) = an],
n = 1,2,... then each Bn is closed, and [LJBnLJ{p]]

is closed in X. Hence X-—[LJBnLJ[p]] is open in X
and is the disconnected open subset we wanted to construct.

If X is discrete the result follows immediately.

Theorem 3.1.3. Let X be a UHOS, metric space with
at least one limit point, and let C c X be compact. Then

there is a homeomorphism h c X such that h(C) c X-C.

Proof: X metric and UHOS implies, as a result of
the lemma above, that X is the union of disjoint open
sets, [Uh]. Now C c X, and C compact implies that fi—
nitely many of these open sets suffice to cover C, say

C C UllJUZLJ...lJUn. Now then,
X: (UlUUzu...uUn) U (Un+1U...),

which is a separation of X, say X = ALJB, where A,B are
open in X. As in a previous theorem, there are homeomor—
phisms hle 4 B, and h2:B 4 A which determine a homeo—
morphism h c X with h(C) c X-C. Simply define h(A) =

hl(A) and h(B) = h2(B).

Theorem 3.1.4. Let X be a compact, nondegenerate,

UHOS—space. Then X is not Hausdorff (T2).

 

 

25

ngpf: Let p be a point in X, and [Un(p)}A the
neighborhood filterbase of p. Then [Ud(P)}A converges
to p, and only to p, if X is T2. (In fact [Ud(p)]
has no accumulation points). NOw, [Ua(p)--p}A is a filter—
base in X-p, but has no point of accumulation. Hence X

cannot be compact [2].

Corollary 3.1.5. There are no infinite compact UHOS

 

metric space.

Proof: Since metric spaces are T2, they cannot be

compact and UHOS!

We note that theorem 4.5.1. asserts that compact UHOS-
spaces are connected. The above corollary is hardly sur—

prising When this result is known

Definition 3.1.6. A topological space X has dimen—

 

sion 0 at a point p if p has arbitrarily small neigh-
borhoods with empty boundaries. A non-empty space X. has
dimension 0 if X has dimension 0 at each of its points
[3]. (This definition of dimension 0 in a space X assumes

X to be a separable metric space.)

Lemma 3.1.7. If U is an open set of real numbers

 

containing a non—countable, O-dimensional, UHOS-space N,
and n is a positive number, then there exists an infinite

sequence of non-overlapping open intervals D1,D2,... ;

 

 

26

each Dn has length <.n, each 5n c‘U, each N11Dn is

non-countable, and the set N-—(D1(JD2lJ...) is empty
(= d).

Proof: N is O—dimensional and separable (because N
is a UHOS-space), hence can be embedded in the space of
irrationals in El since the space of irrationals is a

universal O-dimensional space [3], pg. 64. So without loss

 

of generality, consider an open set, U, of real numbers,
which contains N (as a subset of the irrationals). U is
open in E1, hence can be written as the at most countable
union of disjoint open intervals, [Ui]: .

Let x be any element of N C:U. Then x is in some
Ui = (a,b). Two rational numbers a and B can be found
so that x 6 ((1,6) (2 (a,b), [B-a] < T]: and (3:5) c (a,b).
Now a strictly increasing sequence of rational numbers, say
b1,b2,b3,..., can be chosen in the interval (B,b), con-
verging to b, and a strictly decreasing sequence of rational
numbers, say al,a2,a3,..., can be chosen in the interval
(a,a), converging to a. In addition, sequences can be
chosen so that [a-—al] < n, Ian-an+l] < U and [bu-

bn+l| < n and [bl-—B| < n. Note that ,an) c:(a,b),

(an+l
(bn,bn+l) c:(a,b), and that the set [(an+1,an),(bn,bn+l),

(a,B),(al,a),(B,bl)}m l is countable. Label these intervals
n:

{Ki}

i=1

we sum up the results of our construction as follows.

We have a countable set, [Kn]:, of disjoint open intervals;

27

each Kn has length < h: each R; g Ui = (a,b); each
waKn is noncountable (because waKn = ¢ or is homeo—
morphic to N), and the set (NfiUi)-(D1[JD2LJ-~') = ¢.
This last claim is obvious (and essential in the following
theorem) when we realize that every irrational in (Neri)
is contained in some Ki' (Remember Ns; the irrationals).
But this construction can be duplicated for each one
of the Ui's in [Ui]:, and each Ui gives rise to a
countable number of Kn's. The set of [Kn's] generated

for all the Ui's is countable, and after relabelling them

aSDDD

l’ 2, 3,..., our lemma will be proved.

Definition 3.1.7. A topological space is an absolute
G6 iff it is metrizable and is a G6 in every metric space
in which it is embedded

It can be shown [2], [5] that the irrationals is an
absolute Gé—space, so that the absolute Gé’ UHOS—spaces

hypothesizedj11the following theorem do exist nontrivially.

Theorem 3.1.9. Let X be a O—dimensional, non-
countable, absolute G6 set which satisfies the UHOS—

property. Then X is homeomorphic to the irrational numbers.

Proof: By a theorem of Hurewicz—Wallman [3], X can be

embedded in El (dim n 4 dim 2n-tl), and since X is an

absolute G6 set, it is embedded as a G6 set in E1.

Also call the image of this embedding in E1, X.

 

28

Then there exists an infinite sequence of open sets
G (n = 1,2,...) such that X = Gl nG2 fl"' . Since X
is noncountable etc, and X C G1 and G1 is open, we may
apply our lemma after setting n = 1. Thus we obtain an
infinite sequence D1,D2,... of non—overlapping open in—
tervals, each Dn has length (1., each 5; g G1, each
erDn is non—countable, and the set X-—(DlIJD2LJ---) is
empty (= $20 .

Let n denote a natural number. Since the sets G2

1

and D are open the set G DD is open; since X g G
n1 2 l 2
and XIWD is non—countable, the set XlfiG 0D is no —
n1 2 nl
countable. Applying the lemma to the sets szan , XlfiDn ,
l l

. l . . . .
With n = — we obtain an infinite sequence of non—

2,
overlapping open intervals Dn , Dn , Dn ,...; each
1,1 1,2 1,3
1 _
Dn ,n has length < 2’ each Dn ,n g G2 nDn , each
1 l l
X DD is non—countable, and the set (XIWD ) —(D U
n ,n n n
l l 1,1
Dn U;--) is empty k=¢).
1,2

Further, let nl,n2 be two rational numbers.

Since the sets G3 and Dn are open and the set X11

1’n2

G3 DD is non—countable, we may apply our lemma to
nl,n2

. l
G 0D E nD With n = —.
3 nl,n2’ nl,n2’ 3

Continuing this argument, we obtain for every finite

combination nl,n2,...,nk (abbreviated to ) of natural

“(M

numbers an open interval Dn(k) such that

29

(i) Diameter (Dn(k)) < %,

(ii) Dn(k-1),p “Dn(k—1),q=¢’ P’Iq
(iii)

Dn(k) C Gk r‘Dn(k—1) ,

(iv) X‘WDn(k) is non-countable,

(V) ann(k—l) ' Dn(k—1),l U Dn(k—1),2U"') = 9"

Let N denote the set of all irrational numbers in

the interval (0,1), x a given number of N, and let

LL;

(1) x = ...
ml+ m2+ m3+

be the development of x as a continued fraction. Put

(2) F(x) = 13ml 013ml,m2 “5m1,m2,m3 r‘ -

It follows from (iii) and (iv) that the set (2) is the
intersection of a descending sequence of closed nonempty
intervals and is therefore nonempty; moreover, by (l) F(x)
is contained in an interval of length < i, for k =
l,2,3,... ; hence F(x) consists of a single element which
we denoted by f(x). From (2) and (iii) we have f(x) eGk
for k = l,2,3,... ; so f(x) 6 X. The set T of all the
numbers f(x) for x e N is therefore a subset of the set
X. We next show that the set X-—T is empty h=¢).

To prove this, let

(3) R = (X—S) U (UIXflDn(k) _Sn(k)))

J

30

where the union extends over all finite combinations n(k)

of natural numbers, and where S = DllJDzlJ--- - while

I

(4) Sn(k) = Dn(k),lUDn(k),2UDn(k),3UH'

It is evident from (4) and (v) that the terms of the sum
(union) (3) are empty sets; consequently the set R is
empty.

Let y denote a number of the set X-—R = X. Then
y E X and y f R; so, from (3), y g X-S. But y E X;
therefore y 6 S and since S = DlLJDZlJ--- , there is an

index ml such that y 6 Dn . From y E R and (3), we
1

find that y E (erD -S )7 but since y 6 erD , we
1“1 m1 It‘1

have y E Sm ; hence from (4), there exists an index m2
1

such that y e D .
mi’mz

Continuing this argument, we obtain an infinite sequence

ml,m2,m3,... of indices such that
YEDm(k)’ k=l:213:--~ 0

From (2) we have y e F(x), where x is the number defined
by (l); in virtue of the definition of the set T, this
proves that y E T. Hence X-R = X g T; this gives X-T;
R = ¢ and, X= T. (This proof is due to Mazurkiewicz and
was applied to our particular G6 set X [7]).

It can be shown that N g T, ([7], pg. 239) and the

proof is complete since T = X.

31

Lemma 3.1.10. A UHOS-space, X, containing more than

one point contains no isolated points.

Proof: Let p be an isolated point in X. Then
[p] is open, hence is homeomorphic to X. But Card [p}==1

and Card X > 1, and we are done.
Corollary 3.1.11. Each such UHOS—space is dense—in—itself.
Proof: By definition of "dense-in—itself".

Theorem 3.1.12: Let X be a countably infinite metric
UHOS—space. Then X is homeomorphic to the rational numbers
in El.

Ppppf: By the corollary above, X is dense—in—itself
and countably infinite. The space of rationals R is dense—
in—itself and countably infinite. By a theorem of Kuratowski
[6], pg. 287, countably infinite dense-in—itself spaces are

homeomorphic.

It should be noted that the classical characterization
of the Cantor set C* may be stated as follows: A compact,
totally disconnected perfect metric space is homeomorphic
to C*. If perfect were used in the sense that each point
is a limit point, the above theorem might be stated as fol—
lows: A O—dimensional, absolute Gé’ perfect space is the

irrationals.

32

We remark before proving the next theorem that it is
trivially true for one point spaces, but after this consid—

eration the cardinality of our sets is at least x0.

Theorem 3.1.13. A O—dimensional, UHOS—space is homo—

geneous.

nggfi: Let x,y be two distinct points in X. Be—
cause X is O—dimensional and hence separable and metric
[3], there exist disjoint neighborhoods U,V of x and y
respectively, which contain clopen neighborhoods U1 and

V of x and y.

1

Consider U1 and r1 = d(x,X-—Ul). We can find a

clopen neighborhood U2

spherical neighborhood of x of radius r1. Choose r2 =

d(x,X-—U2), and again we are able to find a clopen neighbor-

of x with U2 C S (x), the
r1

hood, U3, of x with U3 c Sr2(x). Continuing in this
manner we are able to construct a strictly decreasing se—
quence of clopen neighborhoods of x converging to x, say
U1 3 U2 3 ... . (3953: 1. If d(x,X-—Ui) = O for some i,
then the existence of a basis of clopen sets at x is con—
tradicted. 2. If d(x,X-—Ur) = r-—l, i.e., Ur becomes a
spherical clopen set possibly terminating the process above,
we are able to resume the process by considering a clopen

neighborhood of x contained in the spherical neighborhood

Sl(x), where % < r-1, n some positive integer.). Likewise
n

33

we can construct a strictly decreasing sequence of clopen

neighborhoods of y converging to y, say V1 3 V2 3 ... .

Now, Ui--Ui+1 18 open for each 1 = 1,2,..., and Vi-
Vi+l is open for each i = 1,2,... . Because X is a

UHOS—space there eXist homeomorphisms hi:Ui--Ui+l 4 Vi-

V. for each i = 1,2,.

1+1 .. . Define a function H d X

as follows:
1. H(X-(UlUV1))= id(X— (U1UV1))
2' H(Ui‘Ui+1) = hiIUi‘Uiu)

_ -1
3‘ H(Vi—Vi+l) " hi (Vi ‘Vi+1)

 

4. H(x) = y and H(y) = x.

From our construction of the sets U.-—U. and V.-
1 1+1 1
Vi+l’ it is clear that sequences converging to x and y

respectively, converge to H(x) = y and H(y) = x, after

application of H. Hence, H G X is a homeomorphism and

H(x) = y.

 

CHAPTER IV

CONNECTED UHOS—SPACES

In this chapter we study some properties of UHOS-
spaces related to connectedness. The most surprising re—
sult is that compact UHOS—spaces are connected.

1. Closed and Open sets in
UHOS—spaces; Components

Lemma 4.1.1. Let X be a connected UHOS—space, and

let W c X, W # d, W # X be closed. Then W contains no

open sets.

Proof: Suppose U C W is an open set and U # ¢.
Then W is open, WIJU is open and therefore WLJU a X,

contradicting the connectedness of X.

Corollary 4.1.2. Let X be a connected UHOS—space
and let U C X, U # ¢, U # X be open. Then X-U contains

no open sets.
Proof: X-U is closed in X. Apply lemma 4.1.1.
Lemma 4.1.3. Let X be a connected UHOS—space, and

let U c X, U ¢ ¢, U # X be open. Then, 6 = X.

34

 

 

35

Proof: 5 # X implies U is open and H # ¢. Then
UIJH is open, disconnected and UL)? a X, contradicting

the connectedness of X.

As a result of lemma 4.1.3, we can say that non-empty

open sets in a connected UHOS-space, X, are dense in X.
Lemma 4.1.4. All finite UHOS-spaces are connected.
Proof: A finite UHOS—space has the indiscrete topology.

Corollary 4.1.5. Disconnected UHOS—spaces have cardin—

ality at least x0.

Proof: By lemma 4.1.4, if X is a UHOS—space and
disconnected, it has to follow that X cannot have finite

cardinality.

Lemma 4.1.6. Let X be a disconnected UHOS—space.

Then the components of X are infinite in number.

Proof: Let X = UlLJUzlJ"°LJUn be the decomposition

of X into components (i.e., maximal connected sets). Ui
is closed for l = 1,2,...,n, hence Ui = UllJUZIJ-oclJUi_1LJ
Ui+l(J---IJUn is open and UllJ'--(JUi_l(JUi+lLJ--;lJUn a X.

But the number of components in a space is a topological in—

variant, hence we arrive at a contradiction.

36

2. Homeomorphic Closed Sets.

In this section, we study the result of requiring all

nonvoid closed sets in a UHOS—space, X, to be homeomorphic.

Lemma 4.2.1. If X is indiscrete, the topology gives

this property.
Proof: Clear.

Lemma 4.2.2. X and each of its closed sets will be

connected, and the closure of a point is topologically X.

Proof: Let A be a connected subset of X. (There
is at least one, a point for example!). Then A is closed
and connected and homeomorphic to X and every other closed

set in X.

Corollary 4.2.3. No proper closed set in X contains

an open set.

Proof: Let S be a proper closed set in X, and assume
that A g S is open. Then S is open and AIWS = ¢. But

then X a ALJS, which contradicts Lemma 2.

3. Disconnected UHOS-spaces with Clopgp Sets.

The following results assume that X is a disconnected
(hence infinite), UHOS—space in which all open sets are also

closed.

Lemma 4.3.1. No point in X is closed.

37

Proof: If [p] is closed, then X-[p] = [p] is
€25

N
open, and by hypothesis also closed. Then X-[p] = [p] =

{p} is open and homeomorphic to X. This is impossible.

Remark 4.3.2. This shows that no such topological

space can be Tl'

Lemma 4.3.3. All closed sets in X are open.

Proof: Let R c X be closed. Then R is open and

a:
closed in X. Hence R = R is open in X.

Remark 4.3.4. Note that every point, p, in a topolog—
ical space X is connected, and hence [E3 is connected in

X.

Lemma 4.3.5. Let p 6 X. Then [5] is the smallest
open set containing p, (i.e., if p E U, U open, then

[5] c U).

ngpfz Clearly [5] is closed, open and connected,
by lemma 1 and remark 2 above. Suppose there is an open
set U with p E U and {5] ¢ U. Then [in nU is open
and closed, contains p, and is contained in [5]. But
[5] is connected and can contain pp proper open and closed

(clopen) sets.

Theorem 4.3.6. No infinite, UHOS-space, X, has all

open sets closed unless it is ¢, a point, or indiscrete.

38

nggf: Such a space, X, is (i) disconnected, (ii)
the empty set, (iii) a point, or (iv) has the indiscrete
topology. But (i) is impossible since for every point,
p, in X, {5] is open and connected and [E] a X, which

is impossible.

Lemma 4.3.7. A basis for X is a collection of point

closures.

Proof: Let U be an open set in X. Then for each
p E U, {5] is open and [5] C U. Then

U= U [9]-
PEU

4. Miscellaneous Results.

If a UHOS—space has a local or global property, then
it may also have the corresponding global or local property.
Three examples are:

a) X locally connected 4)( connected

b) Each point in X lies in a compact open set 4}{

is compact

c) X arcwise connected 4){ locally arcwise connected.

Lemma 4.4.1. Let X be a connected, T2, UHOS—space.
Let x,y be points in X. X is T2 implies there are disjoint
open sets U,V containing x,y respectively. But ULJV

is open and by hypothesis ULJV a X, which is impossible

 

 

39

since X is connected. Therefore connected, nondegenerate

UHOS-spaces cannot be T2. (Also not T3, not T etc.)

41

Note: This also makes it clear that connected UHOS-spaces

contain no disjoint open sets.

Definition 4.4.2. A topological space X is said
to be rigid if the only homeomorphism from X to itself

is the identity map.

Lemma 4.4.3. If X is a rigid UHOS-space, then X

is connected.

Pppgfz If X is not connected, then let = AIJB
be a separation [5] of X. But A and B are open sets,
hence there are homeomorphisms h and g with h(A) = B
and g(B) = A.

But H:X 4 X defined by H(A) =h(A) =B and H(B) =

g(B)==A is a homeomorphism from X to itself and h # Idx.

Remark: We note that subspaces of UHOS—spaces are
not necessarily UHOS-spaces. For consider the subspace
[1,2,3] in example 5.1. . [1,2,3] and [2,3] are open
in the subspace topology but [1,2,3] é [2,3]. However,
open subsets of UHOS-spaces do inherit the UHOS—property

in the relative topology.

5. The Main Theorem.

Theorem 4.5.1. Compact UHOS—spaces are connected.

Proof: Let X be a compact UHOS-space and assume
that it is not connected. Then X = UIJV, U,V open in
X, Ur1V = ¢. Since V a X, V is not connected, hence

V = UlLJVl, U1,Vl open in V (U1,V open in X also),

1

Ulrwvl = ¢. Since V a X, V is not connected, hence

l 1
Vl = UZLJVZ, U2,V2 open in V1 (X also), szwvz = ¢.
Note that the U's and V's we generate with this process

are compact, clopen sets in X. The V's generated by this

process also give rise to a nested sequence,

Consider a maximal chain of clopen sets containing this

sequence, say {Va}a6T' We observe that,
(IV = D
I. a

is a compact, closed subset of X. D is not open (otherwise

maximality is contradicted).

Lemma 4.5.2. For every open set U with D = 0 Va c U,
I

there is some V with D = n V CV CU.
B T a 5

Proof: Suppose no such VB exists. Then (VBIWE) #
d, for every B e r. Clearly (Variu) # ¢, for every 5.
Note: DcU, Dcva, 0L 6 I‘, gives us Dc: (U nva), a E 1‘,

and Dc:n(Uf1Va). Observe that
T

 

u
D=nv =nanmu(vnm1
1.. (I. I. (1 CI.
= Iowa nU) u Iowa nU).

Now, if n(Vd 05) = ¢, by the finite intersection property
I

n ~ ~

we have 0 (ijwU) = d, n finite, and since the (VafiU)
j=1

are nested, some Vrrifi = ¢, and we are done.

Since (Va nU) # ¢, a E P: we have
D=AUB3 AnB=¢i DCB: A#¢: B7(¢°

But this is impossible. We conclude that VB exists with
D=fl%c%cm

F

We return to the proof of the theorem, and consider

X-D.

Since D is closed in X, X-D is open and is compact.

Construct a net, from the properly nested sequence

{Va}der’ such that vY e vy,

etc. Now this sequence

vY E V0; VO 6 V0, V0 E vy, V6;
{VQ}GET in X-D clusters, but not
in X-—D. This contradicts the compactness of X-D. We

conclude that compact UHOS—spaces are connected.

CHAPTER V

THE INVERTIBLE CASE

In this chapter we begin to study the relationship
between UHOS—spaces and invertible spaces (see Def. 2.1.3.).
The following examples begin to illustrate the problem, but
we are reminded of the result in Corollary 2.4. This re—

sult guarantees the existence of invertible UHOS—spaces.

Example 5.1. A UHOS—space is not necessarily inver—
tible. Consider the integers l,2,3,... with the following
topology: [[n,n-+l,n-+2,...,}| n = l,2,3,...] is the col—

lection of open sets.
Claim: This is a UHOS—space! Clear!

Claim: This space is not invertible! In fact the only
homeomorphism h of X onto X is the identity map (i.e.,
X is rigid). For suppose h is not the identity map,
then there are integers n,t, 3 n # t, and h(n) = t. Sup—
pose n > t. Under h the open set determined by n goes
to an open set, which is determined by an integer of size
t or smaller. But then our homeomorphism is required to
take (n-1) points to (t-—l) points. This is impossible.
The case n < t follows as above by considering h—1 as

the homormorphism in question.

42

43

Example 5.2. An invertible space is not necessarily

 

a UHOS-space.

Proof: The sphere, Sn, is invertible, but is not a
UHOS-space since it is connected and Hausdorff, (See note

in 4.4).
Example 5.3. Invertible, UHOS—spaces do exist.
Proof: See Chapter II, Corollary 2.4.

The next two theorems show that certain subsets of
UHOS—spaces, X, are moved by homeomorphisms h:X 4 X, h
into; h onto respectively, in a manner reminiscent of the

case with invertible spaces.

Theorem 5.4. A topological space X is a UHOS-space

 

iff any nondense set D c X may be taken into X-D by a

homeomorphism h:X 4 X-D.

nggf: If D c X is nondense, then D §.X. Then
X-D is open, and the UHOS property gives a homeomorphism
h:X 4 X-—D. But X-D c X-D, and we are done.

Let U be a proper open set in X. Then X-U is
closed in X and is nondense in X. Therefore, there is
a homeomorphism h:X 4 Xn-(X::53 = X-(X-U) = U. ‘We con—

clude that X has the UHOS prOperty.

Note: This result indicates a priori that the UHOS

property in a space falls short of invertibility. However,

 

44

Theorem 2.1.1 shows the remarkable compatibility of the

two properties.

Theorem 5.5. If X is a metric UHOS—space and C C X
is connected, then there is a homeomorphism h C X 3 h(C) C

X-C.

Proof: 1) Assume X is not connected, so that X =
ULJV, where U,V # ¢ open in X and UriV = ¢. X is a
UHOS—Space, hence there is a homeomorphism h:U 4 V, and
a homeomorphism g:V 4 U. Now C C U or C C V since
C is connected; say C C U. Define H:X 4 X as follows:
h(u) = h(u), u e U; H(v) = g(v), v e V. Clearly H is a

homeomorphism, and H(C) = h(C) C V C X-C.

Note: X cannot be connected since X metric implies
X is Hausdorff, and there exist no non—trivial connected

UHOS-spaces.

Definition 5.6. If X is a topological space and
U,V are open in X, then U and V have the same embed—

ding type if there is an h 3.x and h(U) = V.

We remind the reader that the category of connected

UHOS-spaces is large, and we prove the following theorem.

Theorem 5.7. A topological space X is in C (i.e.,
the category of connected, UHOS—spaces) iff each proper
closed set in X lies in an open set of every embedding

type under homeomorphisms h 3.x.

 

45

Ppggf: Let X e C, and suppose that each proper closed
set W in X lies in an open set of every embedding type.
Let U C X be open. Then X-U is closed. Since X-U
lies in an open set of the same embedding type as U, say
V (i.e., there is a homeomorphism h 3.x with h(V) = U)
and X-—U C V = h_l(U), then h(X-U) C U. Hence X is
invertible.

Now let X be invertible, w a closed set in X,
and U # ¢ an open set in X. By lemma 4.1.1., U ¢ W.
Then (U-W) is open and there is a homeomorphism h C X
with h[X-—(U-—W)] C U-W C U. Since W C [X-(U-W)], the

proof is complete.

 

 

BI BLIOGRAPHY

BI BLOIGRAPHY

P.H. Doyle, J.G. Hocking, Invertible Spaces, Amer.
Math. Monthly 68 (1961), 959-965.

James Dugundji, Topology, Allyn and Bacon, Inc.,
Boston, 1966.

Hurewicz, W., Wallman, H., Dimension Theory, Princeton
University Press, Princeton, N.J., 1940.

Husain, T., Introduction to Topological Groups, W.B.
Saunders Co., 1966.

Kelley, J.L., General Topology, D. Van Nostrand,
Princeton, N.J., 1955.

Kuratowski, K., Topologie I, II, Warsaw (1948).

Sierpinski, W., General Topology, University of Toronto
Press, 1956.

 

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