SELECTION THEOREMS

STEPHANIE HICKS

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS

NIPISSING UNIVERSITY SCHOOL OF GRADUATE STUDIES NORTH BAY, ONTARIO

© Stephanie Hicks August 2012

I hereby declare that I am the sole author of this Major Research Paper.

I authorize Nipissing University to lend this Major Research Paper to other institutions or individuals for the purpose of scholarly research.

I further authorize Nipissing University to reproduce this Major Research Paper by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research.

v Acknowledgements

I would like to thank several important individuals who played an integral role in the development of this paper. Most importantly, I would like to thank my fiancé Dan for his ability to believe in my success at times when I didn’t think possible. Without his continuous support, inspiration and devotion, this paper would not have been possible. I would also like to thank my parents and my sister for their encouragement and love throughout the years I have spent pursuing my post secondary education. Many thanks are due to my advisor Dr. Vesko Valov for his expertise and guidance throughout this project. Additionally, I would like to thank my external examiner Vasil Gochev and my second reader Dr. Logan Hoehn for their time. Lastly, I would like to thank Dr. Wenfeng Chen and Dr. Murat Tuncali for passing on their invaluable knowledge and encouragement to pursue higher education throughout my time as a Mathematics student at Nipissing University. Without these important people, this paper would not have reached completion.

iv

Abstract

The purpose of this paper is to provide a brief introduction to the theory of con- tinuous selections. The theory of continuous selections was founded by E. Mi- chael in his papers [2], [3], [4] and [5]. One of the most important and widely known selection theorems is the Convex-Valued Selection Theorem established by Michael [4], stating that every lower semi-continuous map from a paracompact space into a Banach space with closed convex values admits a continuous selec- tion. The proof of the Convex-Valued Selection Theorem will be presented in Chapter 2 using two different approaches. In Chapter 3 we present another result obtained by Michael [2] called the Zero – Dimensional Selection Theorem. We structure our proofs in a similar manner as the proofs of the theorems found in [7].

vi

Contents

1. Preliminaries and Introduction 1

1.1 Multivalued Maps 1

1.2 Properties of Paracompact Spaces 8

2. The Convex-Valued Selection Theorem 12

2.1 Paracompactness of the Domain 12

2.2 The Convex - Valued Selection Theorem: The Method of Outside 22 Approximations

2.3 The Convex - Valued Selection Theorem: The Method of Inside 28 Approximations

3. The Zero-Dimensional Selection Theorem 37

3.1 Zero-Dimensionality of the Domain 37

3.2 Proof of the Zero-Dimensional Selection Theorem: The Method of 40 Outside Approximations

4. Concluding Remarks 46

References 47

Chapter 1: Preliminaries and Introduction

The following concepts provide a fundamental background for the information that will be discussed throughout the paper. It is assumed however, that the reader has some basic knowledge of Topological Spaces, Metric Spaces, Topo- logical Vector Spaces and Banach Spaces and the properties and basic results of each. We will begin this chapter by introducing the concept of lower semi- continuous maps and some elementary properties of such maps. In the following section we will provide proof of a few properties of paracompact spaces.

1.1 Multivalued Maps To begin this section, we will define the notions of singlevalued and multivalued maps, and continuity of these maps. We will then give proof of important general properties of lower semi-continuous maps which will be of great use in sections to follow.

Definition ([7]): We say that is a multivalued map from a set into a set if for every a nonempty subset is assigned to it. Note that when all the sets consist of only a single point, then the map will be considered in the usual sense as a singlevalued map, and will be denoted by .

2 Preliminaries and Introduction

It will be common practice throughout this paper to represent multivalued maps with uppercase letters while we will represent singlevalued maps with lowercase letters. Next, we will investigate the continuity of multivalued maps.

Definition ([7]): Let and be two topological spaces and be a multivalued map between them. We say that: (1) F is lower semi-continuous if for every open set , the set is open in . (2) F is upper semi-continuous if for every open set , the set is open in . (3) F is continuous if it is lower semi-continuous and upper semi-continuous.

Equivalently, we have the following conditions of lower and upper semi-continuity for closed sets. (1) F is lower semi-continuous For all closed , the set is closed in (2) F is upper semi-continuous For all closed , the set is closed in

Definition ([7]): We say a singlevalued map is a selection of the multivalued map if for every .

Note that by the Axiom of Choice, a selection of a multivalued map always ex- ists. However, the existence of a selection only touches one part of our problem. This problem becomes increasingly difficult because we are concerned with the existence of continuous selections of . Throughout this paper, we will restrict ourselves to finding a continuous selection of lower semi-continuous maps. This brings us to our first theorem which provides an explanation of our restriction to lower semi-continuous maps.

3 Preliminaries and Introduction

Note that all neighbourhoods considered will be open.

Theorem 1.1 ([7]): Let and be topological spaces and be a multivalued map between them. Assume that for every and for every , there exists a neighbourhood and a continuous selection of , such that . Then is lower semi-continuous.

Proof: Let be open in and suppose that . Let and . Let be a neighbourhood of and be a continuous singlevalued selection of such that .

We want to show: (1) is an open subset of and ; Since is a selection, and is open, this follows from the continuity of .

(2) ; Let Show: is a singlevalued selection By assumption

(3) is open in ; Actually, we have proved that for every the open set con- tains and . So, is open.

We will denote the closure of a set in by . Our next theorem demonstrates that the closure of a lower semi-continuous map is also lower semi-continuous. 4 Preliminaries and Introduction

Theorem 1.2 ([7]): Let be lower semi-continuous and for every . Then is also lower semi-continuous.

Proof: Let be open in and suppose . Since by lower semi-continuity of , is open in , it suffices to show that . Obviously, . To show , suppose .

Recall, that for a topological vector space , a set is convex if for all and for every the point . This is equivalent to saying that the segment joining and is contained in .

Definition ([1]): The convex hull of a set denoted by or is the intersection of all convex sets containing . The convex hull can be written

as .

The following theorem shows us that the “convex hull” of a lower semi- continuous map is also lower semi-continuous.

Theorem 1.3 ([7]): Let be a lower semi-continuous map and let be a locally convex topological vector space. Then the map which is defined by , , is lower semi-continuous.

Proof: Let be open in and suppose . Let and . 5 Preliminaries and Introduction

This means is a convex combination such that ,

and .

Take the algebraic difference between and , denoted by . Recall the algebraic difference is defined as . Note that is a neighbourhood of the origin. Consider a convex neighbourhood of the origin such that .

We will show that: (1) For the sets are neighbourhoods of ; By definition we have . Since , then . So we have for every . Then is a neighbourhood of . Hence the sets are non empty neighbourhoods of .

(2) ;

Let . By definition of , for every

there exists . Let . Then and

for all , and we have

because is convex. (Recall that ). From the

hypothesis we have . Hence, and . So,

and . So it follows that

, as desired.

6 Preliminaries and Introduction

(3) is open; Since the sets are non-empty neighbourhoods of and since

, then is a neighbourhood of

. Since was an arbitrarily chosen point from , we have ob- tained that is open.

The following two theorems concern the lower semi-continuity of special multi- valued maps.

Theorem 1.4: Let be a lower semi-continuous map and suppose that is open. Suppose also that for every . Then the map defined by is lower semi-continuous.

Proof: To show lower semi-continuity, it suffices to show that for every open set . Take . By definition of By definition of By definition of

Theorem 1.5 ([7]): Let be a metric space and let be low- er semi-continuous. Suppose is a continuous (singlevalued) map such that there exists with for every , where is the open ball centered at with radius . Then the map , defined by , is lower semi-continuous.

Proof: Let be open in and suppose . Let and . 7 Preliminaries and Introduction

Take such that . Choose such that .

We claim that: (1) ; Since then clearly . For

every , we have . By the triangle inequality

and . So,

Thus, . Hence, , as desired.

(2) ; Recall, that by our definition of , . Let .

By (1)

If .

Because By By the definition of

8 Preliminaries and Introduction

(3) and is a neighbourhood of ; Obviously, and is open because is continuous. Since , and because the last set is open by lower semi-continuity of , we obtain the required result.

(4) The set is an open set of ; By (3) is a nonempty neighbourhood of and by (2) it is contained in . Thus, is open and hence is lower semi-continuous.

1.2 Properties of Paracompact Spaces In this section we are going to prove some elementary facts about para- compact spaces. We will prove that paracompact spaces are both regular and normal. We will also show the equality between the closure of the union of locally finite subsets of and the union of their closures. But, before we can proceed with such proofs, we first require a few definitions.

Definition ([6], [7]): Let and be coverings of the space . We say that is a refinement of if for every , there exists such that . If all the elements of are open then we say that is an open refinement of .

Definition ([6]): Let be a topological space. We say that a family of subsets of is locally finite if for every there exists a neighbourhood of that intersects at most finitely many elements of .

Definition ([6], [7]): Let be a Hausdorff space. We say that is para- compact if each open cover of has a locally finite open refinement.

9 Preliminaries and Introduction

Lemma 1.6 ([7]) : Let be a topological space and a locally finite family of subsets of . Then,

.

Proof: Since is always true without restrictions on ,

we need only to verify the inclusion . Take . Let

be the neighbourhood of such that it intersects at most finitely many el-

ements of . Let .

Let also . By construction, and

. So, and is a nonempty neighbourhood

of that does not contain elements of the set . Hence, .

Lemma 1.7 ([7]): If a topological space is paracompact, then it is regular.

Proof: Recall the definition of a regular space: for all closed and for all , there exist nonempty disjoint open sets and such that and . By the Hausdorff property of , for every there exist neighbourhoods and of and respectively such that . Let be a locally finite covering of which refines the covering . Let and . Let .

10 Preliminaries and Introduction

We will show: (1) and are open and disjoint; By the construction of , it is open. Since is open and the union of open sets is again open, is open. These sets are disjoint by the construction of .

(2) ; Take . Then by construction of , is contained in some such that for every . Hence, according to the construction of . So, .

(3) ; By construction of and Lemma 1.6 we have

. Since for every , the set

is contained in for some and , it follows that

for every . Consequently, . So, .

Lemma 1.8 ([7]): If a topological space is paracompact, then it is normal.

Proof: Recall the definition of a normal space: for any two disjoint closed sets and , there exist nonempty disjoint open sets and such that and . Since is regular, for every there exist disjoint open sets and such that and . Let be a locally finite covering of inscribed into the covering . Let . Let .

11 Preliminaries and Introduction

We will show: (1) and are open and disjoint; Similar to (1) of Lemma 1.7.

(2) ; Similar to (2) of Lemma 1.7.

(3) ; Similar to (3) of Lemma 1.7.

Recall the definition of extension. Let and be a map be- tween sets. The map is said to be an extension of the map if .

Our next theorem is a well known theorem which will be useful in the next chapter. We shall accept this theorem without proof since it can be found in many Topology books.

Theorem 1.9 (The Tietze - Urysohn Theorem) ([6], [7]): For every normal space , any closed subset of and for any continuous function there exists a continuous extension of over the whole space . It follows that and .

Proof: See [6]. Chapter 2: The Convex-Valued Selection Theorem

This chapter is devoted to The Convex-Valued Selection Theorem. We will begin our discussion of this theorem by first showing that the paracompactness of the domain of a lower semi-continuous map into a Banach space with closed convex values is a necessary condition for the existence of a continuous selec- tion. In the remaining sections we will provide two different methods of proving the main theorem. In the first method, we will consider the solution to the selec- tion problem as a uniform limit of continuous - selections using the method of outside approximations to obtain such a uniform limit. We shall conclude this sec- tion with the second method of proving the main theorem in which we will con- struct a selection as a uniform limit of - continuous selections obtained by the method of inside approximations.

2.1 Paracompactness of the Domain In this section we shall show that the paracompactness of a space is a nec- essary condition that every lower semi- continuous map from into a Banach space with closed convex values has a continuous singlevalued selection.

Theorem 2.1 ([7]): Let be a topological space. Suppose that for any Banach space , each lower semi-continuous map with closed convex The Convex-Valued Selection Theorem 13

values has a continuous singlevalued selection. Then any open covering of , has a locally finite open refinement.

Proof: Let be the Banach space of all summable functions. By a summable function we mean a function over the indexing set such that . Let , for every . Let .

We want to show: (1) For every , , where is the character- istic function of ; First, we will show that is closed.

Take a sequence such that . We will show that .

We have and by definition

. So, for every , we have

. Since (recall that ), we have .

By definition of , if , then for every . So, .

We have . Since for all , then

. Thus, , as desired.

Next, let us show that . Denote by the set

. If , then for finitely many

such that and for all . Obviously, and

The Convex-Valued Selection Theorem 14

. If and , then and for all because

. So, for every and thus . This means that

. Therefore, . Since is closed, we obtain

.

To show that , suppose first that is finite. If , then for all with , i.e. for all

. Then . The last equality shows that

i.e. . If is infinite, then implies

. Then there exists a countable set such that

. Let . Then as . For

, we define . Then as a

convex combination of for . Now, we will show that

which will imply . We have

. Since for we

have ,

. Similarly, because for every ,

. Finally,

The Convex-Valued Selection Theorem 15

. So,

. But the series is convergent, so

. Hence , and . Thus,

. So, .

Note that we have actually proved that is a dense subset of .

(2) is lower semi-continuous; It suffices to show that for every , where and , there exists a neighbourhood of contained in .

Consider two cases: Case i) Suppose is a finite subset of .

That is, . Then is a

neighbourhood of . Suppose . If , then

and . Hence, . This yields

. So, is lower semi-continuous.

Case ii) Suppose is infinite. Since , is a dense subset of by (1). There exists . Then is finite. There exists such that . According to the previous case, , and is lower semi-continuous.

Let be the continuous singlevalued selection of .

The Convex-Valued Selection Theorem 16

Let . Let . Let

(3) Each is continuous and for all ;

As a continuous singlevalued selection, . By definition of

Obviously, for every the projection , , is continuous. Since , is also continuous.

(4) The function is positive and continuous; The equality implies that for all .

To show continuity of , let . Pick an index with the property

that . Since , there exists a finite set such

that . This implies the existence of a neighbourhood

of such that for all .

Thus, for every and , we have . Therefore

for all . Since each is continuous

and the supremum of finitely many continuous functions is continuous, the

function is continuous on . This implies that is continuous.

(5) implies for every ; By Contraposition Suppose .

The Convex-Valued Selection Theorem 17

for every By definition of Since ,

By definition of

(6) For every , ; Let . From the definition of and since , we have that . By (5) this yields . So, .

(7) The family of open subsets of is locally finite; Since both and are continuous functions we have that is an open

set itself. Let and be as in the proof of (4). Take , a

finite set, and a neighbourhood of . Then only when

. To see this, if , then

(by (4) and (6)).

Therefore for every , hence . Thus,

is locally finite.

(8) covers ; By Contradiction Suppose .

for every By definition of By definition of But this is a contradiction! So, is a cover of , as desired.

The Convex-Valued Selection Theorem 18

Another important notion we require is that of a locally finite partition of unity, defined below.

Definition ([7]): Let be a topological space. We say that a family of continu- ous nonnegative functions is a locally finite partition of unity if for all , there exists a neighbourhood and a finite subset such that: i) For every ,

ii) Both and imply

Definition ([7]): Let be a topological space. We say that a locally finite parti- tion of unity is inscribed into an open covering of if for every there exists such that .

Note that alternatively we can write: is a locally finite partition of unity is a locally finite open cover of and for all .

Using this notion of inscribed partition of unity, we can improve the statement and proof of Theorem 2.1 as follows.

Theorem 2.2 ([7]): Let be a topological space and let be a Banach space. Suppose that each lower semi-continuous map with closed and convex values has a continuous singlevalued selection. Then every open covering of has a locally finite open partition of unity inscribed into itself.

Proof: We repeat the arguments from the proof of Theorem 2.1 to obtain the functions and the covering refining .

The Convex-Valued Selection Theorem 19

Let . Let .

Let .

We want to show: (1) is continuous; Since both and are continuous functions, then so is .

(2) ; By Contraposition Suppose . Then and, because , we have . Since both and are continuous func- tions, there exists a neighbourhood of such that for all . But by definition of , this means on . And by definition of , . As desired.

(3) is a locally finite covering of , where ; This proof has been omitted because it is identical to parts (7) and (8) of Theorem 2.1.

(4) is a continuous strictly positive function; It follows from the proof of (3), that for a fixed point there exists a neighbourhood of such that is the sum of finitely

many continuous functions in . Thus, is also continuous. Also, by (3) and the definition of , for every there exists such that . Thus, the sum . Therefore, is a continuous

strictly positive function.

The Convex-Valued Selection Theorem 20

(5) is a locally finite partition of unity inscribed into ; By the definition of and since (by (4)), we have that . Which implies . But, So, . Since for each and the covering is locally finite, so is the open covering . This yields that is a locally finite partition of unity inscribed into because .

Our next proposition shows an equivalent definition of paracompactness of a space.

Proposition 2.3 ([7]): Let be a Hausdorff space. Then is paracompact if and only if each open covering of has a locally finite partition of unity inscribed into this covering.

Proof: The reverse implication directly follows from the definition of locally finite partition of unity. Indeed, if is an open cover of and is a lo- cally finite partition of unity inscribed in , then is a locally fi- nite cover of inscribed in .

For the other implication, let be an open cover of . Due to para- compactness of , we may assume that is locally finite.

Since is paracompact, it is also regular, by Proposition 1.7. So, for each and there exists a neighbourhood of such that . Paracompactness of also implies the existence of a locally finite refinement of the covering . Let .

We shall show: (1) is a locally finite open refinement of ;

The Convex-Valued Selection Theorem 21

By construction of , we have that . Let and for some . Then by construction of and , we obtain . So, and is an open covering of . For every there exists a neighbourhood of that intersects only finitely many of the covering . Then intersects only finitely many of the covering . Hence, is locally finite.

(2) ; Lemma 1.6 yields .

Applying the above construction to the covering , we can construct a lo- cally finite open refinement of such that . Since and are disjoint closed sets, there exists a continuous function such that and . By Theorem

1.9 there exists an extension of which yields that .

Let .

(3) For every there is a neighbourhood of in and a finite set such that for each ; This follows from the fact that is locally finite. Indeed, there exists a neighbourhood of intersecting only finitely many . Hence, for every , which implies that .

(4) is a locally finite partition of unity inscribed into ; By (3) it follows that is a continuous function. Since is a cov-

ering and we have that . Hence is a continuous

function.

The Convex-Valued Selection Theorem 22

By the definition of , we have that . This implies , and hence is locally finite. Also, by (2) and the construction of , we obtain . Hence, is inscribed into .

Note that with the above definition of paracompactness Theorems 2.1 and 2.2 are equivalent.

Remark: Let a be topological vector space, a set of points arbitrari- ly chosen from , and a locally finite partition of unity on a space . Then the map defined by is continuous, because

that for a fixed the map is the sum of finitely many continuous maps in a neighbourhood of this point, and thus continuous.

2.2 The Convex – Valued Selection Theorem: The Method of Outside Approximations In this section we are going to prove that every lower semi-continuous map from a paracompact space into a Banach space with closed and convex values has a continuous singlevalued selection. This theorem provides us with sufficient conditions for solving the problem of continuous selections with the domain as a paracompact space. To prove this we will use the method of outside approxima- tions, that is we shall construct a selection as a uniform limit of the sequence of continuous - selections of a lower semi-continuous map . Let us begin with the definition of an - selection, essential for this section.

Definition ([7]): Let be a topological space, be the metric space , and a multivalued map between them. We say that a singlevalued map is an - selection of if , for every , where .

The Convex-Valued Selection Theorem 23

By the above definition, we can equivalently say that for all , where is the open ball centered at and radius . Before we can prove the main theorem of this section, we require two important propositions. The first proposition demonstrates the existence of an - selection of a multivalued map with nonempty convex values.

Proposition 2.4 ([7]): Let be a paracompact topological space, and a normed linear space. Suppose the map is lower semi-continuous with convex values. Then for all , there exists , a continuous singlevalued - selection of .

Proof: Fix . Let the open ball in the normed space with center and radius be de- noted by . Let .

We want to show: (1) covers X; By definition of , and since is lower semi-continuous, we have that is an open set in . To see that is a cover of , take . Then , so there exists . Obviously, . Hence, .

(2) There exists a locally finite partition of unity inscribed into ; Since is paracompact, by Proposition 2.3 we have that the open covering of , has a locally finite partition of unity inscribed into this cover. As desired.

The Convex-Valued Selection Theorem 24

For every we fix such that and define .

(3) is both well defined and continuous; This is a direct result of the Remark at the end of Section 2.1.

(4) for all ; Take . Then has a finite num-

ber of elements. For we have . Hence,

, and let for all . So by

definition of , we have that . Now, define to be the

convex combination . Note that since is a con-

vex set itself. So the distance between and the point can be

represented as

.

Since we are only concerned with finitely many values, the sum can be rewrit-

ten as

because . Thus, we have for every .

So is a continuous singlevalued - selection of , as desired.

The second essential proposition demonstrates the existence of a uniformly Cauchy sequence of - selections of a multivalued map with nonempty convex values.

The Convex-Valued Selection Theorem 25

Proposition 2.5 ([7]): Let be a paracompact topological space and a normed space. Suppose the map is lower semi-continuous with con- vex values. Then for all sequences of positive numbers that con- verge to zero, there exists a uniformly Cauchy sequence of continuous singlevalued - selections of .

Proof: We shall construct by induction a sequence of lower semi-continuous maps with convex values and a sequence of singlevalued continuous maps such that: i) for every ii) iii) For all , is an - selection of Base Step (n=1): We apply Proposition 2.4 for to obtain a continuous singlevalued - selection of . Now let be the open ball in with centre and radius . Let .

We want to show: (1) with and convex; From the Base Step, we have that is a continuous - selection of . So, . is convex as it is the intersection of two convex sets. Recall that open balls in normed spaces are always convex.

(2) ; Note that , so the claim immediately follows.

(3) is lower semi-continuous; This claim immediately follows from Theorem 1.5 when considering the met- ric space as the normed space and with substitutions , and .

The Convex-Valued Selection Theorem 26

Inductive Step (n=m): Let be a sequence of lower semi-continuous maps with convex values and be a sequence of singlevalued contin- uous maps satisfying the above conditions. By Proposition 2.4 there exists a continuous singlevalued - selection of . Now let be the open ball in with centre and radius and .

We want to show: (4) with and convex; Similar to the proof of (1) of Proposition 2.5.

(5) ; Similar to the proof of (2) of Proposition 2.5.

(6) is lower semi-continuous; Similar to the proof of (3) of Proposition 2.5.

(7) is a uniformly Cauchy sequence of continuous singlevalued - selections of ; From the Inductive Step, we have that is a continuous - selection of

. By property i) we have that . So, by the previous inclu-

sion we have that is an - continuous selection of . Note that

. For every we have such that

. Then, for every and , we obtain

because .

Since it follows that is a uniformly Cauchy sequence.

The Convex-Valued Selection Theorem 27

Finally, we are ready to state and prove the main theorem of this section.

Theorem 2.6 (The Convex-Valued Selection Theorem) ([7]): Let be a par- acompact space, and a Banach space. Suppose the map is lower semi-continuous with nonempty, closed and convex values. Then has a con- tinuous singlevalued selection.

Proof: Let be a sequence of positive numbers converging to zero.

By Proposition 2.5, there exists a uniformly Cauchy sequence of con-

tinuous singlevalued - selections of . Take and

such that and for every , and

. Since , then for every and there ex-

ists such that . So, we have the inequality

,

because . It follows that in the complete

subspace the sequence is Cauchy. Hence, by the con-

vergence of Cauchy sequences, . Note that since

and , we have . This

means exists and . It follows that is

continuous since it is the uniform limit of . Hence, is the desired con-

tinuous singlevalued selection of .

The Convex-Valued Selection Theorem 28

2.3 The Convex – Valued Selection Theorem: The Method of In- side Approximations In this section we are going to provide a second proof of Theorem 2.6 using the method of inside approximations. That is, we shall construct a selection as a uniform limit of the sequence of - continuous selections of a lower semi-continuous map .

We begin with the definition of -continuity of a function at a point.

Definition ([7]): Let be a topological space, a metric space, and . We say that a function is - continuous at if for every there exists a neighbourhood , of such that for each . If the function is - continuous at for all , then is called - continuous. Note: i) The usual continuity coincides with 0 - continuity. ii) - continuity implies - continuity for all .

First we must state and prove a technical lemma required for the proof of two important propositions.

Lemma 2.7 ([7]): Let be a paracompact space and a locally finite

partition of unity on . Suppose and

. Let be defined by

.

Then is a neighbourhood of and for every we have

.

The Convex-Valued Selection Theorem 29

Proof: First notice that since is a locally finite partition of unity the set is finite. Let .

We claim that: (1) is a locally finite cover of of closed sets; Take . Then there exists such that . This implies that . So, is a cover of . Since is a locally finite partition of unity, the cover is locally finite. Since for each , we obtain that is also lo- cally finite.

(2) is closed in ; Take and a neighbourhood of , that intersects

only finitely many sets where each . Then

is a neighbourhood of and

. So, it follows that is closed.

Let us note that (1) and (2) imply that is an open set containing .

(3) ; Let and . If then and . Since then implies that . Thus, and .

(4) ; Trivial. This follows from the definition of .

The Convex-Valued Selection Theorem 30

(5) ; Let . Then . Suppose . Then . But which implies , and a contradiction! Hence, and . Thus, .

The following proposition demonstrates the existence of an - continuous se- lection of a multivalued map with nonempty and convex values.

Proposition 2.8 ([7]): Let be a paracompact topological space, a normed space and suppose the map is lower semi-continuous with convex values. Then for every , there exists , a singlevalued - continuous selection of .

Proof: Fix . Let be an arbitrarily chosen, not necessarily continuous, selection of and represents an open ball in with radius . For , let . Since is lower semi-continuous each is open in and is a cover be- cause implies . Suppose is a locally finite partition of unity that is inscribed into . Let . Then and are finite. For each , let such that . For , the set is nonempty. So, for every there exists . The function is singlevalued and well defined.

The Convex-Valued Selection Theorem 31

We shall show: (1) implies ; implies . As constructed, this is the open ball centered at with radius . Thus, it follows directly that .

Define .

(2) is a selection of ; Since for every the set is finite, is a fi-

nite convex combination (i.e. ) of the points . Since

is convex, we have for every . Hence, is a selec- tion of .

(3) The map is - continuous; Fix and suppose is a neighbourhood of as defined in Lemma

2.7. Let , . Since , we have

. Note that

.

and together we have,

.

The Convex-Valued Selection Theorem 32

Now, consider the norms of each term in the above sum.

.

Suppose for all . If and has elements, then

for all , there exists a neighbourhood of such that

implies . Let . Then

implies .

So, can be arbitrarily small in a neighbourhood of . Since

and (Lemma 2.7), for each we

have

.

Finally,

since by Lemma 2.7. Since

is a finite set and for each is a continuous func-

tion with , we can find a neighbourhood of such that

implies . Therefore,

for every . This implies that is

- continuous.

The Convex-Valued Selection Theorem 33

The next proposition demonstrates the existence of a uniformly Cauchy se- quence of - continuous selections in a multivalued map with nonempty and convex values.

Proposition 2.9 ([7]): Let be a paracompact topological space and be a normed space. Suppose the map is lower semi-continuous with con- vex values. Then for all sequences of positive numbers that con-

verge to zero, there exists a uniformly Cauchy sequence of - contin- uous singlevalued selections of .

Proof: Suppose is a monotone decreasing sequence of positive numbers that converges to zero, such that for every and

. We shall construct by induction a sequence of selections of

the map such that: i) is - continuous ii) for all and Since , then - continuity of implies - continuity of . Since

and by property ii), we have that is uniformly Cauchy. In-

deed if there exists such that .

Then for every and , we have

.

We apply Proposition 2.8 to to obtain a - continuous singlevalued se- lection of . Suppose that a sequence with properties i) and ii) has already been constructed. Since is - continuous, for all there exists a neigh-

The Convex-Valued Selection Theorem 34

bourhood of , such that for every , . Let represents an open ball in with radius . For , is open in because is lower semi-continuous and is an open covering of . Suppose is a locally finite partition of unity that is inscribed into . Let , and . Obviously, and are finite. Let be such that . For , the map has nonempty values and we can choose a selection (not necessarily continuous) of that map. So the map is singlevalued and well defined.

We claim: (1) implies ; This proof is similar to (1) from Proposition 2.8.

Define .

(2) is a selection of ; This proof is similar to (2) from Proposition 2.8.

(3) The map is a - continuous map; This proof is similar to (3) from Proposition 2.8.

(4) for all ; Fix . By the construction of we obtain,

The Convex-Valued Selection Theorem 35

because .

Now consider the norm of each summand. By choice of the point , for all we have . This implies . By construction, we obtain . Since is a monotone decreasing sequence and ,

we have

.

Based on the information from the previous two propositions, we can give a proof of the Convex-Valued Selection Theorem different from the original proof.

Alternative Proof of Theorem 2.6: Let be a sequence of positive numbers converging to zero. By Proposition 2.9, there exists a uniformly Cauchy sequence of - continuous singlevalued selections of such that . Since the values of are closed, then there exists , and .

Check the continuity of the limit ;

The Convex-Valued Selection Theorem 36

Take and suppose that for every we have

for every . For all and , take to be a neighbourhood

of such that for every . Without loss

of generality, we may assume that for every . So, for

and ,

.

This shows that is continuous. Hence, is the desired continuous singleval- ued selection of .

Chapter 3: The Zero Dimensional Selection Theorem

In this chapter, we will prove the Zero-Dimensional Selection Theorem. This theorem concerns lower semi-continuous maps with nonempty closed values of zero-dimensional paracompact spaces into completely metrizable spaces. It is the most simple of the Selection Theorems, and the proof is very similar to the proof of The Convex-Valued Selection Theorem without the concept of partition of unity. We will begin our discussion of this theorem by first considering the ze- ro-dimensionality of the domain of a closed valued lower semi-continuous map as a necessary condition for solving the selection problem. Next we will present the proof of the theorem in a similar manner as in the previous chapter.

3.1 Zero-Dimensionality of the Domain In this section we will weaken the hypotheses in the Convex- Valued Selection Theorem by considering as a completely metrizable space rather than a Ba- nach space. But, on the other hand, we now require the domain, , to be a zero - dimensional paracompact space. We begin with an extension of the defini- tion of open refinement given in Chapter 1 required for this chapter.

Definition ([6], [7]): If all the elements of are open and disjoint then we say that is a disjoint open refinement of .

The Zero- Dimensional Selection Theorem 38

This brings us to our first theorem.

Theorem 3.1: Let be a topological space. Suppose that for any completely metrizable space , each lower semi-continuous map with closed values has a continuous singlevalued selection. Then any open covering of , has a disjoint open refinement.

Proof: Let be the index set in the discrete topology that is generated by the complete metric defined by for and for . Let for every .

We shall show: (1) is a nonempty closed subset of ; Since is a covering of , and by construction of , it follows that . Obviously . Since is a discrete metric space, all subsets of it are closed. Hence is a closed subset of .

(2) is lower semi-continuous; Let , then . Note that is open, as it is a point in the dis- crete space . So, for any it follows that . This implies the intersection of the value and the neighbourhood of the point is non – empty. Hence, is lower semi-continuous.

Let be a continuous selection of and .

(3) The family of the sets (without repetition) is a disjoint open re- finement of ; Note that the singleton is open in since is discrete. So, is open because of the continuity of .

The Zero- Dimensional Selection Theorem 39

Let , then and . Therefore , and is a refinement of . We will show is disjoint by contradiction. Take , then implies and implies . Thus, and . But, this is a contradiction! Hence, a disjoint open refinement.

Note we have actually proved the cardinality of is less than or equal to the cardinality of the indexing set of .

Next, we pass to the notion of zero-dimensionality of a topological space.

Definition ([7]): Let be a topological space. We say that is zero- dimensional, i.e. , if every open finite covering of has a finite dis- joint open refinement.

Our next proposition shows the equivalence between zero-dimensionality and paracompactness of a topological space.

Proposition 3.2 ([7]): Let be a paracompact topological space. Then the following are equivalent: (1) (2) Every open covering of has a disjoint open refinement.

Proof: (2) (1) This implication is trivial by the definition of zero-dimensional. (1) (2) Let be an open covering of . By the paracompactness of , we may assume that is locally finite. By Prop-

osition 2.3 there exists an open covering of such that for

The Zero- Dimensional Selection Theorem 40

. By construction, we have that for each the family is

a finite open covering of . Since this finite open covering of has

a finite disjoint open refinement, say . Let .

Obviously, each element of is clopen, so is . By construction it follows

that . Indeed, if then there exists and .

This implies . Then , and we have and hence

.

Let us now take to be a well-ordering on the index set .

Let for .

Since , it follows that is the union of a locally finite family

of closed subsets, and this union is closed by Lemma 1.6. Hence, is open as

the difference of a clopen and closed set and . Therefore,

is a disjoint open refinement of .

3.2 Proof of the Zero-Dimensional Selection Theorem: The Meth- od of Outside Approximations In this section we will provide a proof of the Zero-Dimensional Selection Theo- rem using the method of outside approximations. We will prove that every lower semi-continuous map with closed values from a zero-dimensional paracompact space into a completely metrizable space has a continuous singlevalued selec- tion. We will present the proof in a similar manner as in Section 2.2 and again require two propositions essential to the proof of the theorem. The first proposi- tion demonstrates the existence of an - selection of a multivalued map.

The Zero- Dimensional Selection Theorem 41

Proposition 3.3: Let be a zero-dimensional paracompact space, and a metric space. Suppose the map is lower semi-continuous. Then for all there exists , a continuous singlevalued - selection of .

Proof: Fix . Let the open ball in the metric space with center and radius be denoted by . Let .

We want to show: (1) covers ; Since is lower semi-continuous, we have that each is an open set in . To see that is a cover of , take . Then , so there exists . Obviously, . Hence, .

(2) There exists a disjoint open refinement refining ; Since is a paracompact zero-dimensional space, by Proposition 3.2, has a disjoint open refinement refining this cover.

For every we fix such that and define by where is such that . The definition of is correct

because is disjoint and for every there is a unique with .

(3) is both well defined and continuous; This result directly follows since is a constant function on each set .

(4) for all ;

The Zero- Dimensional Selection Theorem 42

Take and let be the unique index such that . By

construction we have

and

for every . So is a continuous

singlevalued - selection of , as desired.

The second essential proposition demonstrates the existence of a uniformly Cauchy sequence of - selections of a multivalued map.

Proposition 3.4 ([7]): Let be a zero dimensional paracompact space and a metric space. Suppose the map is lower semi-continuous. Then for all sequences of positive numbers that converge to zero, there

exists a uniformly Cauchy sequence of continuous singlevalued - selections of .

Proof: We shall construct by induction a sequence of nonempty lower semi-continuous maps and a sequence of singlevalued continuous maps such that: i) for every ii) iii) For all , is an - selection of Base Step (n=1): We apply Proposition 3.3 for to obtain a continuous singlevalued - selection of . Now, let be the open ball in metric space with centre and radius . Let .

We claim: (1) with ; From the Base Step, we have that is a continuous - selection of .

The Zero- Dimensional Selection Theorem 43

So, .

(2) ; Note that , so the claim immediately follows.

(3) is lower semi-continuous; This claim immediately follows from Theorem 1.5 when considering the met- ric space as the metric space and with substitutions , and .

Inductive Step (n=m): Let be a sequence of lower semi-continuous maps and be a sequence of singlevalued continuous maps satisfying the above conditions. By Proposition 3.3 there exists a continuous singlevalued - selection of . Now let be the open ball in metric space with centre and radius and .

We claim: (4) with ; Similar to the proof of (1) from Proposition 3.4.

(5) ; Similar to the proof of (2) from Proposition 3.4.

(6) is lower semi-continuous; Similar to the proof of (3) from Proposition 3.4.

(7) is a uniformly Cauchy sequence of continuous singlevalued - selections of ;

The Zero- Dimensional Selection Theorem 44

From the Inductive Step, we have that is a continuous - selection of

. By property i) we have that . So, by the previous inclu-

sion we have that is an - continuous selection of . Note that

for every and . Moreover, there exist such

that and such that .

Also note that .

Then, for every and , we obtain

Since then it follows that is a uniformly Cauchy sequence.

Finally, we are ready to state and prove the main theorem of this section.

Theorem 3.5 (The Zero-Dimensional Selection Theorem) ([7]): Let be a zero-dimensional paracompact space, and a completely metrizable space. Suppose the map is lower semi-continuous with nonempty closed values. Then has a continuous singlevalued selection.

Proof: Fix a complete metric on and choose a sequence of

positive numbers converging to zero. By Proposition 3.4, there exists a uniform-

ly Cauchy sequence of continuous singlevalued - selections

of . Take and such that and

for every , and . Since

, then for every and there exists

The Zero- Dimensional Selection Theorem 45

such that . So, by the triangle inequality

, because

. It follows that in the closed subset of

the complete metric space , the sequence is Cauchy. Hence, by

the completeness of this sequence converges and

. Note that since and ,

we have . This means exists

and . It follows that is continuous since it is the uniform lim-

it of . Hence, is the desired continuous singlevalued selection of .

Concluding Remarks 46

Concluding Remarks

Throughout this paper we discussed two important selection theorems, the Convex-Valued Selection Theorem and the Zero-Dimensional Selection Theo- rem. We have shown that paracompactness is necessary if all closed, convex valued lower semi-continuous maps into Banach spaces admit continuous selec- tions. Through the methods of outside and inside approximations we have pro- vided two versions of the proof of the Convex-Valued Selection Theorem. This theorem provides us with sufficient conditions for the existence of a continuous selection of such a lower semi-continuous map. We have shown that zero – dimensionality is necessary if all lower semi-continuous maps into completely metrizable spaces admit continuous selections. In addition, we examined the proof of the Zero-Dimensional Selection Theorem using the method of outside approximations. In summary, we have only touched upon this vast topic of con- tinuous selections through the exploration of two different cases. The theory of continuous selections does not end here, as there are multiple other selection theorems including the Compact-Valued Selection Theorem and the Finite- Dimensional Selection Theorem which are of as much interest as the two pre- sented in this paper.

References 47

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[2] E. Michael, Selected Selection Theorems, Amer. Math. Monthly 63 (1956), 233-238.

[3] E. Michael, Continuous Selections I, Ann. of Math. (2) 63 (1956), 361-382.

[4] E. Michael, Continuous Selections II, Ann. of Math. (2) 64 (1956), 562-580.

[5] E. Michael, Continuous Selections III, Ann. of Math. (2) 65 (1957), 375-390.

[6] James R. Munkres, Topology, 2nd Edition, Pearson Education, Singapore (2000), 1-318.

[7] D. Repovš and P.V. Semenov, Continuous Selections of Multivalued Map- pings, Kluwer Academic Publishers, Netherlands (1998), 5-59.