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Selection Theorems 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.
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