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University Microfilms International300 N INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1.The sign or “target” for pages apparently lacking from the document photographed is “Missing Page(s)”. If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image of the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame, 3. When a map, drawing or chart, etc., is part of the material being photographed, a definite method of “sectioning” the material has been followed. It is customary to begin filming at the upper left hand corner of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed. University M icrofilm s International 300 N. Zeeb Road Ann Arbor, Ml 48106 8526149 Burdick, Bruce Stanley LOCAL COMPACTNESS AND THE COFINE UNIFORMITY WITH APPLICATIONS TO HYPERSPACES The Ohio State University Ph.D. University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 Copyright 1985 by Burdick, Bruce Stanley All Rights Reserved LOCAL COMPACTNESS AND THE COFINE UNIFORMITY WITH APPLICATIONS TO HYPERSPACES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Bruce Stanley Burdick, B.S., M.S. The Ohio State University 1985 Reading Committee: Approved By Philip Huneke Francis Carroll Adviser Department of Mathematics Henry Glover Copyright by Bruce Stanley Burdick 1985 Dedicated to the memory of Dr. Norman Levine, my advisor from 1980 to 1983, whose teaching style and scrupulous attention to detail will be an inspiration to me through out my career. ACKNOWLEDGMENTS I wish first of all to recognize ray debt to my parents, who from early on encouraged ray curiosity and my appetite for the written word. There have also been many other people, both in the academic community and in the various other communities in which I have been involved, who through their friendship and encouragement have helped see this project through to its completion. Since a complete listing of these persons here would be impossible, I assure each of my friends that their contributions, regardless of their irrelevance to the subject matter, is remembered and appreciated. My debt to Dr. Norman Levine is very great. It was he who suggested that I work with uniform spaces; he was the one to whom I first brought each of the results now included in this dissertation and he patiently reviewed each proof, catching my mistakes and suggesting new approaches. I wish to express my heartfelt gratitude to Dr. Philip Huneke, who took over as my advisor in 1984, who gave of his time to let me present to him the results I had found, and who made many helpful suggestions. I iii would like to thank also the other members of my reading committee, Dr. Francis Carroll and Dr. Henry Glover, for their patience and understanding. VITA June 20, 1956 ............. Born - Middletown, Connecticut 1978 ...................... B.S., Heidelberg College, Tiffin, Ohio 1978-1985 ................. Teaching Associate, The Ohio State University, Columbus, Ohio 1984 ...................... Teacher, Punahou School, Honolulu, Hawaii PUBLICATIONS "From Time to Time." Analog Science Fiction / Science Fact, October 1983, pp. 114-122. Reprinted in From Mind to Mind: Tales of Communication from Analog, Stanley Schmidt, ed. New York: David Publications, 1984. "Q.E.D." Analog Science Fiction / Science Fact, December 1984, pp. 96-112. TABLE OF CONTENTS DEDICATION..................... ii ACKNOWLEDGMENTS....................................... i ii VITA ..................................... v FIGURE .............................................. vii INTRODUCTION ....................................... 1 NOTATION, TERMINOLOGY, AND BACKGROUND ........... 5 CHAPTER ONE LOCAL COMPACTNESS AND THE COFINE UNIFORMITY ............................ 21 CHAPTER TWO PROPERTIES OF THE COFINE UNIFORMITY ......... 38 CHAPTER THREE THE HYPERSPACE UNIFORMITY .................... 47 CHAPTER FOUR LOCAL COMPACTNESS OF THE HYPERSPACE......... 56 CHAPTER FIVE CHARACTERIZATION OF HYPERSPACES ............. 73 LIST OF REFERENCES .............................. 85 vi FIGURE 0 Page Figure 1. A Commutative Diagram for Adjointness of m ...................... 46 vii INTRODUCTION The structure known as a uniform space provides the theoretical mathematician with an entity intermediate between metric spaces and topological spaces. Iii the con text of uniform spaces the properties of completeness, total boundedness and uniform continuity can be discussed properties which are usually associated with metric space Each metric on a set generates a uniformity, each unifor­ mity generates a topology, and there is a certain consistency here in that the topology generated by the uniformity generated by the metric is the same as the topology generated directly by the metric. On the other hand, there may be several metrics generating the same uniformity or several uniformities generating the same topology. The collection of uniformities on a set forms a lat­ tice under the inclusion ordering. So there is always both a largest and a smallest uniformity for a given set. The subcollection of uniformities generating a given topology always has a largest element but does not always have a smallest element. As shown by Samuel [14] and Shirota [16] there is such a smallest uniformity if and only if the given topology is that of a locally compact space. Some topologies on a set are compatible with only one uniformity. For example a topology which makes the set compact is generated by a unique uniformity. But Dieudonne [4] gave an example of a non-compact space which admits a unique uniformity. So the question arose as to what topological property is equivalent to the space being generated by a unique uniformity. Different answers to this were given by Doss [5], Newns [13], and Gal [6]. It turns out that this problem of characterizing spaces with unique uniformities is very closely related to the problem of characterizing the smallest uniformity for a space. In Chapter 1 we will give a new proof of the Samuel-Shirota theorem, and we will give nine statements that are equivalent to saying that a uniformity is smallest for its topology. Notable among these is that if a uniformity is minimal among the uniformities generating a given topology then it is actually the smallest of those uniformities. Many of these statements are similar to or even the same as known properties of spaces with unique uniformities. This then generalizes some of the work on unique uniformities since a unique uniformity is auto­ matically •smallest for its topology. Based on what we have found out about smallest uniformities we will give new proofs of some of the characterizations of spaces with unique uniformities using the simple idea that a uniformity is unique if and only if it is both smallest and largest for its topology. Warren [19] has used the Tq identification to show that the results of Doss, Newns, and Gal are true even if one does not assume that the spaces are T 2 . Since we will not assume the T 2 property in Chapter 1 we will be proving some of Warren's claims without using Tg-identifications. In Chapter 2 we will see whether the property of being a smallest uniformity is preserved by some of the construc­ tions of topology, namely, uniformly continuous maps, subspaces, product spaces, sum spaces, and unions. We end the chapter with a discussion of functorial properties. There is a construction in topology called the hyper­ space, whose points are the non-empty closed sets of a given base space. Hausdorff [7] puts a metric on this space, Vietoris [18] worked with a hyperspace topology, and Bourbaki [l] defined a hyperspace uniformity. Many properties of the Bourbaki construction were worked out by Michael [12] and Caulfield [3]. In Chapter 3 we will give an introduction to hyperspaces, proving some properties we will need later. The question of when a hyperspace is locally compact is the theme of Chapter 4. Along the way we will show that a hyperspace uniformity is smallest for its topology if and only if it is compact. Since the union of two non-empty closed sets is a non-empty closed set, the union may be regarded as a binary operation on the points of the hyperspace. In this way, hyperspaces may be viewed as algebraic structures which carry a uniformity. In Chapter 5 we will give necessary and sufficient conditions that such an algebraic structure is unimorphic to a hyperspace. We will wrap up the chapter with some simple properties of such structures. NOTATION, TERMINOLOGY, AND BACKGROUND This section is intended as a broad survey of the assumptions and conventions used in the text. More infor­ mation can be found in Kelly [10], Willard [20], and Bourbaki [1 & 2].
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