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OBJ (Application/Pdf) ON METRIC SPACES AND UNIFORMITIES A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY JOHN H. HARRIS DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA Ml 1966 RrU \ ~ 35 ACKNCMLEDGI*ENTS The most lucid presentation in English, to my knowledge, of the concept of a uniform space is found in John Kelly's classic, "General Topology". This text was trost influential in my choice and arrangement of materials for this thesis. I would also like to acknowledge and thank my instructors for their assistance and suggestions. A special thanks is extended to my parents without whose inspiration and encouragement this thesis would not have been possible. ii CONTENTS Page ACKNOWLEDGMENTS ii INTRODUCTION I Chapter I. Uniform Spaces « 3 1.1 Uniformities and The Uniform Topology 1.2 Uniform Continuity 1.3 The Metrization Theorem 1*JU Compactness II. Uniform Spaces and Topological Groups ... 23 III, Selected Problems ••..,.•.••...2? APPENDIX , . 31 INDEX OF SPECIAL SYMBOLS 3h BIBLIOGRAPHY 36 INTRODUCTION One of the major advantages of the theory of topology is that it lends itself readily to intuitive interpretation. This is especially true of metric spaces because of their close connection with the real number system and because of our intuitive concept of distance associated with that system. However, if we should attempt to generalize metric spaces, and we have, our intuition would often fail us for we would also have to generalize our already intuitive concept of distance. In this thesis, I shall attempt to present in a lucid and in an intuition appealing fashion, the generalized theory of metric spaces,“i.e., the theory of uniform spaces. The theory of uniform spaces"is an outgrowth of the development of the theory of topological spaces which, paradoxically, grew out of- the-attempt to generalize metric spaces. As some important properties of metric spaces were lost in the generalization to topological spaces, it was onljr natural that mathematicians would seek some intermediate structure more general than metric spaces and less general than topological spaces in which these properties were re¬ tained. The pioneering effort along this line was due to the French mathematician Andre Weil, His definitive work on uniform 2 spaces was published in 193? and entitled, "Sur les Espaces a Structure Uniform." In uniform spaces, the important metric concepts of uniform continuity and of Cauchy sequences are retained neither of these concepts are applicable to topological spaces in general. As a result, many of the metric space ideas which depend on the concepts of a uniform distance and of sequential convergence are retained. It is assumed that the reader has a knowledge of metric spaces. However, for purposes of convenience and comparison, I have stated some of the major results of metric spaces in Appendix B—primarily the classical metrization theorem. Chapter I contains the basic theory of uniform spaces. In Chapter II, I have considered an important application of uniform spaces. Chapter II consists of problems from J. Kelley* s “General Topology", Appendix A contains the necessary definitions and theorems on relations along with a very important lemma. CHAPTER I UNIFORM SPACES 1,1 Uniformities And The Uniform Topology,—-A generalization of metric spaces would very likely contain structures related to the open balls of metric spaces. This structure turns out to be a set of relations which we call a uniformity, A uniformity for a set X is a non-empty family U. of sub¬ sets of X x X such that the following hold: 1) each U e U contains the diagonal, ^ j 2) U« IL 4U_1 « U ; 3) if U « It, then VoV C U for some V t Uj U) U and VtU^UOVe U ; and 5) if UtUandUC VC XxX, then V* K. The pair (X, U ) is a uniform space. For example, let X be an arbitrary non-empty set and consider U*fx x x} • The single relation XxX obviously satisfies conditions (l)-(5). (X,U) is called the discrete uniformity for X, The indiscrete uniformity consists of all subsets of X x X which contain the diagonal, A family ft of elements is called a base for a uniformity XL if each U é XL contains an element of • Theorem 1,1,1,—A non-empty family of subsets of X x X is a base for some uniformity for X iff (l1) each member of contains the diagonal? (21) U ( (i> ^ U"1 contains an element of y3 j (3*) if U € y3 , then U3V»V for some V € ? and (1+*) U and V* 0 UflY contains a member of jj . 3 h Proof. —Suppose fi is a base for some uniformity, H of X. Then (l) fi is a subfamily of K and (2) each U è H contains an element of fi . (1^) holds for ficU., (2^) holds for B fj5 ^ B_1€ U => 3 B1 t fi îB'^B1, (31) holds for, 1 1 Bt|îs>3? é UîVoVCB^ But V e Ü.^ J V e /3 3 V C V, thus 1 1 1 V oV C. VoV C B. Finally, B1 and B2 < fi => B±C\ Bg € U 3 B1 H B2 O B for some B^ 6 f$ . Conversely, if we assume (l)-(li) holds then to obtain our uniformity, we simply take all subsets of X x X which contain an element of . QED For example, consider X to be the set of real numbers. Denote by p , all subsets of X x X of the form Ur= { (x,y) : I x-y | < r ] (where r is arbitrary, positive and real). Now Ur D A since for any r and any x l x-y l «o < r <;>x*y. Since | x-y | < r iff |y-xl<r, 1 u _ we have (x,y) € Ur iff (x,y) € Ur”- - or r*Ur ^. If r^ and r2 are given, C. U 0 Ur2 holds if r is any positive number smaller than min (r^r,,). Finally, if r is given, then o U/g C Ur since {(y>z) : I y-z| < r/2 1 o{(x,y): l x-y l <. r/2 }c {(x,z) : | x-z | < r j by the triangular inequality. Hence (3 is a base for some uniformity u on X. This uniformity is called the usual uniformity for the reals. This example is instructive in several ways. Firstly, note that Ur defines a symmetric neighborhood of A i.e. 5 y < x+r, x < y l x-y | < r=> < U > x-r, y < x (cf. fig. 1). Now choose a specifics Xq€ R. Then Ur[xo] is { y: (xQ,y) * Ur 1 and this simply the open ball of radius r about xQ (cf. fig. 2), Hence, » X our uniformity generalizes the open ball concept of metric spaces. Secondly, observe that the condition I x-y I < r can be replaced by an arbitrary metric, p , without altering the proof. Clearly, then, conditions (l)-(3) for a uniformity are generalizations of metric * concepts, i.e., p (x,x)»o condition (Di p (x,y)« p(y,x) (2) and p (x,z) { p(x,y)+ p (y,z) (3) , roughly, i.e., it says for r-spheres 3 r/2 spheres. We next consider the idea of a sub-base. A subfamily $ is a sub-base for U iff the family of finite intersections of members of S is a base for TA. • Theorem 1.1,2,—A non-empty family S of subsets of X x X is a sub-base for some uniformity for X if (lw) each member of S contains A j (2,r) U* S contains a member of S ; and (3”) if Ü € $ , then VoV C U for some V € $ . M 6 It is important to note that the union of any collection of uniformities is a sub-base for some uniformity—such a union may not be a base because we cannot guarantee that (U* ) will hold. The previous example indicates in a very clear way the fact that uniformities induce topologies. If we were to hold *x’ fixed, say x=x0, and let r vary over the positive reals, then Ur [ xQ ] would determine the neighborhood system of xQ relative to the usual topology for the reals. Given a uniform space (X, UL ), we define a topology, 3 on X by defining T « J iff for each x € T 3 a U*U * U txl C T. is called the uniform topology (denoted (X, 3 u)). Theorem 1.1.3.—(X, J is a topological space. Proof.—We have to show (l) the union of any number of elements of 3 u is an element of 3 and (2) the intersection of any two T's in 3 u is in 1 The first follows directly from the definition y ; for if T^C *]u, OC 6 A , then each x € U Trf is in some Tt which implies 3 UtU. » U £ x)c\JT*» Secondly, let T and S be in 3* CC then if x« T fl S, then xfeT and x t S. Hence U £xl C T and V IxlCS for some U, V € IL . Then u[x] nv[xKun»[x]CTOS. But (U n V) € U and we are done. QED We now derive some elementary properties on (X, 3u). Theorem 1.1 .it.—The interior A0 of a subset A of X with uniform topology is the set B, defined as B« { X} U [x] C A for some U fc U. f .
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