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Metrization of Uniform Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by K-State Research Exchange METRIZATION OF UNIFORM SPACES by SAMUEL LELAND LESSSIG S., Fort Hays Kansas State College, 19&1 A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Mathematics KANSAS STATS UNIVERSITY Manhattan, Kansas 1963 Approved by: ft* Docw r-v, */WS TABLE 0? CONTENTS INTRODUCTION 1 RELATIONS 2 DEFINTION OF UNIFORM SPACE 3 DEFINTION OF BASE AMD SUBBASE k DEFINTION OF UNIFORM TOPOLOGY 6 UNIFORM CONTINUITY 9 PRODUCT UNIFORMITIES 12 METRIZATION 15 ACKNOWLEDGEMENT 21 REFERENCES 22 INTRODUCTION thesis in 1906 I 4 Frechet first considered abstract spaces in his J. Much of the developement of the concept of a topological space may be found in 1 6~] Kausdorff s C-rundzur-:e ler Mcngeniehre f and in the early volumes of Fundamenta Kathematica . From this research two fundamental concepts have developed: that of a toplogical space and that of a uniform space. Andre Weil was the first to formalize the notion of a uniform space in a paper in 193? [WJ* The concept of a uniform space developed from the study of topological groups and has presented many new ideas to the study of metric spaces. One of these is the concept that the finest uniformity for a metrizable space is the one in- . using uniform spaces in duced by a metric ^12 J J. R. Isbell has done work ~| the study of the algebra of functions f? • This report is devoted to the examination of several properties of uni- form spaces and the relationship between uniformities and pseudo-metrics. 2 RELATIONS In the study of uniform spaces the subsets of a carteian product X ">C X of a set X with itself arc considered. These subsets are relations or ordered 1 pairs. If is a relation, then U" is the set of all (x,y) such that (y,x) = -1 is in U. The process of taking inverses is involutory. If U U , then U is said to be symmetric. If U and V are relations, then U«V is defined as the set of all pairs (x,z) such that for some y it is true that (y,z) is in U and (x,y) is in V. This operation between relations is called composition. It is easily shown that composition is associative. The inverse of a composition is the composition of the inverses in the reverse order. The identity relation is defined to be the set of all (x,x) such that x is in X. This identity re- lation will be referred to as the diagonal and denoted by £± . For each sub- set A of X, the set U [a] is defined to be the set of all y such that (x,y) is in U for some x in A. If x is a point of X, then U TxJ is defined to be U [fc]] • It can be shown that for each U and V and each A in X it is true that U«V [a] is equal to U [V . | AJ ] To complete the list of needed properties of relations, an important lemma is included. LE^-IA I : If V is symmetric, then V«U"V = jv [x] v : ( x (J X [y] .y) is in . Proof: Ety- definition V«U«V is the set of all pairs (u,v) such that for some x and y in X, (u,x) is in V, (x,y) is in U and (y,v) is in V. Since V is sym- metric, this is the set of all (u,v) such that u is in V [x] and v is in V £yj for some (x,y) in U. However, u is in V £xj and v is in V £yj if and = : only if (u,v) is in V [x] X V [y] . Thus V*U*V £(u,v) (u,v) is in 3 ior some (x,y) in U [ence V«U*V is equal to is in U DEFINITION OF UNIFORM SPACE A uniformity for a set X is a non-empty family Zt of subsets of X X X such that: (a) member of diagonal each 21 contains the ; 1 (b) if U is in 21 , then U" is in U ; (c) if U is in tt, then V«V C U for some V in tl; (d) if U and V are members of Z( , then Ufl V is in Z( ; (e) if U is in # and U C V C X X X, then V is in *Ll . The pair (X, V. ) is called a uniform space. For a given space X there exists many different uniformities, ranging from the set of all subsets of X X X which contain /-±> to the set whose only member is X X X. It follows from condition (a) that each member of ZC is a neighbor- hood of , but the converse is not true. If X is the set of real numbers with the usual uniformity for X of the family of all subsets of X ><• X such that ^(x,y) : | x-y| <T r| is contained in U for some positive number r, then consider the j\x,y) : < 1/(1 + ) 1 . This set is a Jx-yj | y| neighborhood of the diagonal but is not a member of 21 since it fails to satisfy condition (c). 4 DEFINITION OF BASE AND SUBBASE A subfamily Q of a uniformity H is a base for V, if. and only if each member of 2/ contains a member of (3 . From this definition if is evident that C8 determines 21 entirely, for each member of (S is in *L( and a subset U of X X X belongs to 21 if and only if U contains a member of o3 • A sub- family ^ is a subbase for U if and only if the set of all finite inter- sections of members of ^ is a base for H . To characterize a base for a uniformity the following theorem is given. THEOREM I : A family & of subsets of X X X is a base for some uniform- ity for X if and only if: (a) each member of s\ contains the diagonal ; (b) if U is in C3 • then U" contains a member of (3; (c) if U is in , then VoVcU <g for some V in <3 ; (d) the intersection of two members of <S contains a member of (S . Proof: It is obvious that given a family <3 of subsets of X X X satisfying these four conditions, then <3 forms a base for some uniformity 71 . Therefore it suffices to show that if is a base for some uniformity K , then the four conditions are satisfied. Let be a base ^ for some uniformity 2/ , then by definition each member of contains a % member of <g and each member of <S is in 2( . Since each member of % contains the diagonal, it is evident that each member of <3 contains the diagonal. If U is in @ , then U is in It . Since V. is a uniformity, is in U~ U and hence contains a member of (3 . If U is in then U <S , is in 'U and there exists a V in V such that V*V C U. 5 r contained however, V in 11 implies there exists a \ \ in <3 such that V-| is in V. Then V«»V-| C V«V, v:hich implies that V-j »l| is contained in U. For the last condition, let 3-j and B£ be members of <3 . Then Bj and B£ arc in K contains a member of (3 • and hence Ej O B2 is in ^ , and thus Bj Ci Efe The property of being a subbase is not as easy to characterize as the property of being a base. However, the following theorem will be needed in the study of uniform spaces. THSCRZM II : A family <> of subsets of X X X is a subbase for some uniformity of X if: (a) each member of contains the diagonal Z\ ; (b) for each U in £ the set U~' contains a member of S ; (c) for each in £ there is a V in ^ such that y«V C U. In particular, the union of any collection of uniformities for X is a subbase for a uniformity for X. Proof: It suffices to show that the family <S of finite intersections of members of ^ satisfy the conditions of theorem I, If U-j U2, U and V-j Vj?, V are subsets of X XX such that , n , m and Vj are in i and if U = .11 U-? and V = ,f\ V*. then it is obvious that the first condition is satisfied. By observing that V C U " (or V*V C U) whenever Vj_ C Uj_ ' (or V^*. Vj_ C Uj_) for each i, then the remainder of the conditions in Theorem I follow immediately. 6 DEFINITION OF UNIFORM TOPOLOGY If (X, 1L ) is a uniform space, then the topology of the uniformity 71 or the uniform topology, is the family. of all subsets T of Z such that for each x in ? there is a U in % such that U [xj C T. To verify that 3 is a topology it suffices to show that J is closed under an arbitrary union of its members and the intersection of any two members. Let T be any arbitrary set of = u elements of and let T U T , Let x be an element of T; then x is in some T ^ .
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