FUNCTIONS WITH CLOSED GRAPHS • by F. JON LEITCH B.Sc. University of Guelph, 1968 THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of MATHEMATICS . We accept this thesis as conforming to the required standard The University of British Columbia • March 1970 In presenting this thesis in partial fulfilment of the requirements for- an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study* I further agree that permission., for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without ray written permission. Department of The University of British Columbia Vancouver 8, Canada Thesis Supervisor: J. V. Whittaker ABSTRACT This paper concerns itself mainly with those functions from one topological or metric.space to another that have closed graphs in the product space. Their relationship to closed, locally closed, compact, continuous and subcontinuous functions is studied in order to determine the relative strength of the closed graph condition. The paper collects and in some cases extends results found in papers by R. V. Fuller [2], P. E. Long [7] P. Kostyrko'and T. Shalat [4], [5] and [6]. The main theorems deal with; l) the characterization of continuous functions in terms of subcontinuity and the closed graph property; 2) a proof that if f has a closed graph then f is the limit of a sequence of continuous functions; and 3) a study of the operations under which the class of functions with closed graphs is closed. Ill Table of Contents page Chapter 0: Notation and Definitions 1 Chapter 1: Introduction of closed graph 5 • property; open functions with closed- graphs Chapter 2: Subcontinuous functions; 10 relations between functions with various properties Chapter 3: Characterization of continuous 17 functions; limit theorem; statements about domain and range spaces Chapter 4: Points of irregularity; limit 29 numbers Chapter 5: Closed and locally closed 35 functions; topologies determined by compact subsets Chapter 6: Analysis of the class of functions. 43 with closed graph; product spaces. iv Acknowledgements I should like to thank Professor J.V. Whittaker for suggestion of the topic, his patient reading of the original draft and for his helpful criticisms. Also I would like to acknowledge the financial support of the National Research Council of Canada. CHAPTER 0 This chapter is devoted to an explanation of the notation to be used and some definitions. Definitions: , FILTER A class 5 of subsets of a nonempty set X is a filter on X if the following conditions are satisfied: 1. if A,B e 3 then ADB e 3 ' 2. if A e and BsA then B e 5. 3. 0 k * A maximal filter on X is called an ultrafilter. If is any filter on X and U. is an ultrafilter then either $<^lLor and IX. are not comparable. Maximal filters always exist since the class Q = {ACX|Xq e A for a fixed Xq e X} is a principal (fixed) ultrafilter. If X is an infinite set there exist non- principal (free) ultrafilters. TOPOLOGY A class 3" of subsets of a set X is a topology for X if the following conditions are satisfied: 1. If A,B e then ADB e 3\ 00 2. if A e ff n=l,2,... then u A e ff n n=i n 3. 0, X e if - 2 - The • sets belonging to 3" .are called the open sets of X . If . A e 3" then the complement of A is said to be closed. To avoid possible confusion we present our definition of, the four topologies mentioned in this paper. FRECHET (T1): A topology 3" on a set X is T^ if and only if for each x 4 y, x,y e X, there exists M,N e 3" such that x e M, y | M and y e N, x N . HAUSDORFF.(T ) A topology J on a set X is T^ if and only if for each x =j= y, x3y e X, there exist M,,N e 3" such that x e M, y e N and MON = 0 REGULAR: A topology 31 on a set X is regular if and only if for each x | A, where A is a closed subset of X , there exist open M,N such that ACN, x e M and MDN = 0 NORMAL: A topology 3" on a set X is normal if and only if for each pair of closed sets A,BcX with ADB =' $ there are open, disjoint sets M,N with the property that .ACM and B<=N . NEIGHBOURHOOD N Let x e X and let 3" be a topology on X . Then - 3 - N<=X is a ^-neighbourhood or simply a neighbourhood of x if there is an 0 e J such that x e O5N DIRECTED SETS AND NETS A binary relation R on a set X is called a preorder if it is reflexive and transitive: i.e. 1. (x,x) e R for all x e X 2. if (x,y) and (y,z) belong to R then (x3z) e R . A set X together with a definite preorder is called a preordered set. A directed set D is a preordered set with the additional property that for each a,b e D there exists a c e D such that a<c and b<c . A net in a space X is a mapping <p:D -» X of some directed set D into X . We will denote nets either by {xola e A} or, if the directed set A need not be a mentioned,, simply by xQ . a A subnet of x will be denoted by x,- where a J Nb b is a member of the domain of x^ and N is the appropriate function from the domain of x^ to the domain of xa ' PRODUCT DIRECTION Let A and B be two ordered sets. In AxB we define the product direction as follows; (a-.,b-.) _< (ap,b?) - 4 - in A x B if and only if a1<a in A or a± = ag and bl—^2 in ^ * With "this definition A x B is an ordered set, CONVERGENCE x a e A A net C al } converges to a point x in a space X if and only if for each neighbourhood . N of x there exists an a„ e A such that if b>a.T , x, e N . N — N 3 b Throughout the remainder of the paper we will use the following notation: iff if and only if nbhd. neighbourhood X-A complement of A with respect to X A~ the closure of A Gf the graph of the function f X x Y the Cartesian product of X with Y x„ -• x the net xo converges to the point x a a Z+ the positive integers f:X -» Y the function f maps X into Y 3 there exists such that A' the derived set of A , i.e. the set of all accumulation (cluster) points of A . - 5 - CHAPTER 1 Following Fuller [2], we introduce the notion of a function having a closed graph in terms of lim sup (Ls) and lim inf (Li) of nets of sets. We also present a character• ization of open functions and give necessary and sufficient conditions for an open function to have a closed graph. Definition 1.1 A function f: X -* Y has a closed graph (relative to X x Y) iff [(x, f(x))|x e X} is closed in the product topology of X x Y . In terms of nets this becomes Gf is closed iff (x , f(x )) -« (p,q) in X x Y implies that q = f(p) • The following Lemma will be used a great deal in Chapter jj. Lemma 1.2 Let f:X -» Y be given. Then Gf is closed iff for each x e X and y e Y , where y 4 f(x) , there exist open sets • U and V , containing x and y respectively, such' that f (U)nv = 0 . Proof: If the condition holds and (x,y) G^ then U x V is an open set in X x Y such that (U x V)nG^ = 0 which Implies G~ is closed. - 6 - If Gf is closed and (x,y) Gf , then y ^ f(x) , so there exists a basic open set of the form U x V where U and V are open^ containing x and y respectively, and such that (U x V)nGf = 0 . Therefore f(U)HV = 0 . The topological convergence of a net of subsets of a topological space X may be defined in the same manner as the topological convergence of a sequence of sets. Definition 1.3 If C-An|n e D} is a net of subsets of X , then Li A (Ls An) is defined as the set of all x e X such that n v ' every nbhd of x intersects An for almost all (respectively for arbitrarily large) n . We say that a statement S on elements of a directed set D is fulfilled for: 1. almost all n e D if there is an n e D such o that S is fulfilled for all n>n — o 2. arbitrarily large n e D if the set of all n e D for which S if fulfilled is cofinal in .