Properties of almost continuous functions

Abstract

Introduction. Professor T. Husain [2] has defined the concept of an function from one topological space into another and investigated some of their properties. His definition is as follows: DEFINITION 1. The functionf: X-> Y is at xoEX if and only if for each open VC Y containing f (xo), Cl(f-'(V)) is a neighborhood of xo. Iff is at each point of X, then f is called continuous. We prove several results concerning functions and point out that the hypotheses on two of Husain's theorems may be weakened considerably. Functions having closed graphs also play an important role in our investigation. We note that Stallings [6] has also defined a type of function which he calls continuous. His definition is given next. DEFINITION 2. The function f: X-> Y is if and only if given any open set W containing the graph of f, there exists a g: X-> Y such that the graph of g is a subset of W. Of course, any function satisfies both definitions. However, the following examples show that neither of these two functions, in general, imply the other. Example 1 gives a function f which is in the sense of Husain but is not in the sense of Stallings. Furthermore, f is not a connected function (i.e., f does not preserve connected subsets of X) and is not a connectivity function (i.e., the graph map g(x) = (x,,f(x)) from X into X X Y is not a connected function). Example 2 gives a function which is in the sense of Stallings, is connected, is a connectivity function but is not in the sense of Husain. EXAMPLE 1. Let R represent the reals with standard topology. Let f: R-*R be given byf (x) = x if x is rational and f (x) =-x if x is irrational. EXAMPLE 2. With R as in Example 1, define f: R->R as f(x) sin(1/x) if x 5 0, and f(O) = O. When dealing with functions throughout the remainder of this paper, we will be referring only to those of the type described by Husain. Hereafter, we abbreviate almost continuous with a.c. The notation C1(A) will denote the closure of the set A and