Graph topology for function spaces

Abstract

1. Introduction. Most of work in function space topologies concerns continuous functions. In this connection see a remark by Kelley [3, p. 217]. As soon as we begin to consider function spaces of noncontinuous functions we come face to face with some extremely difficult problems. So in order to make a beginning, it is advisable to consider first a subfamily of noncontinuous functions which, in a certain sense, can be approximated by continuous functions. One such subfamily consists of almost continuous functions which were introduced by Stallings [6]. An almost continuous function is one whose graph can be approximated by graphs of continuous functions (see 2.3). The need to introduce a suitable topology for function space of almost continuous functions arose when author was investigating essential fixed points of such functions in his doctoral thesis [4]. The introduction of a new function space called the graph topology, enabled him to tackle almost continuous functions. Let F denote an arbitrary subfamily of functions on a topological space X to a topological space Y and let F be given some topology. Most problems concerning F center round following question, what conditions on X and Y are sufficient to ensure that F has a desired property ? In this paper a few problems of above nature are discussed. This paper has a nonempty intersection with author's doctoral thesis written under supervision of Professor J. G. Hocking of Michigan State University. The author is grateful to his former colleague Professor D. E. Sanderson for valuable suggestions and comments. The referee suggested several improvements, supplied Example 5.1 and references [1] and [5].