A Topology for Spaces of Transformations

Abstract

This paper defines and describes a particular type of topology2 for a class C of continuous functions on one topological space A to another, B; in other words, we topologize the class of transformations of A into (possibly a proper subset of) B. The topology is constructed thus?: Let K be any compact subset of A, and W be any open set of B. Denote by (K, W) the totality of functions f in C for which f(K) C W. By taking the collection of all sets of the form (K, W), and the intersections of all finite subcollections of these, and using them as the neighborhoods in C, one obtains what we shall call the k-topology for the function class C. Some of the properties of the k-topology which we establish are: If the range B is a metric4 space, a sequence f. in C converges, in the k-topology, to a limit f if and only if the distance from f(x) to fn(x) converges to zero uniformly for all x in any fixed compact subset of A. When A is locally compact, f, converges to f if and only if fn(x2n) converges to f(x) whenever x. converges to x. The latter convergence property is connected with a fundamental property of the k-topology: 1) When A is locally compact, the k-topology is admissible, which means that f(x) varies continuously in f and x simultaneously, and, 2) the k-topology is the strongest (i.e., giving the most limit elements) of all the admissible topologies that can be given to C. In fact, we have found that the existence of a strongest admissible topology for the class of real, continuous functions on a completely regular space A is equivalent to A's being locally compact. When the space B has a metric m, the k-topologized function space C can be