On Topologies for Function Spaces

Given topological spaces Xt ÜT, and F and a function h from XXT to F which is continuous in x for each fixed ty there is associated with h a function h* from I t o F = F x , the space whose elements are the continuous functions from X to F. The function h* is defined as follows: h*(t)=ht, where ht(x)~h(x} t) for every x in X. The correspondence between h and h* is obviously one-to-one. Although the continuity of any particular h depends only on the given topological spaces X, 7\ and F, the topology of the function space F is involved in the continuity of h*. It would be desirable to so topologize F that the functions h* which are continuous are precisely those which correspond to continuous functions h. It has been known for a long time that this is possible if X satisfies certain conditions, chief among which is the condition of local compactness (Theorem 1). This condition is often felt to be too restrictive (since it practically excludes the possibility of X itself being a function space), and several years ago, in a letter, Hurewicz proposed to me the problem of defining such a topology for F when X is not locally compact. At that time I showed by an example (essentially Theorem 3) that this is not generally possible. Recently I discovered that, by restricting the range of T in a very reasonable way, one of the standard topologies for F has the desired property even for spaces X which are not locally compact (Theorem 2). In this last result the condition of local compactness is replaced by the first countability axiom and this appeals to me as a less troublesome condition. It should be pointed out that the problem is motivated by the special case in which T is the unit interval. When T is the unit interval, A is a homotopy and h* is a path in the function space; in the topology of deformations, equivalence of the concepts of "homotopy" and of "function-space path" is usually required. Among the various possible topologies for F there is one, which I shall call the compact-open (co.o.) topology, which seems to be the most natural. For any two sets, A in X and W in F, let M(A, W) denote the set of mappings ƒ £ F for which f(A)C.W. The co.o. topology is defined by selecting as a sub-basis for the open sets of F the