Function-space compactifications of function spaces

Abstract

If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X,Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X,Z) has a compact Hausdorff topology whose relative topology on C(X,Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X,Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X×Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.