Function-space compactifications of function spaces

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Colleges, School and Institutes

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 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 a-re pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways. (C) 2002 Elsevier Science B.V. All rights reserved.

Details

Original languageEnglish
Pages (from-to)441-463
Number of pages23
JournalTopology and its Applications
Volume120
Issue number3
Publication statusPublished - 15 May 2002