The continuity properties of compact-preserving functions

Abstract

A function f:X→Y between topological spaces is called compact-preserving if the image f(K) of each compact subset K⊂X is compact. We prove that a function f:X→Y defined on a strong Fréchet space X is compact-preserving if and only if for each point x∈X there is a compact subset Kx⊂Y such that for each neighborhood Of(x)⊂Y of f(x) there is a neighborhood Ox⊂X of x such that f(Ox)⊂Of(x)∪Kx and the set Kx∖Of(x) is finite. This characterization is applied to give an alternative proof of a classical characterization of continuous functions on locally connected metrizable spaces as functions that preserve compact and connected sets. Also we show that for each compact-preserving function f:X→Y defined on a (strong) Fréchet space X, the restriction f|LIf′ (resp. f|LIf) is continuous. Here LIf is the set of points x∈X of local infinity of f and LIf′ is the set of non-isolated points of the set LIf. Suitable examples show that the obtained results cannot be improved.