The continuity properties of compact-preserving functions

TL;DR

This paper characterizes compact-preserving functions on strong Frechet spaces, linking their properties to local behavior, and explores their continuity on specific subsets, providing new insights into their structure and classical function properties.

Contribution

It provides a new characterization of compact-preserving functions on strong Frechet spaces and analyzes their continuity on certain subsets, extending classical results.

Findings

Characterization of compact-preserving functions via local neighborhoods and finite set conditions.

Proof that restrictions of such functions to specific subsets are continuous.

Examples demonstrating the optimality of the results.

Abstract

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is compact-preserving if and only if for each point $x\in X$ there is a compact subset $K_x\subset Y$ such that for each neighborhood $O_{f(x)}\subset Y$ of $f(x)$ there is a neighborhood $O_x\subset X$ of $x$ such that $f(O_x)\subset O_{f(x)}\cup K_x$ and the set $K_x\setminus O_{f(x)}$ 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 $f:X\to Y$ defined on a (strong) Fr\'echet space $X$, the restriction $f|LI'_f$ (resp. $f|LI_f)$ is continuous. Here $LI_f$ is the set of points $x\in X$ of local infinity of $f$ and $LI'_f$ is the set of non-isolated points of the set $LI_f$. Suitable examples show that the obtained results cannot be improved.