On maps preserving connectedness and /or compactness

TL;DR

This paper investigates when maps that preserve properties like connectedness or compactness are necessarily continuous, showing that non-trivial product functions with connected domains and T1 ranges are continuous if they preserve connectedness.

Contribution

It proves that non-trivial product functions with connected domains and T1 ranges that preserve connectedness are necessarily continuous, and explores related properties and examples.

Findings

Non-trivial product functions with connected domain and T1 range preserving connectedness are continuous.

The analogous statement for compactness-preserving functions does not hold.

Provides examples and results on maps that preserve compactness and continuum properties.

Abstract

We call a function $f: X\to Y$ $P$-preserving if, for every subspace $A \subset X$ with property $P$, its image $f(A)$ also has property $P$. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions is such a map continuous, has a long history. Our main result is that any non-trivial product function, i.e. one having at least two non-constant factors, that has connected domain, $T_1$ range, and is connectedness-preserving must actually be continuous. The analogous statement badly fails if we replace in it the occurrences of "connected" by "compact". We also present, however, several interesting results and examples concerning maps that are compactness-preserving and/or continuum-preserving.