Abstract:We call a function $f\colon X\to Y\,$ 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 such a map is continuous, has a long history. Our main result is that any nontrivial product function, i.e.\ one having at least two nonconstant 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.
Keywords: compactness; connectedness; preserving compactness; preserving connectedness
DOI: DOI 10.14712/1213-7243.2015.263
AMS Subject Classification: 54C05 54D05 54F05 54B10