## István Juhász, Jan van MillOn maps preserving connectedness and/or compactness

Comment.Math.Univ.Carolin. 59,4 (2018) 513-521.

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

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