On maps preserving connectedness and /or compactness

DOI10.14712/1213-7243.2015.263arXiv1801.06212WikidataQ128635856 ScholiaQ128635856MaRDI QIDQ6296655

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Publication date: 18 January 2018

Abstract: We call a function $f:XoY$ $P$-preserving if, for every subspace $AsubsetX$ 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.

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