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Let $f\colon X\to Y$ be a continuous map between topological spaces, which you can assume to be Hausdorff if you like. Say that $f$ has property $P$ if for every compact subset $L\subseteq Y$, there exists a compact subset $K\subseteq X$ with $f(K)=L$. Is there a more standard name for this?

Clearly a map with property $P$ must be surjective. Any proper surjective map has property $P$. A product projection $Y\times Z\to Y$ has property $P$ iff $Z\neq\emptyset$, but is only proper if $Z$ is compact. More generally, I think that any surjective fibre bundle projection has property $P$. Any map that has a continuous section has property $P$.

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  • $\begingroup$ In the context of continuous linear operators between topological vector spaces one often says the $f$ lifts compact sets (similarly, $f$ may lift bounded sets, null sequences, etc.). $\endgroup$
    – Jochen Wengenroth
    Commented Jun 6, 2022 at 11:53

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A map $f\colon X\to Y$ is a compact-covering map if every compact $K\subseteq Y$ is the image of some compact $C\subseteq X$.

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    $\begingroup$ Do you have a reference for this definition? $\endgroup$
    – Jochen Wengenroth
    Commented Jun 7, 2022 at 6:02
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    $\begingroup$ Encyclopedia of General Topology - Authors: K.P. Hart, Jun-iti Nagata, J.E. Vaughan-1st Edition - October 1, 2003 - p. $ {\bf c-03-5}$. $\endgroup$
    – Alexander Osipov
    Commented Jun 7, 2022 at 7:09

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