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 $P$.



What is this property of surjective continuous maps called?
A map $f\colon X\to Y$ is a *compact-covering* map if every compact $K\subseteq Y$ is the image of some compact $C\subseteq X$.


