TITLE: A property of continuous maps with respect to compact subsets
QUESTION [6 upvotes]: I'm interested in continuous maps between topological spaces $f:X\to Y$ such that for any compact subset $L$ of $Y$ contained in $f(X)$, there is a compact subset $K$ of $X$ such that $L$ is contained in $f(K)$.
Proper maps satisfy this, but there are examples of continuous maps which don't, for example with discrete spaces, taking a non-stationary convergent sequence extended at infinity.
I would like to know if there are characterizations for those topological spaces which have enough compact subsets in the sense that: each real-valued continuous function satisfy this property.

REPLY [3 votes]: Such maps are called compact-covering maps, and are a somewhat well-known and well studied object. They came up naturally in many contexts, and if you look for that keyword in mathscinet, you will find many matches. As Bruno mentions, it is very easy to construct examples of maps in very ordinary settings that fail to be CC.