A property of continuous maps with respect to compact subsets 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.
 A: Every path-connected space $X$ satisfies this property. The image $f(X)$ is a connected subset of $\mathbb R$ and is hence a (possibly infinite, or trivial) interval. Every compact sub-interval $[y_0,y_1]$ is contained in the image of a compact path in $X$ connecting a point in $f^{-1}(y_0)$ to another point in $f^{-1}(y_1)$. Every compact subspace $L$ of $f(X)$ is contained in one such sub-interval, so you are done.
Note however that it is easy to construct an $X$ with two path-connected components which does not satisfies the property. For instance, take $X$ as two copies of $\mathbb R$ and define $f$ as constantly $\pi/2$ on one copy and as $f(x) = \arctan x$ on the other. 
 Added.  If $Y = \mathbb C$ then this is no longer true. For instance, a (non-embedding) immersion $f:\mathbb R \to \mathbb C$ with compact image is a counterexample: simply take $L= f(\mathbb R)$. 

A: 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.
