# Is there a name for this property of maps between topological spaces?

Let $X$, $Y$ be Hausdorff locally compact spaces and let $f \colon X \to Y$ be a proper continuous map. Consider the following property (P): $$\text{for any compact set K in Y and for any open neighborhood \mathcal O of f^{-1}(K)}\\ \text{there exists an open neighborhood \mathcal U of K such that f^{-1}(K) \subset f^{-1}(\mathcal U) \subseteq \mathcal O.}$$ In particular, this property holds if we additionally require $f$ to be open or bijective. But does this property (P) hold in general or, otherwise, does it have some simple characterization?

The property you describe is equivalent to closedness of the map $f$. To show that it holds for a closed map $f$, we introduce the function $$f_\forall: \mathcal P(X) \to \mathcal P(Y) \\ A \mapsto \{y\mid f^{-1}(y)\subseteq A\},$$ Note that $f_\forall$ sends open sets to open sets since $f_\forall(O)=Y\setminus f(X\setminus O)$. Now given $f^{-1}(K)\subseteq O$, we can take $U$ to be $f_\forall(O)$. Compactness of $K$ is not needed at all.
However, if the property holds for compact sets $K$, then it holds for points, and one can show that this implies that $f$ is closed.