Proper map

Given topological spaces $𝑋, 𝑌$ , a map $𝑓 : 𝑋 \to 𝑌$ is called proper iff the preïmage $𝑓 - 1 (𝐾)$ of every compact $𝐾 \subseteq 𝑌$ is compact. #m/def/topology

Properties

If $X$ and $Y$ are locally compact
- If $𝑋$ is second-countable: a continuous map $𝑓 : 𝑋 \to 𝑌$ is proper iff every sequence without limit points maps to a sequence without limit points.
- A continuous map $𝑓 : 𝑋 \to 𝑌$ is compact iff the map
$\begin{align*} f^\star : X^\star &\to Y^\star \\ \omega &\mapsto \omega \\ x \in X &\mapsto f(x) \end{align*}$ between Alexandroff extensions is continuous.
If $𝑋$ is compact: all continuous maps are proper, since all compact subsets of $𝑌$ are closed and all closed subsets of $𝑋$ are compact.

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