Proper map

2020 Mathematics Subject Classification: Primary: 54C05 [MSN][ZBL]

Let $X$ and $Y$ be two topological spaces. A continuous map $f:X\to Y$ is called proper if $f^{-1} (K)$ is compact for every $K\subset Y$ compact.

When $X$ is Hausdorff and $Y$ locally compact the properness of $f$ is equivalent to the requirement that $f^{-1} (\{y\})$ is compact for every $y\in Y$.

If $X$ is a compact space and $Y$ is Hausdorff, then any continuous map $f:X\to Y$ is proper and closed.