Proper map

Definition

There are several competing definitions of a "proper function". Some authors call a function ${\displaystyle f:X\to Y}$ between two topological spaces proper if the preimage of every compact set in ${\displaystyle Y}$ is compact in ${\displaystyle X.}$ Other authors call a map ${\displaystyle f}$ proper if it is continuous and closed with compact fibers; that is if it is a continuous closed map and the preimage of every point in ${\displaystyle Y}$ is compact. The two definitions are equivalent if ${\displaystyle Y}$ is locally compact and Hausdorff.

If ${\displaystyle X}$ is Hausdorff and ${\displaystyle Y}$ is locally compact Hausdorff then proper is equivalent to universally closed. A map is universally closed if for any topological space ${\displaystyle Z}$ the map ${\displaystyle f\times {\mathrm{id}}_Z:X\times Z\to Y\times Z}$ is closed. In the case that ${\displaystyle Y}$ is Hausdorff, this is equivalent to requiring that for any map ${\displaystyle Z\to Y}$ the pullback ${\displaystyle X{\times }_{Y}Z\to Z}$ be closed, as follows from the fact that ${\displaystyle X{\times }_{Y}Z}$ is a closed subspace of ${\displaystyle X\times Z.}$

An equivalent, possibly more intuitive definition when ${\displaystyle X}$ and ${\displaystyle Y}$ are metric spaces is as follows: we say an infinite sequence of points ${\displaystyle \{p_i\}}$ in a topological space ${\displaystyle X}$ escapes to infinity if, for every compact set ${\displaystyle S\subseteq X}$ only finitely many points ${\displaystyle p_i}$ are in ${\displaystyle S.}$ Then a continuous map ${\displaystyle f:X\to Y}$ is proper if and only if for every sequence of points ${\displaystyle \left\{p_i\right\}}$ that escapes to infinity in ${\displaystyle X,}$ the sequence ${\displaystyle \left\{f\left(p_i\right)\right\}}$ escapes to infinity in ${\displaystyle Y.}$