In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.^[1] In algebraic geometry, the analogous concept is called a proper morphism.

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.

Partial proof of equivalence

Let ${\displaystyle f:X\to Y}$ be a closed map, such that ${\displaystyle f^{-1}(y)}$ is compact (in ${\displaystyle X}$) for all ${\displaystyle y\in Y.}$ Let ${\displaystyle K}$ be a compact subset of ${\displaystyle Y.}$ It remains to show that ${\displaystyle f^{-1}(K)}$ is compact. Let ${\displaystyle \left\{U_{a}:a\in A\right\))$ be an open cover of ${\displaystyle f^{-1}(K).}$ Then for all ${\displaystyle k\in K}$ this is also an open cover of ${\displaystyle f^{-1}(k).}$ Since the latter is assumed to be compact, it has a finite subcover. In other words, for every ${\displaystyle k\in K,}$ there exists a finite subset ${\displaystyle \gamma _{k}\subseteq A}$ such that ${\displaystyle f^{-1}(k)\subseteq \cup _{a\in \gamma _{k))U_{a}.}$ The set ${\displaystyle X\setminus \cup _{a\in \gamma _{k))U_{a))$ is closed in ${\displaystyle X}$ and its image under ${\displaystyle f}$ is closed in ${\displaystyle Y}$ because ${\displaystyle f}$ is a closed map. Hence the set $${\displaystyle V_{k}=Y\setminus f\left(X\setminus \cup _{a\in \gamma _{k))U_{a}\right)}$$ is open in ${\displaystyle Y.}$ It follows that ${\displaystyle V_{k))$ contains the point ${\displaystyle k.}$ Now ${\displaystyle K\subseteq \cup _{k\in K}V_{k))$ and because ${\displaystyle K}$ is assumed to be compact, there are finitely many points ${\displaystyle k_{1},\dots ,k_{s))$ such that ${\displaystyle K\subseteq \cup _{i=1}^{s}V_{k_{i)).}$ Furthermore, the set ${\displaystyle \Gamma =\cup _{i=1}^{s}\gamma _{k_{i))}$ is a finite union of finite sets, which makes ${\displaystyle \Gamma }$ a finite set. Now it follows that ${\displaystyle f^{-1}(K)\subseteq f^{-1}\left(\cup _{i=1}^{s}V_{k_{i))\right)\subseteq \cup _{a\in \Gamma }U_{a))$ and we have found a finite subcover of ${\displaystyle f^{-1}(K),}$ which completes the proof.

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 ${\displaystyle f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z}$ 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 ${\displaystyle Y.}$

Properties

• Every continuous map from a compact space to a Hausdorff space is both proper and closed.
• Every surjective proper map is a compact covering map.
  • A map ${\displaystyle f:X\to Y}$ is called a compact covering if for every compact subset ${\displaystyle K\subseteq Y}$ there exists some compact subset ${\displaystyle C\subseteq X}$ such that ${\displaystyle f(C)=K.}$
• A topological space is compact if and only if the map from that space to a single point is proper.
• If ${\displaystyle f:X\to Y}$ is a proper continuous map and ${\displaystyle Y}$ is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then ${\displaystyle f}$ is closed.^[2]

Generalization

It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see (Johnstone 2002).