Proper map

This article is about the concept in topology. For the concept in convex analysis, see proper convex function.

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. 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 $f:X\to Y$ between two topological spaces proper if the preimage of every compact set in $Y$ is compact in $X.$ Other authors call a map $f$ 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 $Y$ is compact. The two definitions are equivalent if $Y$ is locally compact and Hausdorff.

Partial proof of equivalence

Let $f:X\to Y$ be a closed map, such that $f^{-1}(y)$ is compact (in $X$) for all $y\in Y.$ Let $K$ be a compact subset of $Y.$ It remains to show that $f^{-1}(K)$ is compact.

Let $\{U_a:a\in A\}$ be an open cover of $f^{-1}(K).$ Then for all $k\in K$ this is also an open cover of $f^{-1}(k).$ Since the latter is assumed to be compact, it has a finite subcover. In other words, for every $k\in K,$ there exists a finite subset ${\gamma }_k\subseteq A$ such that $f^{-1}(k)\subseteq {\cup }_{a\in {\gamma }_k}{U}_a.$ The set $X\setminus {\cup }_{a\in {\gamma }_k}{U}_a$ is closed in $X$ and its image under $f$ is closed in $Y$ because $f$ is a closed map. Hence the set $$V_k=Y\setminus f(X\setminus {\cup }_{a\in {\gamma }_k}{U}_a)$$ is open in $Y.$ It follows that $V_k$ contains the point $k.$ Now $K\subseteq {\cup }_{k\in K}{V}_k$ and because $K$ is assumed to be compact, there are finitely many points $k_1,\dots ,k_s$ such that $K\subseteq \underset{i=1}{\overset{s}{\cup }}{V}_{k_i}.$ Furthermore, the set $\Gamma =\underset{i=1}{\overset{s}{\cup }}{\gamma }_{k_i}$ is a finite union of finite sets, which makes $\Gamma$ a finite set.

Now it follows that $f^{-1}(K)\subseteq f^{-1}\left(\underset{i=1}{\overset{s}{\cup }}{V}_{k_i}\right)\subseteq {\cup }_{a\in \Gamma }{U}_a$ and we have found a finite subcover of $f^{-1}(K),$ which completes the proof.

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

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

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

Generalization

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

See also

• Almost open map – Map that satisfies a condition similar to that of being an open map
• Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
• Perfect map – Continuous closed surjective map, each of whose fibers are also compact sets
• Topology glossary

References

• , esp. section C3.2 "Proper maps"
• , esp. p. 90 "Proper maps" and the Exercises to Section 3.6.