Proper map

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 $f:X\to Y$ 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 $\left\{U_{a}:a\in A\right\}$ 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 $V_{k}=Y\setminus f\left(X\setminus \cup _{a\in \gamma _{k}}U_{a}\right)$ 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 $K\subseteq \cup _{i=1}^{s}V_{k_{i}}.$ Furthermore, the set $\Gamma =\cup _{i=1}^{s}\gamma _{k_{i}}$ is a finite union of finite sets, which makes $\Gamma$ a finite set.

Now it follows that $f^{-1}(K)\subseteq f^{-1}\left(\cup _{i=1}^{s}V_{k_{i}}\right)\subseteq \cup _{a\in \Gamma }U_{a}$ 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 $f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z$ 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.^[2]

Generalization

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

Citations

^ Lee 2012, p. 610, above Prop. A.53.
^ Palais, Richard S. (1970). "When proper maps are closed". Proceedings of the American Mathematical Society. 24 (4): 835–836. doi:10.1090/s0002-9939-1970-0254818-x. MR 0254818.

References

Bourbaki, Nicolas (1998). General topology. Chapters 5–10. Elements of Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-64563-4. MR 1726872.
Johnstone, Peter (2002). Sketches of an elephant: a topos theory compendium. Oxford: Oxford University Press. ISBN 0-19-851598-7., esp. section C3.2 "Proper maps"
Brown, Ronald (2006). Topology and groupoids. North Carolina: Booksurge. ISBN 1-4196-2722-8., esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
Brown, Ronald (1973). "Sequentially proper maps and a sequential compactification". Journal of the London Mathematical Society. Second series. 7 (3): 515–522. doi:10.1112/jlms/s2-7.3.515.
Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771.