Proper Map
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Proper Map

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

A function ${\displaystyle f\colon X\to Y}$ between two topological spaces is proper if the preimage of every compact set in Y is compact in X.

There are several competing descriptions. For instance, a continuous map f is proper if it is closed with compact fibers, i.e. if it is a 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 ${\displaystyle f\colon X\to Y}$ be a closed map, such that ${\displaystyle f^{-1}(y)}$ is compact (in X) for all ${\displaystyle y\in Y}$. Let ${\displaystyle K}$ be a compact subset of ${\displaystyle Y}$. We will show that ${\displaystyle f^{-1}(K)}$ is compact.

Let ${\displaystyle \{U_{\lambda }\vert \lambda \ \in \ \Lambda \}}$ 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 all ${\displaystyle k\ \in K}$ there is a finite set ${\displaystyle \gamma _{k}\subset \Lambda }$ such that ${\displaystyle f^{-1}(k)\subset \cup _{\lambda \in \gamma _{k}}U_{\lambda }}$. The set ${\displaystyle X\setminus \cup _{\lambda \in \gamma _{k}}U_{\lambda }}$ is closed. Its image is closed in Y, because f is a closed map. Hence the set

${\displaystyle V_{k}=Y\setminus f(X\setminus \cup _{\lambda \in \gamma _{k}}U_{\lambda })}$ is open in Y. It is easy to check that ${\displaystyle V_{k}}$ contains the point ${\displaystyle k}$. Now ${\displaystyle K\subset \cup _{k\in K}V_{k}}$ and because K is assumed to be compact, there are finitely many points ${\displaystyle k_{1},\dots ,k_{s}}$ such that ${\displaystyle K\subset \cup _{i=1}^{s}V_{k_{i}}}$. Furthermore the set ${\displaystyle \Gamma =\cup _{i=1}^{s}\gamma _{k_{i}}}$ is a finite union of finite sets, thus ${\displaystyle \Gamma }$ is finite.

Now it follows that ${\displaystyle f^{-1}(K)\subset f^{-1}(\cup _{i=1}^{s}V_{k_{i}})\subset \cup _{\lambda \in \Gamma }U_{\lambda }}$ and we have found a finite subcover of ${\displaystyle f^{-1}(K)}$, 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 ${\displaystyle f\times \operatorname {id} _{Z}\colon X\times Z\to Y\times Z}$ 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 X and Y are metric spaces is as follows: we say an infinite sequence of points ${\displaystyle \{p_{i}\}}$ in a topological space X escapes to infinity if, for every compact set ${\displaystyle S\subseteq X}$ only finitely many points ${\displaystyle p_{i}}$ are in S. Then a continuous map ${\displaystyle f\colon X\to Y}$ is proper if and only if for every sequence of points ${\displaystyle \{p_{i}\}}$ that escapes to infinity in X, the sequence ${\displaystyle \{f(p_{i})\}}$ escapes to infinity in Y.

## Generalization

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