# Proper map

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In mathematics, a continuous 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 f : XY 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 a closed map and the pre-image of every point in Y is compact. The two definitions are equivalent if Y is compactly generated and Hausdorff. For a proof of this fact see the end of this section. More abstractly, f is proper if f is universally closed, i.e. if for any topological space Z the map

f × idZ: X × ZY × Z

is closed. These definitions are equivalent to the previous one if X is Hausdorff and Y is locally compact Hausdorff.

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

This last sequential idea looks like being related to the notion of sequentially proper, see a reference below.

### Proof of fact

Let ${\displaystyle f:X\to Y}$ be a continuous 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.

## Generalization

It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see Template:Harv.

## References

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• Brown, R. "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7 (1973) 515-522.
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