# 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.

## Generalization

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