Proper Map - Definition

Definition

A function f : X → Y between two topological spaces is proper if and only 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. For a proof of this fact see the end of this section. More abstractly, f is proper if for any space Z the map

f × idZ: X × Z → Y × 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 is as follows: we say an infinite sequence of points {pi} in a topological space X escapes to infinity if, for every compact set S ⊂ X only finitely many points pi are in S. Then a map f : X → Y 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.

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