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.

A function f : X → 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 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. 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

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 {p_i} in a topological space X escapes to infinity if, for every compact set S ⊂ X only finitely many points p_i are in S. Then a continuous map f : X → Y is proper if and only if for every sequence of points {p_i} that escapes to infinity in X, {f(p_i)} escapes to infinity in Y.

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

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

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