Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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. 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 it is a closed map, and for any space Z the map (f, idZ): X x Z -> Y x Z is closed. These definitions are equivalent to the previous one if the space X is locally compact. 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.

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