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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. == Definition == 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 pre-image 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 :''f'' × id''Z'': ''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 when ''X'' and ''Y'' are metric spaces is as follows: we say an infinite sequence of points 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 that escapes to infinity in ''X'', escapes to infinity in ''Y''. This last sequential idea looks like being related to the notion of sequentially proper, see a reference below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Proper map」の詳細全文を読む スポンサード リンク
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