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Concept
Proper map
Summary
In mathematics, a function between topological spaces is called proper if s of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
There are several competing definitions of a "proper function".
Some authors call a function between two topological spaces if the of every compact set in is compact in
Other authors call a map if it is continuous and ; that is if it is a continuous closed map and the preimage of every point in is compact. The two definitions are equivalent if is locally compact and Hausdorff.
Let be a closed map, such that is compact (in ) for all Let be a compact subset of It remains to show that is compact.
Let be an open cover of Then for all this is also an open cover of Since the latter is assumed to be compact, it has a finite subcover. In other words, for every there exists a finite subset such that
The set is closed in and its image under is closed in because is a closed map. Hence the set
is open in It follows that contains the point
Now and because is assumed to be compact, there are finitely many points such that Furthermore, the set is a finite union of finite sets, which makes a finite set.
Now it follows that and we have found a finite subcover of which completes the proof.
If is Hausdorff and is locally compact Hausdorff then proper is equivalent to . A map is universally closed if for any topological space the map is closed. In the case that is Hausdorff, this is equivalent to requiring that for any map the pullback be closed, as follows from the fact that is a closed subspace of
An equivalent, possibly more intuitive definition when and are metric spaces is as follows: we say an infinite sequence of points in a topological space if, for every compact set only finitely many points are in Then a continuous map is proper if and only if for every sequence of points that escapes to infinity in the sequence escapes to infinity in
Every continuous map from a compact space to a Hausdorff space is both proper and closed.
Official source
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