Proper map

A proper map is a continuous map that is in the set-theoretic topology, a branch of mathematics studied.

Definitions

The actual definition pictures varies from author to author. Therefore here two common definitions are presented.

  • A continuous map between two locally compact spaces is actually called, if the inverse image of each compact set is compact.

A wider and more general definition is:

  • A continuous map between two topological spaces is actually called, if and only if is complete for any topological space Z, the picture.

The second definition is equivalent to the first, if X is a Hausdorff space and Y is a locally compact Hausdorff space.

Examples

  • Is the definition of quantity compact, so the picture is always actually.
  • Every homeomorphism is actually, that is every diffeomorphism and every biholomorphic map.

Properties

  • An actual figure is complete, that is, the image of each closed set is complete.
  • The actual restriction mapping completed on a subspace is always actually.
  • The actual composition of mappings is again actually. Topological spaces together with the actual pictures so form a subcategory of the category of continuous functions.
  • Are topological spaces and are actual pictures, so is a proper map again.
  • Is a proper map between topological spaces is compact and so is compact.
  • Is a compact space and an arbitrary topological space and the topological product, then the projection is a proper map.

Applications

Actual figures provide a criterion for the compactness of a topological space: Be a singleton topological space with the only existing topology. Then: A topological space is compact if and only if the constant map actually is. From this follow the latter two properties.

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