In mathematics, especially topology, a perfect map is a particular kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps.

Formal definition

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Let   and   be topological spaces and let   be a map from   to   that is continuous, closed, surjective and such that each fiber   is compact relative to   for each   in  . Then   is known as a perfect map.

Examples and properties

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  1. If   is a perfect map and   is compact, then   is compact.
  2. If   is a perfect map and   is regular, then   is regular. (If   is merely continuous, then even if   is regular,   need not be regular. An example of this is if   is a regular space and   is an infinite set in the indiscrete topology.)
  3. If   is a perfect map and if   is locally compact, then   is locally compact.
  4. If   is a perfect map and if   is second countable, then   is second countable.
  5. Every injective perfect map is a homeomorphism. This follows from the fact that a bijective closed map has a continuous inverse.
  6. If   is a perfect map and if   is connected, then   need not be connected. For example, the constant map from a compact disconnected space to a singleton space is a perfect map.
  7. A perfect map need not be open. Indeed, consider the map   given by   if   and   if  . This map is closed, continuous (by the pasting lemma), and surjective and therefore is a perfect map (the other condition is trivially satisfied). However, p is not open, for the image of [1, 2] under p is [1, 2] which is not open relative to [1, 3] (the range of p). Note that this map is a quotient map and the quotient operation is 'gluing' two intervals together.
  8. Notice how, to preserve properties such as local connectedness, second countability, local compactness etc. ... the map must be not only continuous but also open. A perfect map need not be open (see previous example), but these properties are still preserved under perfect maps.
  9. Every homeomorphism is a perfect map. This follows from the fact that a bijective open map is closed and that since a homeomorphism is injective, the inverse of each element of the range must be finite in the domain (in fact, the inverse must have precisely one element).
  10. Every perfect map is a quotient map. This follows from the fact that a closed, continuous surjective map is always a quotient map.
  11. Let G be a compact topological group which acts continuously on X. Then the quotient map from X to X/G is a perfect map.
  12. Perfect maps are proper. Surjective proper maps are perfect, provided the topology of Y is Hausdorff and compactly generated.[1]

See also

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  • Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
  • Quotient space – Topological space construction
  • Proper map – Map between topological spaces with the property that the preimage of every compact is compact

References

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  1. ^ "ProperCoverings.pdf" (PDF).