Perfect map


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

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

Examples and properties

1. If is a perfect map and is compact, then is compact.
2. If is a perfect map and is regular, then is regular.
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, and surjective and therefore is a perfect map. However, p is not open, for the image of under p is which is not open relative to . 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, 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.
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.