Perfect map
In mathematics, particularly topology, a perfect map is a map which preserves "inverse-like" properties. Just as the continuous image of a connected space is always connected, if the perfect image (image under a perfect map) of a certain space X is connected, then X must be connected. Perfect maps are good in this sense as they are weaker than a homeomorphism, but strong enough to behave like homeomorphisms.
Formal definition
Let X and Y be topological spaces and let p be a map from X to Y that is continuous, closed, surjective and such that p −1(y) is compact relative to X for each y in Y. Then p is known as a perfect map.
Examples and properties
A continuous function f is an open map if and only if the inverse of f is continuous.
1. If p : X→Y is a perfect map and Y is compact, then X is compact.
2. If p : X→Y is a perfect map and X is regular, then Y is regular. (If p is merely continuous, then even if X is regular, Y need not be regular. An example of this is if X is a regular space and Y is an infinite set in the indiscrete topology.)
3. If p : X → Y is a perfect map and if X is locally compact, then Y is locally compact.
4. If p : X → Y is a perfect map and if X is second countable, then Y 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 p : X → Y is a perfect map and if Y is connected, then X 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, as the following map shows:
- p(x) = x if x belongs to [1, 2]
- p(x) = x − 1 if x belongs to [3, 4]
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… we require that the map 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.
See also
References
- James Munkres (1999). Topology, 2nd edition, Prentice Hall. ISBN 0-13-181629-2