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. 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. Let X {displaystyle X} and Y {displaystyle Y} be topological spaces and let p {displaystyle p} be a map from X {displaystyle X} to Y {displaystyle Y} that is continuous, closed, surjective and such that p − 1 ( y ) {displaystyle p^{-1}(y)} is compact relative to X {displaystyle X} for each y {displaystyle y} in Y {displaystyle Y} . Then p {displaystyle p} is known as a perfect map. 1. If p : X → Y {displaystyle pcolon X o Y} is a perfect map and Y {displaystyle Y} is compact, then X {displaystyle X} is compact. 2. If p : X → Y {displaystyle pcolon X o Y} is a perfect map and X {displaystyle X} is regular, then Y {displaystyle Y} is regular. (If p {displaystyle p} is merely continuous, then even if X {displaystyle X} is regular, Y {displaystyle Y} need not be regular. An example of this is if X {displaystyle X} is a regular space and Y {displaystyle Y} is an infinite set in the indiscrete topology.) 3. If p : X → Y {displaystyle pcolon X o Y} is a perfect map and if X {displaystyle X} is locally compact, then Y {displaystyle Y} is locally compact. 4. If p : X → Y {displaystyle pcolon X o Y} is a perfect map and if X {displaystyle X} is second countable, then Y {displaystyle 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 {displaystyle pcolon X o Y} is a perfect map and if Y {displaystyle Y} is connected, then X {displaystyle 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 p : [ 1 , 2 ] ∪ [ 3 , 4 ] → [ 1 , 3 ] {displaystyle pcolon cup o } shows: