2A06:C701:4C4D:BC00:D9BF:24B9:7DDD:9369 at 16:50, 25 September 2024

2024-09-25T16:50:18Z

New page

{{short description|Continuous closed surjective map, each of whose fibers are also compact sets}}
In [[mathematics]], especially [[topology]], a '''perfect map''' is a particular kind of [[continuous function]] between [[topological space]]s. Perfect maps are weaker than [[homeomorphism]]s, but strong enough to preserve some topological properties such as [[locally compact space|local compactness]] that are not always preserved by continuous maps.

== Formal definition ==
Let <math>X</math> and <math>Y</math> be [[topological space]]s and let <math>p</math> be a map from <math>X</math> to <math>Y</math> that is [[Continuous function|continuous]], [[Closed map|closed]], [[Surjective function|surjective]] and such that each [[Fiber (mathematics)|fiber]] <math>p^{-1}(y)</math> is [[Compact space|compact]] relative to <math>X</math> for each <math>y</math> in <math>Y</math>. Then <math>p</math> is known as a perfect map.

== Examples and properties ==
# If <math>p\colon X \to Y</math> is a perfect map and <math>Y</math> is [[compact space|compact]], then <math>X</math> is compact.
# If <math>p\colon X \to Y</math> is a perfect map and <math>X</math> is [[regular space|regular]], then <math>Y</math> is regular. (If <math>p</math> is merely continuous, then even if <math>X</math> is regular, <math>Y</math> need not be regular. An example of this is if <math>X</math> is a regular space and <math>Y</math> is an infinite set in the indiscrete topology.)
# If <math>p\colon X \to Y</math> is a perfect map and if <math>X</math> is [[locally compact]], then <math>Y</math> is locally compact.
# If <math>p\colon X \to Y</math> is a perfect map and if <math>X</math> is second countable, then <math>Y</math> is [[second countable]].
# Every [[injective]] perfect map is a [[homeomorphism]]. This follows from the fact that a bijective closed map has a continuous inverse.
# If <math>p\colon X \to Y</math> is a perfect map and if <math>Y</math> is [[Connected space|connected]], then <math>X</math> need not be connected. For example, the constant map from a compact disconnected space to a singleton space is a perfect map.
# A perfect map need not be open. Indeed, consider the map <math>p\colon [1, 2] \cup [3, 4] \to [1, 3]</math> given by <math>p(x) = x</math> if <math>x\in [1, 2]</math> and <math>p(x) = x - 1</math> if <math>x\in{}[3, 4]</math>. 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 {{nowrap|[1, 2]}} under ''p'' is {{nowrap|[1, 2]}} which is not open relative to {{nowrap|[1, 3]}} (the range of ''p''). Note that this map is a [[Quotient space (topology)|quotient map]] and the quotient operation is 'gluing' two intervals together.
# Notice how, to preserve properties such as [[Locally connected space|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.
# 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).
# Every perfect map is a quotient map. This follows from the fact that a closed, continuous surjective map is always a quotient map.
# 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.
# Perfect maps are [[Proper map|proper]]. Surjective proper maps are perfect, provided the topology of ''Y'' is [[Hausdorff space|Hausdorff]] and compactly generated.<ref>{{Cite web |title=ProperCoverings.pdf |url=https://web.math.utk.edu/~freire/teaching/m562s21/ProperCoverings.pdf}}</ref>

== See also ==
* {{annotated link|Open and closed maps}}
* {{annotated link|Quotient space (topology)|Quotient space}}
* {{annotated link|Proper map}}

== References ==
{{Reflist}}
* {{cite book|first=James|last= Munkres|authorlink=James Munkres |year=1999|title=Topology|edition= 2nd |publisher=[[Prentice Hall]]|isbn=0-13-181629-2}}

{{DEFAULTSORT:Perfect Map}}
[[Category:Theory of continuous functions]]

Perfect map - Revision history

2A06:C701:4C4D:BC00:D9BF:24B9:7DDD:9369 at 16:50, 25 September 2024