View source for Point-finite collection

{{short description|Topological concept for collections of sets}}
In [[mathematics]], a collection or [[family of sets|family]] <math>\mathcal{U}</math> of subsets of a [[topological space]] <math>X</math> is said to be '''point-finite''' if every point of <math>X</math> lies in only finitely many members of <math>\mathcal{U}.</math>{{sfn|Willard|2012|p=145–152}}<ref name="w">{{citation|title=General Topology|series=Dover Books on Mathematics|first=Stephen|last=Willard|publisher=Courier Dover Publications|year=2012|pages=145–152|url=https://books.google.com/books?id=UrsHbOjiR8QC&pg=PA145|isbn=9780486131788|oclc=829161886}}.</ref>
A [[metacompact space]] is a topological space in which every [[open cover]] admits a point-finite open [[Refinement (topology)|refinement]]. Every [[locally finite collection]] of subsets of a topological space is also point-finite.
A topological space in which every open cover admits a locally finite open refinement is called a [[paracompact space]]. Every paracompact space is therefore metacompact.<ref name="w"/>
== Dieudonné's theorem ==
Since a paracompact (Hausdorff) space is normal, the next theorem applies in particular to a paracompact space.
{{math_theorem|math_statement=<ref>{{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Une généralisation des espaces compacts | mr=0013297 | year=1944 | journal=[[Journal de Mathématiques Pures et Appliquées]]|series= Neuvième Série | issn=0021-7824 | volume=23 | pages=65–76}}, Théorème 6.</ref>{{sfn|Willard|2012|loc=Theorem 15.10}} A topological space <math>X</math> is [[normal space|normal]] if and only if each point-finite open cover of <math>X</math> has a [[shrinking (topology)|shrinking]]; that is, if <math>\{ U_i \mid i \in I \}</math> is an open cover indexed by a set <math>I</math>, there is an open cover <math>\{ V_i \mid i \in I \}</math> indexed by the same set <math>I</math> such that <math>\overline{V_i} \subset U_i</math> for each <math>i \in I</math>.}}
The original proof uses Zorn's lemma, while Willard uses transfinite recursion.
==References==
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{{PlanetMath attribution|id = 8398|title = point finite}}
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[[Category:General topology]]
[[Category:Families of sets]]