Point-finite collection

{{short description|Topological concept for collections of sets}}

In mathematics, a collection or family $\mathcal{U}$ of subsets of a topological space $X$ is said to be point-finite if every point of $X$ lies in only finitely many members of $\mathcal{U}.$ {{sfn|Willard|2012|p=145–152}}{{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}}.

A metacompact space is a topological space in which every open cover admits a point-finite open 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.

Dieudonné's theorem

{{math_theorem|math_statement={{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.{{sfn|Willard|2012|loc=Theorem 15.10}} A topological space $X$ is normal if and only if each point-finite open cover of $X$ has a shrinking; that is, if $\{ U_i \mid i \in I \}$ is an open cover indexed by a set $I$ , there is an open cover $\{ V_i \mid i \in I \}$ indexed by the same set $I$ such that $\overline{V_i} \subset U_i$ for each $i \in I$ .}}

The original proof uses Zorn's lemma, while Willard uses transfinite recursion.

References

Category:General topology

Category:Families of sets