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In mathematics, a collection or family ${\displaystyle {\mathcal {U}}}$ of subsets of a topological space ${\displaystyle X}$ is said to be point-finite if every point of ${\displaystyle X}$ lies in only finitely many members of ${\displaystyle {\mathcal {U}}.}$

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

Theorem —  A topological space ${\displaystyle X}$ is normal if and only if each point-finite open cover of ${\displaystyle X}$ has a shrinking; that is, if ${\displaystyle \{U_{i}\mid i\in I\}}$ is an open cover indexed by a set ${\displaystyle I}$, there is an open cover ${\displaystyle \{V_{i}\mid i\in I\}}$ indexed by the same set ${\displaystyle I}$ such that ${\displaystyle {\overline {V_{i}}}\subset U_{i}}$ for each ${\displaystyle i\in I}$.

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

References

1. Willard 2012, p. 145–152.
2. ^ Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, pp. 145–152, ISBN 9780486131788, OCLC 829161886.
3. Dieudonné, Jean (1944), "Une généralisation des espaces compacts", Journal de Mathématiques Pures et Appliquées, Neuvième Série, 23: 65–76, ISSN 0021-7824, MR 0013297, Théorème 6.
4. Willard 2012, Theorem 15.10.

This article incorporates material from point finite on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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