In mathematics, a collection or family of subsets of a topological space is said to be point-finite if every point of lies in only finitely many members of [1][2]
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.[2]
Dieudonné's theorem
editTheorem — [3][4] A topological space is normal if and only if each point-finite open cover of has a shrinking; that is, if is an open cover indexed by a set , there is an open cover indexed by the same set such that for each .
The original proof uses Zorn's lemma, while Willard uses transfinite recursion.
References
edit- ^ Willard 2012, p. 145–152.
- ^ a b Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, pp. 145–152, ISBN 9780486131788, OCLC 829161886.
- ^ 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.
- ^ Willard 2012, Theorem 15.10.
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