Point-finite collection

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 ${\mathcal {U}}.$ ^[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]

References

^ Willard 2004, p. 145–152.
^ Jump up to: ^a ^b Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, pp. 145–152, ISBN 9780486131788, OCLC 829161886.

Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
Willard, Stephen (2012) [1970]. General Topology. Mineola, N.Y.: Courier Dover Publications. ISBN 9780486131788. OCLC 829161886.

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[FOOTNOTEWillard2004145–152-1] Willard 2004, p. 145–152.

[w-2] Jump up to: ^a ^b Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, pp. 145–152, ISBN 9780486131788, OCLC 829161886.

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