Point-finite collection

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 ${\displaystyle {\mathcal {U}}.}$^[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, 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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This page was last edited on 17 June 2024, at 19:53