Point-finite collection

Template:Short description In mathematics, a collection or family $𝒰$ 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 $𝒰.$Template:Sfn

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

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

References

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