In mathematics, a collection or family U {\displaystyle {\mathcal {U}}} of subsets of a topological space X {\displaystyle X} is said to be point-finite if every point of X {\displaystyle X} lies in only finitely many members of U . {\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
The original proof uses Zorn's lemma, while Willard uses transfinite recursion.
References
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