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In mathematics, a collection 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 topological space in which every open cover admits a point-finite open refinement is called metacompact. 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 paracompact. Every paracompact space is therefore metacompact.
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