Point-finite collection

In mathematics, a collection ${\mathcal {U}}$ of subsets of a topological space $X$ is said to be point-finite or a point-finite collection if every point of $X$ lies in only finitely many members of ${\mathcal {U}}$.

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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