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 metacompact.

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• This article incorporates material from point finite on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.