# Point-finite collection

In mathematics, a collection ${\displaystyle {\mathcal {U}}}$ of subsets of a topological space ${\displaystyle X}$ is said to be point finite or a point finite collection if every point of ${\displaystyle X}$ lies in only finitely many members of ${\displaystyle {\mathcal {U}}}$.[1]

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.[1]

## References

1. Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, pp. 145–152, ISBN 9780486131788.