Point-finite collection

In mathematics, a collection or family ${\displaystyle {\mathcal {U}}}$ of subsets of a topological space ${\displaystyle X}$ is said to be point-finite if every point of ${\displaystyle X}$ lies in only finitely many members of ${\displaystyle {\mathcal {U}}.}$[1][2]

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.[2]