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]

Dieudonné's theorem

Theorem—^[3][4] A topological space ${\displaystyle X}$ is normal if and only if each point-finite open cover of ${\displaystyle X}$ has a shrinking; that is, if ${\displaystyle \{U_{i}\mid i\in I\}}$ is an open cover indexed by a set ${\displaystyle I}$, there is an open cover ${\displaystyle \{V_{i}\mid i\in I\}}$ indexed by the same set ${\displaystyle I}$ such that ${\displaystyle {\overline {V_{i}}}\subset U_{i}}$ for each ${\displaystyle i\in I}$.

The original proof uses Zorn's lemma, while Willard uses transfinite recursion.