Point-finite collection

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

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.

Dieudonné's theorem

Since a paracompact (Hausdorff) space is normal, the next theorem applies in particular to a paracompact space.

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

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

References

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