Point-finite collection explained

of subsets of a topological space is said to be point-finite if every point of lies in only finitely many members of ^[1]

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.^[1]

Dieudonné's theorem

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

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

Notes and References

1. .