nLab fully normal spaces are equivalently paracompact

Statement

Let $X$ be a $T_1$ topological space.

Assuming the axiom of choice then the following are equivalent:

1. $X$ is a fully normal topological space;

2. $X$ is a paracompact Hausdorff topological space.

(Stone 48)

Since metric spaces are fully normal it follows as a corollary that metric spaces are paracompact. Accordingly, this statement is now also known as Stone’s theorem.

Note that without a separation axiom such as $T_1$, the result fails to hold. For example, any compact topological space is paracompact, and any fully normal topological space is normal, so any non-normal compact space is a paracompact space that’s not fully normal.

The Hausdorff condition from statement 2 cannot be dropped, again because there are compact $T_1$ spaces that are not normal, such as an infinite set with the cofinite topology.

References

• A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. (Euclid)