nLab second-countable regular spaces are paracompact

Statement

Lemma

(Michael's theorem, Michael 53, theorem 1)

Let $X$ be a topological space such that

1. $X$ is regular;

2. every open cover of $X$ has a refinement by a union of a countable set of locally finite sets of open subsets (not necessarily covering).

Then $X$ is paracompact topological space.

Note that while in Michael’s paper he assumes that paracompact spaces are Hausdorff, the assumption is not necessary, and the proof goes through for regular non-Hausdorff spaces. See Kelley, p. 156.

Proposition

(second-countable regular spaces are paracompact)

Let $X$ be a topological space which is

1. second-countable;

2. regular.

Then $X$ is paracompact topological space.

Proof

Let $\{U_i \subset X\}_{i \in I}$ be an open cover. By Michael's theorem (lemma ) it is sufficient that we find a refinement by a countable cover.

But second countability implies precisely that every open cover has a countable subcover:

Every open cover has a refinement by a cover consisting of base elements, and if there is only a countable set of these, then the resulting refinement necessarily contains at most this countable set of distinct open subsets.

Related statements

• locally compact and sigma-compact spaces are paracompact

• locally compact and second-countable spaces are sigma-compact

• CW-complexes are paracompact Hausdorff spaces

References

• Kelley, General Topology, 1955.

• Ernest Michael, A note on paracompact spaces, Proc. Amer. Math. Soc. 1953 (jstor, pdf)

• Akhil Mathew, A theorem of Michael on paracompactness