nLab metric spaces are paracompact

Statement

Every separable metrisable topological space is paracompact.

With the axiom of choice we have more generally that:

Every metrisable topological space is paracompact.

The orignal proof due to (Stone 48) used that metric spaces are fully normal and showed that fully normal spaces are equivalently paracompact (“Stone’s theorem”).

A direct and short proof was later given in (Rudin 68).

Related statements

• paracompact Hausdorff spaces are normal

• locally compact and sigma-compact spaces are paracompact

• second-countable regular spaces are paracompact

• regular Lindelöf spaces are paracompact

• Hausdorff spaces are sober

• compact subspaces of Hausdorff spaces are closed

• closed subspaces of compact Hausdorff spaces are equivalently compact subspaces

• CW-complexes are paracompact Hausdorff spaces

• Hausdorff spaces are sober

• maps from compact spaces to Hausdorff spaces are closed and proper

• continuous images of compact spaces are compact

• paracompact Hausdorff spaces equivalently admit subordinate partitions of unity

References

Historically, first it was shown that fully normal spaces are equivalently paracompact in

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

Since it is easy to see that metric spaces are fully normal this implies that metric spaces and hence metriable topological spaces are paracompact. Accordingly this statement came to be known as Stone’s theorem.

A direct and short proof that metric spaces are paracompact was given in

• Mary Ellen Rudin?, A new proof that metric spaces are paracompact, AMS 1968 (pdf)