nLab
metric spaces are paracompact

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

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Basic concepts

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Extra stuff, structure, properties

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Theorems

Analysis Theorems

topological homotopy theory

Contents

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).

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)

Last revised on May 23, 2017 at 15:12:12. See the history of this page for a list of all contributions to it.