fully normal spaces are equivalently paracompact



Let XX be a topological space.

Assuming the axiom of choice then the following are equivalent:

  1. XX is a fully normal topological space;

  2. XX is a paracompact 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.


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

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