nLab fully normal spaces are equivalently paracompact




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

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Let XX be a T 1T_1 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 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 1T_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 1T_1 spaces that are not normal, such as an infinite set with the cofinite topology.


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

Last revised on June 16, 2021 at 04:07:44. See the history of this page for a list of all contributions to it.