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fully normal spaces are equivalently paracompact
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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 statements

Theorems

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Contents
Statement
Let $X$ be a $T_1$ topological space .

Assuming the axiom of choice then the following are equivalent:

$X$ is a fully normal topological space ;

$X$ 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 $T_1$ , 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 $T_1$ spaces that are not normal, such as an infinite set with the cofinite topology .

References
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.
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