# Fully Normal Space is Paracompact

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## Theorem

Let $T = \struct {S, \tau}$ be a fully normal space.

Then $T$ is paracompact.

## Proof

From the definition, $T$ is fully normal if and only if:

- $T$ is fully $T_4$
- $T$ is a $T_1$ (Fréchet) space.

Then $T$ is fully $T_4$ if and only if every open cover of $S$ has a star refinement.

Let $\UU$ be an open cover for $T$.

Then from the definition, there exists a cover $\VV$ for $T$ such that:

- $\ds \forall x \in S: \exists U \in \UU: \paren {\bigcup \set {V \in \VV: x \in V} } \subseteq U$

Recall from the definition of paracompact:

- $T$ is paracompact if and only if every open cover of $S$ has an open refinement which is locally finite.

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## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness