Fully Normal Space is Paracompact
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Theorem
Let $T = \left({X, \vartheta}\right)$ be a fully normal space.
Then $T$ is paracompact.
Proof
From the definition, $T$ is fully normal iff:
- $T$ is fully $T_4$
- $T$ is a $T_1$ (Fréchet) space.
Then $T$ is fully $T_4$ iff every open cover of $X$ has a star refinement.
Let $\mathcal U$ be an open cover for $T$.
Then from the definition, there exists a be a cover $\mathcal V$ for $T$ such that:
- $\displaystyle \forall x \in S: \exists U \in \mathcal U: \left({\bigcup \left\{{V \in \mathcal V: x \in V}\right\} }\right) \subseteq U$
$T$ is paracompact iff every open cover of $X$ has an open refinement which is locally finite.
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 3$: Paracompactness
