# Fortissimo Space is Paracompact

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## Theorem

Let $T = \left({S, \tau_p}\right)$ be a Fortissimo space.

Then $T$ is a paracompact space.

## Proof

Let $\mathcal C$ be an open cover of $T$.

Let $U_p \in \mathcal C$ be an open set in that cover which contains $p$.

All the points of $S \setminus U_p$ are open points by definition of Fortissimo space.

Then $\mathcal D = \left\{{\left\{{s}\right\}: s \in S \setminus U_p}\right\} \cup \left\{{U_p}\right\}$ is an open refinement of $\mathcal C$.

Take $k \in S$:

- If $k = p$, then $U_p$ is a neighbourhood of $k$ which intersects only with $U_p$ from all the subsets in $\mathcal D$
- If $k \ne p$, then $\left\{{k}\right\}$ is a neighbourhood of $k$ which intersects only with $\left\{{k}\right\}$ from all the subsets in $\mathcal D$

Thus $\mathcal D$ is by definition locally finite.

Hence, by definition, $T$ is a paracompact space.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 25: \ 1$