# Fortissimo Space is Paracompact

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## Theorem

Let $T = \struct {S, \tau_p}$ be a Fortissimo space.

Then $T$ is a paracompact space.

## Proof

Let $\CC$ be an open cover of $T$.

Let $U_p \in \CC$ 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 $\DD = \set {\set s: s \in S \setminus U_p} \cup \set {U_p}$ is an open refinement of $\CC$.

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 $\DD$
- If $k \ne p$, then $\set k$ is a neighbourhood of $k$ which intersects only with $\set k$ from all the subsets in $\DD$

Thus $\DD$ is by definition locally finite.

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

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $25$. Fortissimo Space: $1$