Fortissimo Space is not Sigma-Compact

Theorem

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

Then $T$ is not a $\sigma$-compact space.

Proof

From Compact Sets in Fortissimo Space we have that the only compact sets in $T$ are finite.

A union of countably many finite sets can not be an uncountable set.

Hence the result by definition of $\sigma$-compact 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: $2$