Fortissimo Space is Lindelöf

Theorem

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

Then $T$ is a Lindelöf space.

Proof

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

Then $\exists U \in \CC$ such that $p \in U$ and so $\relcomp S U$ is countable.

So $U$, together with an open neighborhood of each of the elements of $\relcomp S U$, is a countable subcover of $\CC$.

Hence the result by definition of Lindelöf 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: $1$