Double Pointed Fortissimo Space is Lindelöf

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

Let $T \times D$ be the double pointed topology on $T$.

Then $T \times D$ is a Lindelöf space.

Proof
Let $D = \set {0, 1}$.

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

Then $\exists A \times B \in \CC$ such that $\tuple {p, 0} \in A \times B$.

We must have $\relcomp S A$ is countable and $B = D$.

Hence $\relcomp S A \times D$, a product of countable sets, must be countable.

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

Hence the result by definition of Lindelöf space.