Discrete Subspace of Fortissimo Space

Theorem

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

Let $T' = \left({S \setminus \left\{{p}\right\}, \tau_p}\right)$ be the topological subspace induced on $T$ by the subset $S \setminus \left\{{p}\right\}$.

Then $T'$ is a discrete topological space.

Proof

By the definition of Fortissimo space, any $A \subset S \setminus \left\{{p}\right\}$ is open in $S$, because $p \notin A$.

Thus by definition of topological subspace, $A \subset S \setminus \left\{{p}\right\}$ is open in $S \setminus \left\{{p}\right\}$.

The result follows by the definition of discrete space.

$\blacksquare$