Either-Or Topology is First-Countable

Theorem
Let $T = \left({S, \tau}\right)$ be the either-or space.

Then $T$ is a first-countable space.

Proof
Let $x \in S$ such that $x \ne 0$.

Then $\left\{{x}\right\}$ is open in $T$ and so on its own forms a local basis of $x$ which is (trivially) countable.

Now if $x = 0$ any open set $U \in \tau: 0 \in U$, contains the open set $\left({-1 \,.\,.\, 1}\right)$.

So $\left({-1 \,.\,.\, 1}\right)$ forms a local basis of $0$ which is (trivially) countable.

Hence the result by definition of first-countable space.