Discrete Space is First-Countable

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Then $T$ is first-countable.

Proof
From Points in Discrete Space are Neighborhoods, every point $x \in S$ is contained in an open set $\left\{{x}\right\}$.

From the definition of local basis, it is clear that $\left\{{\left\{{x}\right\}}\right\}$ is (trivially) a local basis at $x$.

That is, that every open set of $X$ containing $x$ also contains at least one of the sets of $\left\{{\left\{{x}\right\}}\right\}$.

Equally trivially, we have that $\left\{{\left\{{x}\right\}}\right\}$ is countable.

Hence the result by definition of first-countable.