Uncountable Discrete Space is not Second-Countable

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

Let $S$ be an uncountable set, thereby making $\vartheta$ the uncountable discrete topology on $S$.

Then $T$ is not second-countable.

Proof
Let $T = \left({S, \vartheta}\right)$ be the uncountable discrete topology on $S$.

We have that an Uncountable Discrete Space is Not Separable.

From Second-Countable Space is First-Countable and Separable, it follows that $T$ can not be second-countable.

Also see

 * Countable Discrete Space is Second-Countable