Uncountable Discrete Space is not Second-Countable

Theorem
Let $T = \left({S, \tau}\right)$ be an uncountable discrete topological space.

Then $T$ is not second-countable.

Proof
We have that an Uncountable Discrete Space is not Separable.

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

Also see

 * Countable Discrete Space is Second-Countable