Uncountable Particular Point Space is not Second-Countable

Theorem
Let $T = \left({S, \vartheta_p}\right)$ be an uncountable particular point space.

Then $T$ is not second-countable.

Proof
Let $H = S \setminus \left\{{p}\right\}$ where $\setminus$ denotes set difference.

Every subset $V \subseteq H$ is a closed set from Subset of Particular Point Space is either Open or Closed.

Thus we can consider $H$ as an uncountable discrete space.

The result follows from Uncountable Discrete Space is Not Second-Countable.