# Uncountable Particular Point Space is not Second-Countable

## Theorem

Let $T = \struct {S, \tau_p}$ be an uncountable particular point space.

Then $T$ is not second-countable.

## Proof

Let $H = S \setminus \set p$ 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.

$\blacksquare$