Separability in Uncountable Particular Point Space

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

Let $H = S \setminus \set p$ where $\setminus$ denotes set difference.

Then $H$ is not separable.

Proof
By definition, $H$ is separable there exists a countable subset of $S$ which is everywhere dense in $T$.

Let $V \subseteq H$ where $V$ is countable.

$V$ is not open in $T$ as it does not contain $p$.

From Subset of Particular Point Space is either Open or Closed it follows that $V$ is closed.

From Closed Set Equals its Closure, $V^- = V$.

But $V^- \ne H$ as $V$ is countable and $H$ is uncountable.

So whatever $V$ is, if it is countable it is not everywhere dense.

The result follows from definition of separable.