Subset of Excluded Point Space is not Dense-in-itself

Theorem
Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton.

Let $H \subseteq S$.

Then $H$ is not dense-in-itself.

Proof
From Limit Points in Excluded Point Space, the only limit point of $H$ is $p$.

So by definition, all points of $H$ are isolated in $H$ except $p$.

So if $H \ne \set p$, $H$ contains at least one point which is isolated in $H$.

As for $p$ itself, from Singleton Point is Isolated we have that $p$ is itself isolated in $\set p$.

So if $H = \set p$, $H$ also contains one point which is isolated in $H$.

Hence the result, by definition of dense-in-itself.