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

Theorem
Let $T = \left({S, \tau_{\bar p}}\right)$ 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 $p$.

From Point is Isolated Iff Not a Limit Point, all points of $H$ are isolated except $p$.

Hence the result.