Discrete Space is not Dense-In-Itself

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

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

Proof
A space is dense-in-itself iff it contains no isolated points.

The result follows from Topological Space is Discrete iff All Points are Isolated.