Singleton Set is not Dense-in-itself

Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x \in S$.

Then the singleton set $\left\{{x}\right\}$ is not dense-in-itself.

Proof

From Singleton Point is Isolated, $x$ is isolated in $\left\{{x}\right\}$.

So by definition $\left\{{x}\right\}$ is not dense-in-itself.

$\blacksquare$