Underlying Set of Topological Space is Everywhere Dense

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

Then $S$ is everywhere dense in $T$.

Proof
From Underlying Set of Topological Space is Closed, $S$ is closed in $T$.

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

The result follows from definition of everywhere dense.