Underlying Set of Topological Space is Everywhere Dense

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Then the underlying set $S$ of $T$ 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.

$\blacksquare$