Everywhere Dense iff Interior of Complement is Empty

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

Let $A \subset S$.

Then $A$ is everywhere dense :
 * $\paren {\relcomp S A}^\circ = \O$

where $A^\circ$ is the interior of $A$.

Proof
By definition of everywhere dense, $A$ is everywhere dense :
 * $A^- = S$

where $A^-$ is the closure of $A$.

That happens :

Hence the result.