Definition:Everywhere Dense/Definition 1

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

Let $H \subseteq S$ be a subset.

The subset $H$ is (everywhere) dense in $T$ :
 * $H^- = S$

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

That is, every point in $S$ is a point or a limit point of $H$.

Also see

 * Equivalence of Definitions of Everywhere Dense