Definition:Everywhere Dense

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

Let $H \subseteq S$.

Then $H$ is (everywhere) dense in $T$ iff:
 * $\operatorname{cl}\left({H}\right) = S$

where $\operatorname{cl}\left({H}\right)$ is the closure of $H$.

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

Also see

 * Nowhere dense
 * Dense-in-itself