Definition:Everywhere Dense/Definition 2

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

Let $H \subseteq S$ be a subset.

The subset $H$ is (everywhere) dense in $T$ the intersection of $H$ with every non-empty open set of $T$ is non-empty:
 * $\forall U \in \tau \setminus \set \O: H \cap U \ne \O$

Also see

 * Equivalence of Definitions of Everywhere Dense