Definition:Everywhere Dense/Definition 2

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 $T$ $H$ has nonempty intersection with every open subset of $T$:
 * $\forall U \in \tau : H\cap U \neq \varnothing$

Also see

 * Equivalence of Definitions of Everywhere Dense