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 $T$ if and only if the intersection of $H$ with every open subset of $T$ is non-empty:

$\forall U \in \tau: H \cap U \ne \varnothing$