\begin{definition}[Definition:Everywhere Dense/Definition 2]
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a subset.
The subset $H$ is '''(everywhere) dense in $T$''' {{iff}} the intersection of $H$ with every non-empty open set of $T$ is non-empty:
:$\forall U \in \tau \setminus \set \O: H \cap U \ne \O$
\end{definition}