\begin{definition}[Definition:Everywhere Dense/Real Numbers]
Let $S$ be a subset of the real numbers.
Then $S$ is '''(everywhere) dense''' in $\R$ {{iff}}:
:$\forall x \in \R: \forall \epsilon \in \R_{>0}: \exists s \in S: x - \epsilon < s < x + \epsilon$.
That is, {{iff}} in every neighborhood of every real number lies an element of $S$.
Category:Definitions/Denseness
Category:Definitions/Real Numbers
\end{definition}