Nowhere Dense Subsets of Metric Spaces
 Definition: Let $(X, d)$ be a metric space. A set $E \subseteq X$ is said to be Nowhere Dense in $X$ if $\mathrm{int} (\bar{E}) = \emptyset$.
There are many examples of nowhere dense sets. For example, the set $\mathbb{N}$ is nowhere dense in $\mathbb{R}$ with the usual Euclidean metric defined on $\mathbb{R}$ .
However, $\mathbb{Q}$ is NOT nowhere dense in $\mathbb{R}$ because: