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:

(1)

\begin{align} \quad \mathrm{int}(\bar{\mathbb{Q}}) = \mathrm{int} (\mathbb{R}) = \mathbb{R} \neq \emptyset \end{align}