Nowhere dense set

Definition

Let $X$ be a topological space and let $A \subseteq X$. The set $A$ is nowhere dense in $X$ when the interior of its closure is empty, that is, $\operatorname{int} \overline{A} = \emptyset$.

Remarks

The set $\mathbb{Z}$ is nowhere dense in $\mathbb{R}$. So is each finite subset of $\mathbb{R}$. The Cantor set is nowhere dense in $\mathbb{R}$, though it is uncountable. The set $\mathbb{Q}$ is not nowhere dense in $\mathbb{R}$, because its closure is $\mathbb{R}$. A closed set is nowhere dense if and only if its complement is open and dense.

Explanations

L1 human

A nowhere dense set is thin everywhere: inside each open set, you can find a smaller open set that misses it completely. The integers on the real line are an example, since there are gaps between them.