Nowhere dense set - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A nowhere dense set in a topological space $X$ is a subset $A\subseteq X$ such that the interior of its closure is empty, i.e. $\operatorname{int}(\overline{A})=\varnothing$ .

Equivalent characterizations

Equivalently, $A$ is nowhere dense if $\overline{A}$ contains no nonempty open set.

Remarks

Nowhere dense sets are the basic building blocks of meager sets in the Baire category viewpoint.

Examples

The set of integers $\mathbb{Z}\subseteq \mathbb{R}$ (with the usual topology) is nowhere dense: it is closed and has empty interior.
The set $\{1/n : n\ge1\}\subseteq \mathbb{R}$ is nowhere dense: its closure is $\{1/n:n\ge1\}\cup\{0\}$ , which has empty interior.