Nowhere dense set

Definition · applies to sets

Graduate (topology)

Consider a set $A$ in a topological space, with its closure $\overline{A}$. The set $A$ is nowhere dense if the interior of its closure is empty:

\[\operatorname{int}(\overline{A}) = \emptyset.\]

Examples

The Cantor set Undergraduate
The Cantor set is nowhere dense, because the interior of its closure is empty. The Cantor set is closed, so its closure is the Cantor set itself. After $n$ steps of the construction, what remains is a union of intervals of length $3^{-n}$, so the Cantor set contains no interval of positive length, and therefore no open set: its interior is empty.
A “fat” Cantor set Graduate
A fat Cantor set is nowhere dense, because it contains no interval.

Things that may be mistaken for examples.

The set of rational numbers, $\mathbb{Q}$ High school
$\mathbb{Q}$ is not nowhere dense, because it is dense in $\reals$.

$A$ is nowhere dense if and only if every nonempty open set contains a nonempty open subset that misses $A$.
Subsets and closures of nowhere dense sets are nowhere dense, and a finite union of nowhere dense sets is nowhere dense.

Types of set sparsity: Several terms describe a set that is “small” or “negligible,” even when it has infinitely many points. “Nowhere dense” means small in a topological sense (and a meager set is a countable union of nowhere dense sets), and “measure zero” means small in a size sense. Neither implies the other. An event of probability zero is a set of measure zero for a probability measure.