Nowhere dense set

In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. The order of operations is important. For example, the set of rational numbers, as a subset of R, has the property that the interior has an empty closure, but it is not nowhere dense; in fact it is dense in R. Equivalently, a nowhere dense set is a set that is not dense in any nonempty open set.

The surrounding space matters: a set A may be nowhere dense when considered as a subspace of a topological space X but not when considered as a subspace of another topological space Y. A nowhere dense set is always dense in itself.

Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense. That is, the nowhere dense sets form an ideal of sets, a suitable notion of negligible set. The union of countably many nowhere dense sets, however, need not be nowhere dense. (Thus, the nowhere dense sets need not form a sigma-ideal.) Instead, such a union is called a meagre set or a set of first category. The concept is important to formulate the Baire category theorem.

Examples

• $\mathbb Z$ is nowhere dense in $\mathbb R$.
• $S = \left\{ \frac{1}{n} : n \in \mathbb{N} \right\}$ is also nowhere dense in $\mathbb R$: although the points get arbitrarily close to 0, the closure of the set is $S \cup \{0\}$, which has empty interior.
• $\mathbb Z \cup \left[(0,1) \cap \mathbb Q\right]$ is not nowhere dense in $\mathbb R$: it is dense in the interval $[0,1]$, and in particular the interior of its closure is $(0,1)$.

Open and closed

• A nowhere dense set need not be closed (for instance, the set $\left\{\frac{1}{n} \mid n \in \mathbb{N} \right\}$ is nowhere dense in the reals), but is properly contained in a nowhere dense closed set, namely its closure (which would add 0 to the set). Indeed, a set is nowhere dense if and only if its closure is nowhere dense.
• The complement of a closed nowhere dense set is a dense open set, and thus the complement of a nowhere dense set is a set with dense interior.
• The boundary of every open set is closed and nowhere dense.
• Every closed nowhere dense set is the boundary of an open set.

Nowhere dense sets with positive measure

A nowhere dense set is not necessarily negligible in every sense. For example, if X is the unit interval [0,1], not only is it possible to have a dense set of Lebesgue measure zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure.

For one example (a variant of the Cantor set), remove from [0,1] all dyadic fractions, i.e. fractions of the form a/2ⁿ in lowest terms for positive integers a and n, and the intervals around them: (a/2ⁿ − 1/2²ⁿ⁺¹, a/2ⁿ + 1/2²ⁿ⁺¹). Since for each n this removes intervals adding up to at most 1/2ⁿ⁺¹, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1/2 (in fact just over 0.535... because of overlaps) and so in a sense represents the majority of the ambient space [0,1]. This set is nowhere dense, as it is closed and has an empty interior: any interval (a, b) is not contained in the set since the dyadic fractions in (a, b) have been removed.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1, although the measure cannot be exactly one (else its complement would be a nonempty open set with measure zero, which is impossible).

See also

• Baire space
• Fat Cantor set

External links

• Some nowhere dense sets with positive measure