In mathematics, a subset of a topological space is called nowhere dense or a rare if its closure has emptyinterior. In a very loose sense, it is a set whose elements are not tightly clustered anywhere. The order of operations is important. For example, the set of rational numbers, as a subset of the real numbers,, has the property that its interior has an empty closure, but it is not nowhere dense; in fact it is dense in. The surrounding space matters: a set may be nowhere dense when considered as a subset of a topological space, but not when considered as a subset of another topological space. Notably, a set is always dense in its own subspace topology. A countable union of nowhere dense sets is called a meagre set. Meager sets play an important role in the formulation of the Baire category theorem.
Characterizations
Let be a topological space and a subset of. Then the following are equivalent:
is nowhere dense in ;
the interior of the closure of is empty;
the closure of in does not contain any non-empty open subset of ;
If is nowhere dense in and is an open subset of then is nowhere dense in.
Every subset of a nowhere dense set is nowhere dense.
The union of finitely many nowhere dense sets is nowhere dense.
Thus 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.
Instead, such a union is called a meagre set or a set of first category'.
Examples
The boundary of every open and every closed set is nowhere dense.
A nowhere dense set need not be closed, but is properly contained in a nowhere dense closed set, namely its closure. 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 is the unit interval, not only is it possible to have a dense set of Lebesgue measure zero, but it is also possible to have a nowhere dense set with positive measure. For one example, remove from all dyadic fractions, i.e. fractions of the form in lowest terms for positive integers and, and the intervals around them:,. Since for each this removes intervals adding up to at most, the nowhere dense set remaining after all such intervals have been removed has measure of at least and so in a sense represents the majority of the ambient space. This set is nowhere dense, as it is closed and has an empty interior: any interval is not contained in the set since the dyadic fractions in 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 1.
