in mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. 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&nbsp;r. 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. == 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&nbsp;&minus;&nbsp;1/22n+1, a/2n&nbsp;+&nbsp;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 h
General_topology