In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. For example, the integers form a nowhere dense subset of the real line R. A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... The empty set is the set containing no elements. ... The integers are commonly denoted by the above symbol. ... In mathematics, the real line is simply the set of real numbers. ...

Note that the order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it is dense in R, which is the opposite notion. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of...

Note also that the surrounding space matters: a set A may be nowhere dense when considered as a subspace of X but not when considered as a subspace of Y.

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 set of first category or meagre. The concept is important to formulate the Baire category theorem. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). ... In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read sigma, means countable in this context) is a subset with certain desirable closure properties. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... The Baire category theorem is an important tool in general topology and functional analysis. ...

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. In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...

For one example (a variant of the Cantor set), remove from [0,1] all dyadic 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]. The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In mathematics, a dyadic fraction or dyadic rational is a rational number that when written as a fraction has denominator a power of two, i. ... An irreducible fraction is a fraction a/b, where the numerator a is an integer and the denominator b is a positive integer, such that there is not another fraction c/d with c smaller in absolute value than a and 0<d<b, and c and d are integers...

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1.

External links

• Some nowhere dense sets with positive measure