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In mathematics, a '''nowhere dense set''' in a [[topological space]] is a set whose [[closure (topology)|closure]] has empty [[interior (topology)|interior]]. The order of operations is important. For example, the set of [[rational number]]s, 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 set|dense]] in '''R'''.
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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.
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Every subset of a nowhere dense set is nowhere dense, and the [[union (set theory)|union]] of [[finite set|finite]]ly 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 [[countable|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 set|closed]] (for instance, the set <math>\left\{\frac{1}{n} \mid n \in \mathbb{N} \right\}</math> is nowhere dense in the reals), but is properly contained in a nowhere dense closed set, namely its [[closure (topology)|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 (set theory)|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 (topology)|interior]].
 
* The [[boundary (topology)|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 fraction]]s, i.e. fractions of the form ''a''/2<sup>''n''</sup> in [[lowest terms]] for positive integers ''a'' and ''n'', and the intervals around them: (''a''/2<sup>''n''</sup>&nbsp;&minus;&nbsp;1/2<sup>2''n''+1</sup>, ''a''/2<sup>''n''</sup>&nbsp;+&nbsp;1/2<sup>2''n''+1</sup>). Since for each ''n'' this removes intervals adding up to at most  1/2<sup>''n''+1</sup>, 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&nbsp;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==
*[http://www.se16.info/hgb/nowhere.htm Some nowhere dense sets with positive measure]
*[http://www.encyclopediaofmath.org/index.php/Nowhere_dense_set Nowhere dense set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
 
[[Category:General topology]]
 
[[de:Dichte Teilmenge#Nirgends dichte Teilmenge]]
[[ru:Глоссарий общей топологии#Н]]

Latest revision as of 21:13, 23 November 2014

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