NOWHERE DENSE SET

(Redirected from Nowhere dense)
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'.
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.
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.

Contents

Nowhere dense sets with positive measure

External links

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 of the form ''a''/2^''n'' in lowest terms for positive integers ''a'' and ''n'' and the intervals around them [''a''/2^''n'' − 1/2^2''n''+1, ''a''/2^''n'' + 1/2^2''n''+1]; since for each ''n'' this removes intervals adding up to at most 1/2^''n''+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].
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