Nowhere dense set

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In topology, a subset A of a topological space X is called nowhere dense (in X) if there is no neighborhood in X on which A is dense. For example, the integers form a nowhere dense subset of the real line R.

A subset A of a topological space X is nowhere dense in X if and only if the interior of the closure of A is empty. 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.

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.

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Open and closed

  • A nowhere dense set need not be closed (for instance, the set \{1,1/2,1/3,\dots\} is nowhere dense in the reals), but is 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 an 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 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].

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