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Nowhere Dense SetIn 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. The concept is important to formulate the Baire category theorem. 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; 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. Generalising this method, one can construct in the unit interval nowhere dense sets of any measure less than 1.
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