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In topology, a subset A of
a topological space 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 essentially 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. The union of countably many nowhere dense
sets, however, need not be nowhere dense. Thus, the nowhere dense sets need not form a σ-ideal.
The concept is mainly important to formulate the Baire
category theorem.
Nowhere dense does not necessarily mean rare. In the interval [0,1], not only is it possible to have a dense set of Lebesgue measure zero (such as the rationals), but it is also possible to
have a nowhere dense set with positive measure (such as variants on the Cantor
set). One example would be to remove from [0,1] all dyadic
fractions in lowest terms of the form
a/2n 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 1/2n+1 or less because of overlaps, the nowhere dense
set remaining after all such intervals have been removed has measure of more than 1/2 and so in a sense represents the majority
of [0,1].
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