In mathematics, a subset of a topological space is called nowhere dense or a rare[1] if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. The order of operations is important. For example, the set of rational numbers, as a subset of the real numbers, ℝ, has the property that its interior has an empty closure, 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 subset of a topological space X, but not when considered as a subset of another topological space Y. Notably, a set is always dense in its own subspace topology.
A countable union of nowhere dense sets is called a meagre set. Meager sets play an important role in the formulation of the Baire category theorem.
Characterizations
Let X be a topological space and S a subset of X. Then the following are equivalent:
- S is nowhere dense in X;
- (definition) the interior of the closure of S (both taken in X) is empty;
- the closure of S in X does not contain any non-empty open subset of X;
- S ∩ U is not dense in any nonempty open subset U of X;
- the complement in X of the closure of S is dense in X;[1]
- every non-empty open subset V of X contains a non-empty open subset U of X such that U ∩ S = ∅;[1]
- the closure of S is nowhere dense in X (according to any defining condition other than this one);[1]
- to see this, recall that a subset of X has empty interior if and only if its complement is dense in X.
- (only for the case S closed) S is equal to its boundary.[1]
Properties and sufficient conditions
- Suppose A ⊆ B ⊆ X.
- If A is nowhere dense in B then A is nowhere dense in X.
- If A is nowhere dense in X and B is an open subset of X then A is nowhere dense in B.[1]
- Every subset of a nowhere dense set is nowhere dense.[1]
- The union of finitely many nowhere dense sets is nowhere dense.
Thus 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.
Examples
- The boundary of every open set and of every closed set is nowhere dense.[1]
- The empty set is nowhere dense and in a discrete space, the empty set is the only nowhere dense subset.[1]
- In a T1 space, any singleton set that is not an isolated point is nowhere dense.
- ℝ is nowhere dense in ℝ2.[1]
- ℤ is nowhere dense in ℝ but the rationals ℚ are not.[1]
- S = { 1/n : n ∈ ℕ} is nowhere dense in ℝ: although the points get arbitrarily close to 0, the closure of the set is S ∪ { 0 }, which has empty interior (and is thus also nowhere dense in ℝ).[1]
- ℤ ∪ [(a, b) ∩ ℚ] is not nowhere dense in ℝ: it is dense in the interval [a, b], and in particular the interior of its closure is (a, b).
- A vector subspace of a topological vector space is either dense or nowhere dense.[1]
Open and closed
- A nowhere dense set need not be closed (for instance, the set { 1/n : n ∈ ℕ } is nowhere dense in the reals), but is properly contained in a nowhere dense closed set, namely its closure (which would add 0 to the example set). 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 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 fractions, i.e. 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]. 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 1, although the measure cannot be exactly 1 (else the complement of its closure would be a nonempty open set with measure zero, which is impossible).
See also
Baire space
Fat Cantor set
References
Narici 2011, pp. 371-423.
Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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