In mathematics, a subset of a topological space is called nowhere dense or rare 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. For example, the integers are nowhere dense among the reals, whereas the interval (0, 1) is not nowhere dense.
A countable union of nowhere dense sets is called a meagre set. Meagre sets play an important role in the formulation of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis.
Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density:
Expanding out the negation of density, it is equivalent to require that each nonempty open set contains a nonempty open subset disjoint from  It suffices to check either condition on a base for the topology on In particular, density nowhere in is often described as being dense in no open interval.
Definition by closure
The notion of nowhere dense set is always relative to a given surrounding space. Suppose where has the subspace topology induced from The set may be nowhere dense in but not nowhere dense in Notably, a set is always dense in its own subspace topology. So if is nonempty, it will not be nowhere dense as a subset of itself. However the following results hold:
- If is nowhere dense in then is nowhere dense in
- If is open in , then is nowhere dense in if and only if is nowhere dense in
- If is dense in , then is nowhere dense in if and only if is nowhere dense in
A set is nowhere dense if and only if its closure is.
Every subset of a nowhere dense set is nowhere dense, and a finite union of nowhere dense sets is nowhere dense. Thus the nowhere dense sets form an ideal of sets, a suitable notion of negligible set. In general they do not form a 𝜎-ideal, as meager sets, which are the countable unions of nowhere dense sets, need not be nowhere dense. For example, the set is not nowhere dense in
The boundary of every open set and of every closed set is closed and nowhere dense. A closed set is nowhere dense if and only if it is equal to its boundary, if and only if it is equal to the boundary of some open set (for example the open set can be taken as the complement of the set). An arbitrary set is nowhere dense if and only if it is a subset of the boundary of some open set (for example the open set can be taken as the exterior of ).
- The set and its closure are nowhere dense in since the closure has empty interior.
- viewed as the horizontal axis in the Euclidean plane is nowhere dense in
- is nowhere dense in but the rationals are not (they are dense everywhere).
- is not nowhere dense in : it is dense in the open interval and in particular the interior of its closure is
- The empty set is nowhere dense. In a discrete space, the empty set is the only nowhere dense set.
- In a T1 space, any singleton set that is not an isolated point is nowhere dense.
- A vector subspace of a topological vector space is either dense or nowhere dense.
Nowhere dense sets with positive measure
A nowhere dense set is not necessarily negligible in every sense. For example, if is the unit interval 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 all dyadic fractions, i.e. fractions of the form in lowest terms for positive integers and the intervals around them: Since for each this removes intervals adding up to at most the nowhere dense set remaining after all such intervals have been removed has measure of at least (in fact just over because of overlaps) and so in a sense represents the majority of the ambient space This set is nowhere dense, as it is closed and has an empty interior: any interval is not contained in the set since the dyadic fractions in have been removed.
Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than although the measure cannot be exactly 1 (because otherwise the complement of its closure would be a nonempty open set with measure zero, which is impossible).
For another simpler example, if is any dense open subset of having finite Lebesgue measure then is necessarily a closed subset of having infinite Lebesgue measure that is also nowhere dense in (because its topological interior is empty). Such a dense open subset of finite Lebesgue measure is commonly constructed when proving that the Lebesgue measure of the rational numbers is This may be done by choosing any bijection (it actually suffices for to merely be a surjection) and for every letting
- Baire space – Concept in topology
- Fat Cantor set – set that is nowhere dense (in particular it contains no intervals), yet has positive measure
- Meagre set – "Small" subset of a topological space
- Bourbaki 1989, ch. IX, section 5.1.
- Willard 2004, Problem 4G.
- Narici & Beckenstein 2011, section 11.5, pp. 387-389.
- Fremlin 2002, 3A3F(a).
- Oxtoby, John C. (1980). Measure and Category (2nd ed.). New York: Springer-Verlag. pp. 1–2. ISBN 0-387-90508-1.
A set is nowhere dense if it is dense in no interval; although note that Oxtoby later gives the interior-of-closure definition on page 40.
- Natanson, Israel P. (1955). Teoria functsiy veshchestvennoy peremennoy [Theory of functions of a real variable]. Vol. I (Chapters 1-9). Translated by Boron, Leo F. New York: Frederick Ungar. p. 88. hdl:2027/mdp.49015000681685. LCCN 54-7420.
- Steen, Lynn Arthur; Seebach Jr., J. Arthur (1995). Counterexamples in Topology (Dover republication of Springer-Verlag 1978 ed.). New York: Dover. p. 7. ISBN 978-0-486-68735-3.
A subset of is said to be nowhere dense in if no nonempty open set of is contained in
- Gamelin, Theodore W. (1999). Introduction to Topology (2nd ed.). Mineola: Dover. pp. 36–37. ISBN 0-486-40680-6 – via ProQuest ebook Central.
- Rudin 1991, p. 41.
- Narici & Beckenstein 2011, Theorem 11.5.4.
- Haworth & McCoy 1977, Proposition 1.3.
- Fremlin 2002, 3A3F(c).
- Narici & Beckenstein 2011, Example 11.5.3(e).
- Narici & Beckenstein 2011, Example 11.5.3(a).
- Narici & Beckenstein 2011, Example 11.5.3(f).
- "Some nowhere dense sets with positive measure and a strictly monotonic continuous function with a dense set of points with zero derivative".
- Folland, G. B. (1984). Real analysis: modern techniques and their applications. New York: John Wiley & Sons. p. 41. hdl:2027/mdp.49015000929258. ISBN 0-471-80958-6.
- Bourbaki, Nicolas (1989) . General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063.
- Fremlin, D. H. (2002). Measure Theory. Lulu.com. ISBN 978-0-9566071-1-9.
- Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk
- Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Trèves, François (2006) . Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Willard, Stephen (2004) . General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.