Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density:

A subset ${\displaystyle S}$  of a topological space ${\displaystyle X}$  is said to be dense in another set ${\displaystyle U}$  if the intersection ${\displaystyle S\cap U}$  is a dense subset of ${\displaystyle U.}$  ${\displaystyle S}$  is nowhere dense or rare in ${\displaystyle X}$  if ${\displaystyle S}$  is not dense in any nonempty open subset ${\displaystyle U}$  of ${\displaystyle X.}$ 

Expanding out the negation of density, it is equivalent to require that each nonempty open set ${\displaystyle U}$  contains a nonempty open subset disjoint from ${\displaystyle S.}$ ^[4] It suffices to check either condition on a base for the topology on ${\displaystyle X.}$  In particular, density nowhere in ${\displaystyle \mathbb {R} }$  is often described as being dense in no open interval.^[5][6]

Definition by closure edit

The second definition above is equivalent to requiring that the closure, ${\displaystyle \operatorname {cl} _{X}S,}$  cannot contain any nonempty open set.^[7] This is the same as saying that the interior of the closure of ${\displaystyle S}$  is empty; that is,

${\displaystyle \operatorname {int} _{X}\left(\operatorname {cl} _{X}S\right)=\varnothing .}$ ^[8][9]

Alternatively, the complement of the closure ${\displaystyle X\setminus \left(\operatorname {cl} _{X}S\right)}$  must be a dense subset of ${\displaystyle X;}$ ^[4][8] in other words, the exterior of ${\displaystyle S}$  is dense in ${\displaystyle X.}$ 

The notion of nowhere dense set is always relative to a given surrounding space. Suppose ${\displaystyle A\subseteq Y\subseteq X,}$  where ${\displaystyle Y}$  has the subspace topology induced from ${\displaystyle X.}$  The set ${\displaystyle A}$  may be nowhere dense in ${\displaystyle X,}$  but not nowhere dense in ${\displaystyle Y.}$  Notably, a set is always dense in its own subspace topology. So if ${\displaystyle A}$  is nonempty, it will not be nowhere dense as a subset of itself. However the following results hold:^[10][11]

• If ${\displaystyle A}$  is nowhere dense in ${\displaystyle Y,}$  then ${\displaystyle A}$  is nowhere dense in ${\displaystyle X.}$ 
• If ${\displaystyle Y}$  is open in ${\displaystyle X}$ , then ${\displaystyle A}$  is nowhere dense in ${\displaystyle Y}$  if and only if ${\displaystyle A}$  is nowhere dense in ${\displaystyle X.}$ 
• If ${\displaystyle Y}$  is dense in ${\displaystyle X}$ , then ${\displaystyle A}$  is nowhere dense in ${\displaystyle Y}$  if and only if ${\displaystyle A}$  is nowhere dense in ${\displaystyle X.}$ 

A set is nowhere dense if and only if its closure is.^[1]

Every subset of a nowhere dense set is nowhere dense, and a finite union of nowhere dense sets is nowhere dense.^[12] 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 ${\displaystyle \mathbb {Q} }$  is not nowhere dense in ${\displaystyle \mathbb {R} .}$ 

The boundary of every open set and of every closed set is closed and nowhere dense.^[13][2] A closed set is nowhere dense if and only if it is equal to its boundary,^[13] if and only if it is equal to the boundary of some open set^[2] (for example the open set can be taken as the complement of the set). An arbitrary set ${\displaystyle A\subseteq X}$  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 ${\displaystyle A}$ ).

• The set ${\displaystyle S=\{1/n:n=1,2,...\}}$  and its closure ${\displaystyle S\cup \{0\}}$  are nowhere dense in ${\displaystyle \mathbb {R} ,}$  since the closure has empty interior.
• ${\displaystyle \mathbb {R} }$  viewed as the horizontal axis in the Euclidean plane is nowhere dense in ${\displaystyle \mathbb {R} ^{2}.}$ 
• ${\displaystyle \mathbb {Z} }$  is nowhere dense in ${\displaystyle \mathbb {R} }$  but the rationals ${\displaystyle \mathbb {Q} }$  are not (they are dense everywhere).
• ${\displaystyle \mathbb {Z} \cup [(a,b)\cap \mathbb {Q} ]}$  is not nowhere dense in ${\displaystyle \mathbb {R} }$ : it is dense in the open interval ${\displaystyle (a,b),}$  and in particular the interior of its closure is ${\displaystyle (a,b).}$ 
• The empty set is nowhere dense. In a discrete space, the empty set is the only nowhere dense set.^[14]
• 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.^[15]

A nowhere dense set is not necessarily negligible in every sense. For example, if ${\displaystyle X}$  is the unit interval ${\displaystyle [0,1],}$  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 ${\displaystyle [0,1]}$  all dyadic fractions, i.e. fractions of the form ${\displaystyle a/2^{n}}$  in lowest terms for positive integers ${\displaystyle a,n\in \mathbb {N} ,}$  and the intervals around them: ${\displaystyle \left(a/2^{n}-1/2^{2n+1},a/2^{n}+1/2^{2n+1}\right).}$  Since for each ${\displaystyle n}$  this removes intervals adding up to at most ${\displaystyle 1/2^{n+1},}$  the nowhere dense set remaining after all such intervals have been removed has measure of at least ${\displaystyle 1/2}$  (in fact just over ${\displaystyle 0.535\ldots }$  because of overlaps^[16]) and so in a sense represents the majority of the ambient space ${\displaystyle [0,1].}$  This set is nowhere dense, as it is closed and has an empty interior: any interval ${\displaystyle (a,b)}$  is not contained in the set since the dyadic fractions in ${\displaystyle (a,b)}$  have been removed.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than ${\displaystyle 1,}$  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).^[17]

For another simpler example, if ${\displaystyle U}$  is any dense open subset of ${\displaystyle \mathbb {R} }$  having finite Lebesgue measure then ${\displaystyle \mathbb {R} \setminus U}$  is necessarily a closed subset of ${\displaystyle \mathbb {R} }$  having infinite Lebesgue measure that is also nowhere dense in ${\displaystyle \mathbb {R} }$  (because its topological interior is empty). Such a dense open subset ${\displaystyle U}$  of finite Lebesgue measure is commonly constructed when proving that the Lebesgue measure of the rational numbers ${\displaystyle \mathbb {Q} }$  is ${\displaystyle 0.}$  This may be done by choosing any bijection ${\displaystyle f:\mathbb {N} \to \mathbb {Q} }$  (it actually suffices for ${\displaystyle f:\mathbb {N} \to \mathbb {Q} }$  to merely be a surjection) and for every ${\displaystyle r>0,}$  letting

• Baire space – Concept in topology
• Fat Cantor set – set that is nowhere dense (in particular it contains no intervals), yet has positive measurePages displaying wikidata descriptions as a fallback
• Meagre set – "Small" subset of a topological space

• Bourbaki, Nicolas (1989) [1967]. 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) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

• Some nowhere dense sets with positive measure