To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Nowhere dense set

In mathematics, a subset of a topological space is called nowhere dense or rare[1] if its closure has empty interior.[note 1] 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 an open ball is not.

The surrounding space matters: a set ${\displaystyle A}$ may be nowhere dense when considered as a subset of a topological space ${\displaystyle X,}$ but not when considered as a subset of another topological space ${\displaystyle Y.}$ Notably, a set is always dense in its own subspace topology.

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.

## Characterizations

Density nowhere can be characterized in three 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 ${\displaystyle S.}$[2] It suffices to check either condition on a base for the topology on ${\displaystyle X,}$ and density nowhere in ${\displaystyle \mathbb {R} }$ is often described as being dense in no open interval.[3][4]

### Definition by closure

The second definition above is equivalent to requiring that the closure, ${\displaystyle \operatorname {cl} _{X}S,}$ cannot contain any nonempty open set.[5] This is the same as saying that the interior of the closure of ${\displaystyle S}$ (both taken in ${\displaystyle X}$) is empty; that is,

${\displaystyle \operatorname {int} _{X}\left(\operatorname {cl} _{X}S\right)=\varnothing .}$[6][7]

Alternatively, the complement of the closure ${\displaystyle X\setminus \left(\operatorname {cl} _{X}S\right)}$ must be a dense subset of ${\displaystyle X.}$[2][6]

### Definition by boundaries

From the previous remark, ${\displaystyle S}$ is nowhere dense in ${\displaystyle X}$ if and only if ${\displaystyle S}$ is a subset of the boundary of a dense open subset: namely, ${\displaystyle X\setminus \left(\operatorname {cl} _{X}S\right).}$ In fact, one can remove the denseness condition:

${\displaystyle S}$ is nowhere dense if and only if there exists some open subset ${\displaystyle U}$ of ${\displaystyle X}$ such that ${\displaystyle S\subseteq \operatorname {bd} _{X}U,}$

Alternatively, one can strengthen the containment to equality by taking the closure:

${\displaystyle S}$ is nowhere dense if and only if there exists some open subset ${\displaystyle U}$ of ${\displaystyle X}$ such that ${\displaystyle \operatorname {cl} _{X}S=\operatorname {bd} _{X}U.}$[8]

If ${\displaystyle S}$ is closed, this implies by trichotomy that ${\displaystyle S}$ is nowhere dense if and only if ${\displaystyle S}$ is equal to its topological boundary.[1]

## Properties and sufficient conditions

• A set is nowhere dense if and only if its closure is.[1] Thus a nowhere dense set need not be closed (for instance, the set ${\displaystyle \left\{{\frac {1}{n}}:n\in \mathbb {N} \right\}}$ is nowhere dense in the reals), but is then properly contained in a nowhere dense closed set.
• Suppose ${\displaystyle A\subseteq B\subseteq X.}$
• If ${\displaystyle A}$ is nowhere dense in ${\displaystyle B}$ then ${\displaystyle A}$ is nowhere dense in ${\displaystyle X.}$
• If ${\displaystyle A}$ is nowhere dense in ${\displaystyle X}$ and ${\displaystyle B}$ is an open subset of ${\displaystyle X}$ then ${\displaystyle A}$ is nowhere dense in ${\displaystyle B.}$[1]
• Every subset of a nowhere dense set is nowhere dense.[9]
• The union of finitely many nowhere dense sets is nowhere dense.[9]

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 do not, in general, form a 𝜎-ideal.) Instead, such a union is called a meagre set or a set of first category.

## Examples

• ${\displaystyle S:=\left\{{\frac {1}{n}}:n\in \mathbb {N} \right\}}$ is nowhere dense in ${\displaystyle \mathbb {R} }$: although the points get arbitrarily close to ${\displaystyle 0,}$ the closure of the set is ${\displaystyle S\cup \{0\},}$ which has empty interior (and is thus also nowhere dense in ${\displaystyle \mathbb {R} }$).[1]
• ${\displaystyle \mathbb {R} }$ is nowhere dense in ${\displaystyle \mathbb {R} ^{2}.}$[1]
• ${\displaystyle \mathbb {Z} }$ is nowhere dense in ${\displaystyle \mathbb {R} }$ but the rationals ${\displaystyle \mathbb {Q} }$ are not (they are dense everywhere).[1]
• ${\displaystyle \mathbb {Z} \cup [(a,b)\cap \mathbb {Q} ]}$ is not nowhere dense in ${\displaystyle \mathbb {R} }$: it is dense in the 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 such subset.[1]
• In a T1 space, any singleton set that is not an isolated point is nowhere dense.
• The boundary of every open set and of every closed set is nowhere dense.[1]
• A vector subspace of a topological vector space is either dense or nowhere dense.[1]

## Nowhere dense sets with positive measure

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[10]) 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).[11]

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

${\displaystyle U_{r}~:=~\bigcup _{n\in \mathbb {N} }\left(f(n)-r/2^{n},f(n)+r/2^{n}\right)~=~\bigcup _{n\in \mathbb {N} }f(n)+\left(-r/2^{n},r/2^{n}\right)}$
(here, the Minkowski sum notation ${\displaystyle f(n)+\left(-r/2^{n},r/2^{n}\right):=\left(f(n)-r/2^{n},f(n)+r/2^{n}\right)}$ was used to simplify the description of the intervals). The open subset ${\displaystyle U_{r}}$ is dense in ${\displaystyle \mathbb {R} }$ because this is true of its subset ${\displaystyle \mathbb {Q} }$ and its Lebesgue measure is no greater than ${\displaystyle \sum _{n\in \mathbb {N} }2r/2^{n}=2r.}$ Taking the union of closed, rather than open, intervals produces the F𝜎-subset
${\displaystyle S_{r}~:=~\bigcup _{n\in \mathbb {N} }f(n)+\left[-r/2^{n},r/2^{n}\right]}$
that satisfies ${\displaystyle S_{r/2}\subseteq U_{r}\subseteq S_{r}\subseteq U_{2r}.}$ Because ${\displaystyle \mathbb {R} \setminus S_{r}}$ is a subset of the nowhere dense set ${\displaystyle \mathbb {R} \setminus U_{r},}$ it is also nowhere dense in ${\displaystyle \mathbb {R} .}$ Because ${\displaystyle \mathbb {R} }$ is a Baire space, the set
${\displaystyle D:=\bigcap _{m=1}^{\infty }U_{1/m}=\bigcap _{m=1}^{\infty }S_{1/m}}$
is a dense subset of ${\displaystyle \mathbb {R} }$ (which means that like its subset ${\displaystyle \mathbb {Q} ,}$ ${\displaystyle D}$ cannot possibly be nowhere dense in ${\displaystyle \mathbb {R} }$) with ${\displaystyle 0}$ Lebesgue measure that is also a nonmeager subset of ${\displaystyle \mathbb {R} }$ (that is, ${\displaystyle D}$ is of the second category in ${\displaystyle \mathbb {R} }$), which makes ${\displaystyle \mathbb {R} \setminus D}$ a comeager subset of ${\displaystyle \mathbb {R} }$ whose interior in ${\displaystyle \mathbb {R} }$ is also empty; however, ${\displaystyle \mathbb {R} \setminus D}$ is nowhere dense in ${\displaystyle \mathbb {R} }$ if and only if its closure in ${\displaystyle \mathbb {R} }$ has empty interior. The subset ${\displaystyle \mathbb {Q} }$ in this example can be replaced by any countable dense subset of ${\displaystyle \mathbb {R} }$ and furthermore, even the set ${\displaystyle \mathbb {R} }$ can be replaced by ${\displaystyle \mathbb {R} ^{n}}$ for any integer ${\displaystyle n>0.}$

## Notes

1. ^ The order of operations is important. For example, the set of rational numbers, as a subset of the real numbers, ${\displaystyle \mathbb {R} ,}$ has the property that its interior has an empty closure, but it is not nowhere dense; in fact it is dense in ${\displaystyle \mathbb {R} .}$

## References

1. Narici & Beckenstein 2011, pp. 371–423.
2. ^ a b Fremlin 2002, 3A3F(a).
3. ^ 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.
4. ^ Natanson, Israel P. (1955). Teoria functsiy veshchestvennoy peremennoy [Theory of functions of a real variable]. Volume I (Chapters 1-9). Translated by Boron, Leo F. New York: Frederick Ungar. p. 88. hdl:2027/mdp.49015000681685. LCCN 54-7420. |volume= has extra text (help)
5. ^ 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 ${\displaystyle A}$ of ${\displaystyle X}$ is said to be nowhere dense in ${\displaystyle X}$ if no nonempty open set of ${\displaystyle X}$ is contained in ${\displaystyle {\overline {A}}.}$
6. ^ a b Gamelin, Theodore W. (1999). Introduction to Topology (2nd ed.). Mineola: Dover. pp. 36–37. ISBN 0-486-40680-6 – via ProQuest ebook Central.
7. ^ Rudin 1991, p. 41.
8. ^ Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley. p. 37. hdl:2027/mdp.49015000696204. LCCN 74-100890.CS1 maint: date and year (link) See 4G(2-3).
9. ^ a b Fremlin 2002, 3A3F(c).
10. ^ Some nowhere dense sets with positive measure and a strictly monotonic continuous function with a dense set of points with zero derivative
11. ^ 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.CS1 maint: date and year (link)

## Bibliography

Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.