View source for Nowhere dense set

{{short description|Mathematical set whose closure has empty interior}}

In [[mathematics]], a [[Set (mathematics)|subset]] of a [[topological space]] is called '''nowhere dense'''{{sfn|Bourbaki|1989|loc=ch. IX, section 5.1}}{{sfn|Willard|2004|loc=Problem 4G}} or '''rare'''{{sfn|Narici|Beckenstein|2011|loc=section 11.5, pp. 387-389}} if its [[closure (topology)|closure]] has [[Empty set|empty]] [[interior (topology)|interior]].  In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the [[Topological space#Definitions|topology]] on the space) anywhere.  For example, the [[integer]]s are nowhere dense among the [[real number|real]]s, whereas the [[interval (mathematics)|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]].

== Definition ==

Density nowhere can be characterized in different (but equivalent) ways.  The simplest definition is the one from density:
<blockquote>A subset <math>S</math> of a [[topological space]] <math>X</math> is said to be '''''dense''''' in another set <math>U</math> if the intersection <math>S \cap U</math> is a [[Dense set|dense subset]] of <math>U.</math>  <math>S</math> is '''{{em|nowhere dense}}''' or '''{{em|rare}}''' in <math>X</math> if <math>S</math> is not dense in any nonempty open subset <math>U</math> of <math>X.</math>  </blockquote>
Expanding out the negation of density, it is equivalent that each nonempty open set <math>U</math> contains a nonempty open subset disjoint from <math>S.</math>{{sfn|Fremlin|2002|loc=3A3F(a)}}  It suffices to check either condition on a [[Base (topology)|base]] for the topology on <math>X.</math>  In particular, density nowhere in <math>\R</math> is often described as being dense in no [[Open Interval|open interval]].<ref>{{Cite book|last=Oxtoby|first=John C.|title=Measure and Category|publisher=Springer-Verlag|year=1980|isbn=0-387-90508-1|edition=2nd|location=New York|pages=1–2|quote=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.</ref><ref>{{Cite book|last=Natanson|first=Israel P.|url=http://hdl.handle.net/2027/mdp.49015000681685|title=Teoria functsiy veshchestvennoy peremennoy|publisher=Frederick Ungar|year=1955|volume=I (Chapters 1-9)|location=New York|pages=88|hdl=2027/mdp.49015000681685|language=English|translator-last=Boron|translator-first=Leo F.|trans-title=Theory of functions of a real variable|lccn=54-7420}}</ref>

=== Definition by closure ===

The second definition above is equivalent to requiring that the closure, <math>\operatorname{cl}_X S,</math> cannot contain any nonempty open set.<ref>{{Cite book|last1=Steen|first1=Lynn Arthur|title=Counterexamples in Topology|last2=Seebach Jr.|first2=J. Arthur|publisher=Dover|year=1995|isbn=978-0-486-68735-3|edition=Dover republication of Springer-Verlag 1978|location=New York|pages=7|quote=A subset <math>A</math> of <math>X</math> is said to be nowhere dense in <math>X</math> if no nonempty open set of <math>X</math> is contained in <math>\overline{A}.</math>}}</ref>  This is the same as saying that the [[interior (topology)|interior]] of the [[Closure (topology)|closure]] of <math>S</math> is empty; that is,<blockquote><math>\operatorname{int}_X \left(\operatorname{cl}_X S\right) = \varnothing.</math><ref name=":0">{{Cite book|last=Gamelin|first=Theodore&nbsp;W.|title=Introduction to Topology|publisher=Dover|year=1999|isbn=0-486-40680-6|edition=2nd|location=Mineola|pages=36–37|via=ProQuest ebook Central}}</ref>{{sfn|Rudin|1991|p=41}}  </blockquote>Alternatively, the complement of the closure <math>X \setminus \left(\operatorname{cl}_X S\right)</math> must be a dense subset of <math>X;</math>{{sfn|Fremlin|2002|loc=3A3F(a)}}<ref name=":0" /> in other words, the [[exterior (topology)|exterior]] of <math>S</math> is dense in <math>X.</math>

== Properties ==

The notion of ''nowhere dense set'' is always relative to a given surrounding space.  Suppose <math>A\subseteq Y\subseteq X,</math> where <math>Y</math> has the [[subspace topology]] induced from <math>X.</math>  The set <math>A</math> may be nowhere dense in <math>X,</math> but not nowhere dense in <math>Y.</math>  Notably, a set is always dense in its own subspace topology.  So if <math>A</math> is nonempty, it will not be nowhere dense as a subset of itself.  However the following results hold:{{sfn|Narici|Beckenstein|2011|loc=Theorem 11.5.4}}{{sfn|Haworth|McCoy|1977|loc=Proposition 1.3}}
* If <math>A</math> is nowhere dense in <math>Y,</math> then <math>A</math> is nowhere dense in <math>X.</math>
* If <math>Y</math> is open in <math>X</math>, then <math>A</math> is nowhere dense in <math>Y</math> if and only if <math>A</math> is nowhere dense in <math>X.</math>
* If <math>Y</math> is dense in <math>X</math>, then <math>A</math> is nowhere dense in <math>Y</math> if and only if <math>A</math> is nowhere dense in <math>X.</math>

A set is nowhere dense if and only if its closure is.{{sfn|Bourbaki|1989|loc=ch. IX, section 5.1}}

Every subset of a nowhere dense set is nowhere dense, and a finite [[union (set theory)|union]] of nowhere dense sets is nowhere dense.{{sfn|Fremlin|2002|loc=3A3F(c)}}{{sfn|Willard|2004|loc=Problem 25A}}  Thus the nowhere dense sets form an [[ideal of sets]], a suitable notion of [[negligible set]].  In general they do not form a [[sigma-ideal|𝜎-ideal]], as [[meager set]]s, which are the countable unions of nowhere dense sets, need not be nowhere dense.  For example, the set <math>\Q</math> is not nowhere dense in <math>\R.</math>

The [[boundary (topology)|boundary]] of every open set and of every closed set is closed and nowhere dense.{{sfn|Narici|Beckenstein|2011|loc=Example 11.5.3(e)}}{{sfn|Willard|2004|loc=Problem 4G}}  A closed set is nowhere dense if and only if it is equal to its boundary,{{sfn|Narici|Beckenstein|2011|loc=Example 11.5.3(e)}} if and only if it is equal to the boundary of some open set{{sfn|Willard|2004|loc=Problem 4G}} (for example the open set can be taken as the complement of the set).  An arbitrary set <math>A\subseteq X</math> 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 (topology)|exterior]] of <math>A</math>).

== Examples ==

* The set <math>S=\{1/n:n=1,2,...\}</math> and its closure <math>S\cup\{0\}</math> are nowhere dense in <math>\R,</math> since the closure has empty interior.
* The [[Cantor set]] is an uncountable nowhere dense set in <math>\R.</math>
* <math>\R</math> viewed as the horizontal axis in the Euclidean plane is nowhere dense in <math>\R^2.</math>
* <math>\Z</math> is nowhere dense in <math>\R</math> but the rationals <math>\Q</math> are not (they are dense everywhere).
* <math>\Z \cup [(a, b) \cap \Q]</math> is '''{{em|not}}''' nowhere dense in <math>\R</math>: it is dense in the open interval <math>(a,b),</math> and in particular the interior of its closure is <math>(a,b).</math>
* The empty set is nowhere dense.  In a [[discrete space]], the empty set is the {{em|only}} nowhere dense set.{{sfn|Narici|Beckenstein|2011|loc=Example 11.5.3(a)}}
* In a [[T1 space|T<sub>1</sub> 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.{{sfn|Narici|Beckenstein|2011|loc=Example 11.5.3(f)}}

== Nowhere dense sets with positive measure ==

A nowhere dense set is not necessarily negligible in every sense.  For example, if <math>X</math> is the [[unit interval]] <math>[0, 1],</math> 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. One such example is the [[Smith–Volterra–Cantor set]].

For another example (a variant of the [[Cantor set]]), remove from <math>[0, 1]</math> all [[dyadic fraction]]s, i.e. fractions of the form <math>a/2^n</math> in [[lowest terms]] for positive integers <math>a, n \in \N,</math> and the intervals around them: <math>\left(a/2^n - 1/2^{2n+1}, a/2^n + 1/2^{2n+1}\right).</math> 
Since for each <math>n</math> this removes intervals adding up to at most <math>1/2^{n+1},</math> the nowhere dense set remaining after all such intervals have been removed has measure of at least <math>1/2</math> (in fact just over <math>0.535\ldots</math> because of overlaps<ref>{{cite web| url = http://www.se16.info/hgb/nowhere.htm| title = Some nowhere dense sets with positive measure and a strictly monotonic continuous function with a dense set of points with zero derivative}}</ref>) and so in a sense represents the majority of the ambient space <math>[0, 1].</math> 
This set is nowhere dense, as it is closed and has an empty interior: any interval <math>(a, b)</math> is not contained in the set since the dyadic fractions in <math>(a, b)</math> have been removed.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than <math>1,</math> 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).<ref>{{Cite book|last=Folland|first=G. B.|url=http://hdl.handle.net/2027/mdp.49015000929258|title=Real analysis: modern techniques and their applications|publisher=John Wiley & Sons|year=1984|isbn=0-471-80958-6|location=New York|pages=41|hdl=2027/mdp.49015000929258}}</ref>

For another simpler example, if <math>U</math> is any dense open subset of <math>\R</math> having finite [[Lebesgue measure]] then <math>\R \setminus U</math> is necessarily a closed subset of <math>\R</math> having infinite Lebesgue measure that is also nowhere dense in <math>\R</math> (because its topological interior is empty). Such a dense open subset <math>U</math> of finite Lebesgue measure is commonly constructed when proving that the Lebesgue measure of the rational numbers <math>\Q</math> is <math>0.</math> This may be done by choosing any [[bijection]] <math>f : \N \to \Q</math> (it actually suffices for <math>f : \N \to \Q</math> to merely be a [[surjection]]) and for every <math>r > 0,</math> letting 
<math display="block">U_r ~:=~ \bigcup_{n \in \N} \left(f(n) - r/2^n, f(n) + r/2^n\right) ~=~ \bigcup_{n \in \N} f(n) + \left(- r/2^n, r/2^n\right)</math>
(here, the [[Minkowski sum]] notation <math>f(n) + \left(- r/2^n, r/2^n\right) := \left(f(n) - r/2^n, f(n) + r/2^n\right)</math> was used to simplify the description of the intervals). 
The open subset <math>U_r</math> is dense in <math>\R</math> because this is true of its subset <math>\Q</math> and its Lebesgue measure is no greater than <math>\sum_{n \in \N} 2 r / 2^n = 2 r.</math> 
Taking the union of closed, rather than open, intervals produces the [[Fσ set|F<sub>{{sigma}}</sub>-subset]]
<math display="block">S_r ~:=~ \bigcup_{n \in \N} f(n) + \left[- r/2^n, r/2^n\right]</math>
that satisfies <math>S_{r/2} \subseteq U_r \subseteq S_r \subseteq U_{2r}.</math> Because <math>\R \setminus S_r</math> is a subset of the nowhere dense set <math>\R \setminus U_r,</math> it is also nowhere dense in <math>\R.</math> 
Because <math>\R</math> is a [[Baire space]], the set
<math display="block">D := \bigcap_{m=1}^{\infty} U_{1/m} = \bigcap_{m=1}^{\infty} S_{1/m}</math>
is a dense subset of <math>\R</math> (which means that like its subset <math>\Q,</math> <math>D</math> cannot possibly be nowhere dense in <math>\R</math>) with <math>0</math> Lebesgue measure that is also a [[Nonmeager set|nonmeager subset]] of <math>\R</math> (that is, <math>D</math> is of the [[second category]] in <math>\R</math>), which makes <math>\R \setminus D</math> a [[Comeager set|comeager subset]] of <math>\R</math> whose interior in <math>\R</math> is also empty; however, <math>\R \setminus D</math> is nowhere dense in <math>\R</math> if and only if its {{em|closure}} in <math>\R</math> has empty interior. 
The subset <math>\Q</math> in this example can be replaced by any countable dense subset of <math>\R</math> and furthermore, even the set <math>\R</math> can be replaced by <math>\R^n</math> for any integer <math>n > 0.</math>

== See also ==

* {{annotated link|Baire space}}
* {{annotated link|Smith–Volterra–Cantor set}}
* {{annotated link|Meagre set}}

== References ==

{{reflist}}

== Bibliography ==

* {{Bourbaki General Topology Part II Chapters 5-10}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{Cite book|last=Fremlin|first=D.&nbsp;H.|title=Measure Theory|publisher=Lulu.com|year=2002|isbn=978-0-9566071-1-9}}  <!--Do NOT indicate quotes to this by page number.  The book's TeX source code is available for free, so most readers will have access to it that way, but reproducing the PDF from the TeX changes page numbers (quite dramatically).  {{sfn|Fremlin|2002|loc=???}}-->
* {{Citation|last1=Haworth|first1=R. C.|last2=McCoy|first2=R. A.|title=Baire Spaces|location=Warszawa|publisher=Instytut Matematyczny Polskiej Akademi Nauk|year=1977|url=http://eudml.org/doc/268479}}
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Willard General Topology}} <!--{{sfn|Willard|2004|p=}}-->

== External links ==

* [http://www.se16.info/hgb/nowhere.htm Some nowhere dense sets with positive measure]

[[Category:General topology]]

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