Editing Nowhere dense set (section)

== 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>