Editing Nowhere dense set (section)

=== 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>