Editing Closed set (section)

{{Short description|Complement of an open subset}}
{{About|the complement of an [[open set]]|a set closed under an operation|closure (mathematics)|other uses|Closed (disambiguation)}}

In [[geometry]], [[topology]], and related branches of [[mathematics]], a '''closed set''' is a [[Set (mathematics)|set]] whose [[complement (set theory)|complement]] is an [[open set]].<ref>{{cite book|last=Rudin|first=Walter|author-link=Walter Rudin|title=Principles of Mathematical Analysis|url=https://archive.org/details/principlesofmath00rudi|url-access=registration|publisher=[[McGraw-Hill]]|year=1976|isbn=0-07-054235-X}}</ref><ref>{{cite book|last=Munkres|first=James R.|author-link=James Munkres|title=Topology|edition=2nd|publisher=[[Prentice Hall]]|year=2000|isbn=0-13-181629-2}}</ref> In a [[topological space]], a closed set can be defined as a set which contains all its [[limit point]]s. In a [[complete metric space]], a closed set is a set which is [[Closure (mathematics)|closed]] under the [[limit of a sequence|limit]] operation. This should not be confused with [[closed manifold]].

Sets that are both open and closed and are called [[clopen sets]].