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In [[topology]], a '''clopen set''' (a [[portmanteau]] of '''closed-open set''') in a [[topological space]] is a set which is both [[open set|open]] and [[closed set|closed]]. That this is possible may seem counter-intuitive, as the common meanings of ''open'' and ''closed'' are antonyms. But their mathematical definitions are not [[mutually exclusive]]. A set is closed if its [[Complement_(set_theory)|complement]] is open, which leaves the possibility of an open set whose complement is itself also open, making both sets both open ''and'' closed, and therefore clopen.
 
== Examples ==
 
In any topological space ''X'', the [[empty set]] and the whole space ''X'' are both clopen.<ref>{{cite book|author=Bartle, Robert G.|coauthors=Sherbert, Donald R.|title=Introduction to Real Analysis|edition=2nd|publisher=John Wiley & Sons, Inc.|origyear=1982|year=1992|pages=348 (regarding the real numbers and the empty set in R)}}</ref><ref>{{cite book|author=Hocking, John G.|coauthors=Young, Gail S.|title=Topology|publisher=Dover Publications, Inc.|location=NY|year=1961|pages=5 and 6 (regarding topological spaces)}}
</ref>
 
Now consider the space ''X'' which consists of the union of the two open [[interval (mathematics)|interval]]s (0,1) and (2,3) of '''R'''. The topology on ''X'' is inherited as the [[topological subspace|subspace topology]] from the ordinary topology on the [[real line]] '''R'''. In ''X'', the set (0,1) is clopen, as is the set (2,3). This is a quite typical example: whenever a space is made up of a finite number of disjoint [[connected space|connected component]]s in this way, the components will be clopen.
 
As a less trivial example, consider the space '''Q''' of all [[rational number]]s with their ordinary topology, and the set ''A'' of all positive rational numbers whose square is bigger than 2. Using the fact that <math>\sqrt 2</math> is not in '''Q''', one can show quite easily that ''A'' is a clopen subset of '''Q'''. (Note also that ''A'' is ''not'' a clopen subset of the real line '''R'''; it is neither open nor closed in '''R'''.)
 
==Properties==
 
* A topological space ''X'' is [[connected space|connected]] if and only if the only clopen sets are the empty set and ''X''.
* A set is clopen if and only if its [[boundary (topology)|boundary]] is empty.
* Any clopen set is a union of (possibly infinitely many) [[Connected space|connected components]].
* If all connected components of ''X'' are open (for instance, if ''X'' has only finitely many components, or if ''X'' is [[locally connected]]), then a set is clopen in ''X'' if and only if it is a union of connected components.
* A topological space ''X'' is [[discrete space|discrete]] if and only if all of its subsets are clopen.
* Using the [[union (set theory)|union]] and [[intersection (set theory)|intersection]] as operations, the clopen subsets of a given topological space ''X'' form a [[Boolean algebra (structure)|Boolean algebra]]. ''Every'' Boolean algebra can be obtained in this way from a suitable topological space: see [[Stone's representation theorem for Boolean algebras]].
 
==Notes==
{{reflist}}
 
==References==
*{{cite web|url=http://uob-community.ballarat.edu.au/~smorris/topology.htm|title=Topology Without Tears|author=Sidney A. Morris|archiveurl=http://web.archive.org/web/20130419134743/http://uob-community.ballarat.edu.au/~smorris/topology.htm|archivedate=19 April 2013}}
 
[[Category:General topology]]

Revision as of 17:19, 9 February 2014

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