Editing Clopen set

{{short description|Subset which is both open and closed}}
{{Distinguish|Half-open interval}}
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 counterintuitive, as the common meanings of {{em|open}} and {{em|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 also open, making both sets both open {{em|and}} closed, and therefore clopen. As described by topologist [[James Munkres]], unlike a [[door]], "a set can be open, or closed, or both, or neither!"{{sfn|Munkres|2000|p=91}} emphasizing that the meaning of "open"/"closed" for {{em|doors}} is unrelated to their meaning for {{em|sets}} (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "[[door space]]s" their name. 

== Examples ==

In any topological space <math>X,</math> the [[empty set]] and the whole space <math>X</math> are both clopen.<ref>{{cite book|last1=Bartle|first1=Robert G.|author-link1=Robert G. Bartle |last2=Sherbert|first2=Donald R.|date=1992|orig-year=1982|title=Introduction to Real Analysis|edition=2nd|publisher=John Wiley & Sons, Inc.|page=348}} (regarding the real numbers and the empty set in R)</ref><ref>{{cite book|last1=Hocking|first1=John G.|last2=Young|first2=Gail S. |date=1961|title=Topology|publisher=Dover Publications, Inc.|location=NY|page=56}} (regarding topological spaces)</ref>

Now consider the space <math>X</math> which consists of the [[union (set theory)|union]] of the two open [[Interval (mathematics)|interval]]s <math>(0, 1)</math> and <math>(2, 3)</math> of <math>\R.</math> The [[topological space|topology]] on <math>X</math> is inherited as the [[Topological subspace|subspace topology]] from the ordinary topology on the [[real line]] <math>\R.</math> In <math>X,</math> the set <math>(0, 1)</math> is clopen, as is the set <math>(2, 3).</math> This is a quite typical example: whenever a space is made up of a finite number of [[disjoint sets|disjoint]] [[Connected space|connected components]] in this way, the components will be clopen.

Now let <math>X</math> be an infinite set under the [[discrete metric]]{{snd}}that is, two points <math>p, q \in X</math> have distance 1 if they're not the same point, and 0 otherwise. Under the resulting [[metric space]], any [[singleton set]] is open; hence any set, being the union of single points, is open. Since any set is open, the complement of any set is open too, and therefore any set is closed. So, all sets in this metric space are clopen.

As a less trivial example, consider the space <math>\Q</math> of all [[rational number]]s with their ordinary topology, and the set <math>A</math> of all positive rational numbers whose [[square (algebra)|square]] is bigger than 2. Using the fact that <math>\sqrt 2</math> is not in <math>\Q,</math> one can show quite easily that <math>A</math> is a clopen subset of <math>\Q.</math> (<math>A</math> is {{em|not}} a clopen subset of the real line <math>\R</math>; it is neither open nor closed in <math>\R.</math>)

== Properties ==

* A topological space <math>X</math> is [[Connected space|connected]] if and only if the only clopen sets are the empty set and <math>X</math> itself.
* A set is clopen if and only if its [[Boundary (topology)|boundary]] is empty.<ref>{{cite book|last=Mendelson|first=Bert|date=1990|orig-year=1975|title=Introduction to Topology|edition=Third|publisher=Dover|isbn=0-486-66352-3|page=87|quote=Let <math>A</math> be a subset of a topological space. Prove that <math>\operatorname{Bdry}(A) = \varnothing</math> if and only if <math>A</math> is open and closed.}} (Given as Exercise 7)</ref>
* Any clopen set is a union of (possibly infinitely many) connected components.
* If all [[Connected component (topology)|connected component]]s of <math>X</math> are open (for instance, if <math>X</math> has only finitely many components, or if <math>X</math> is [[locally connected]]), then a set is clopen in <math>X</math> if and only if it is a union of connected components.
* A topological space <math>X</math> is [[Discrete space|discrete]] if and only if all of its subsets are clopen.
* Using the union and [[Intersection (set theory)|intersection]] as operations, the clopen subsets of a given topological space <math>X</math> form a [[Boolean algebra (structure)|Boolean algebra]]. {{em|Every}} Boolean algebra can be obtained in this way from a suitable topological space: see [[Stone's representation theorem for Boolean algebras]].

==See also==

* {{annotated link|List of set identities and relations}}

== Notes ==

{{reflist|group=note}}
{{reflist}}

== References ==

* {{Munkres Topology|edition=2}} <!-- {{sfn|Munkres|2000|p=}} -->
* {{cite web|last=Morris|first=Sidney A.|title=Topology Without Tears|url=http://uob-community.ballarat.edu.au/~smorris/topology.htm|archive-url=https://web.archive.org/web/20130419134743/http://uob-community.ballarat.edu.au/~smorris/topology.htm|archive-date=19 April 2013}}

[[Category:General topology]]