Editing Clopen set (section)

== 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]].