Editing Clopen set (section)

== 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>)