clopen subset

A subset of a topological spaceMathworldPlanetmath X is called clopen if it is both open and closed.

Theorem 1.

The clopen subsets form a Boolean algebraMathworldPlanetmath under the operationMathworldPlanetmath of union, intersectionMathworldPlanetmath and complement. In other words:

  • X and are clopen,

  • the complement of a clopen set is clopen,

  • finite unions and intersections of clopen sets are clopen.


The first follows by the definition of a topologyMathworldPlanetmath, the second by noting that complements of open sets are closed, and vice versa, and the third by noting that this property holds for both open and closed setsPlanetmathPlanetmath. ∎

One application of clopen sets is that they can be used to describe connectness. In particular, a topological space is connectedPlanetmathPlanetmath if and only if its only clopen subsets are itself and the empty setMathworldPlanetmath.

If a space has finitely many connected componentsMathworldPlanetmathPlanetmath then each connected component is clopen. This may not be the case if there are infinitely many components, as the case of the rational numbersPlanetmathPlanetmathPlanetmath demonstrates.

Title clopen subset
Canonical name ClopenSubset
Date of creation 2013-03-22 13:25:29
Last modified on 2013-03-22 13:25:29
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Definition
Classification msc 54D05
Synonym clopen set
Synonym clopen
Synonym closed and open
Related topic IdentityTheorem