closed set


closed under

Let (X,τ) be a topological spaceMathworldPlanetmath. Then a subset CX is closed if its complement XC is open under the topologyMathworldPlanetmath τ.


  • In any topological space (X,τ), the sets X and are always closed.

  • Consider with the standard topology. Then [0,1] is closed since its complement (-,0)(1,) is open (being the union of two open sets).

  • Consider with the lower limit topology. Then [0,1) is closed since its complement (-,0)[1,) is open.

Closed subsets can also be characterized as follows:

A subset CX is closed if and only if C contains all of its cluster pointsPlanetmathPlanetmath, that is, CC.

So the set {1,1/2,1/3,1/4,} is not closed under the standard topology on since 0 is a cluster point not contained in the set.

Title closed set
Canonical name ClosedSet
Date of creation 2013-03-22 12:30:23
Last modified on 2013-03-22 12:30:23
Owner yark (2760)
Last modified by yark (2760)
Numerical id 10
Author yark (2760)
Entry type Definition
Classification msc 54-00
Synonym closed subset
Defines closed