Cantor-Bendixson derivative

Let A be a subset of a topological spaceMathworldPlanetmath X. Its Cantor-Bendixson derivative A is defined as the set of accumulation pointsMathworldPlanetmathPlanetmath of A. In other words


Through transfinite inductionMathworldPlanetmath, the Cantor-Bendixson derivative can be defined to any order α, where α is an arbitrary ordinalMathworldPlanetmathPlanetmath. Let A(0)=A. If α is a successor ordinal, then A(α)=(A(α-1)). If λ is a limit ordinal, then A(λ)=α<λA(α). The Cantor-Bendixson rank of the set A is the least ordinal α such that A(α)=A(α+1). Note that A=A implies that A is a perfect setMathworldPlanetmath.

Some basic properties of the Cantor-Bendixson derivative include

  1. 1.


  2. 2.


  3. 3.


  4. 4.


  5. 5.


  6. 6.


  7. 7.


The last property requires some justification. Obviously, AA¯. Suppose aA¯, then every neighborhoodMathworldPlanetmathPlanetmath of a contains some points of A distinct from a. But by definition of A, each such neighborhood must also contain some points of A. This implies that a is an accumulation point of A, that is aA. Therefore A¯A and we have A¯=A.

Finally, from the definition of the Cantor-Bendixson rank and the above properties, if A has Cantor-Bendixson rank α, the sets


form a strictly decreasing chain of closed setsPlanetmathPlanetmath.

Title Cantor-Bendixson derivative
Canonical name CantorBendixsonDerivative
Date of creation 2013-03-22 15:01:37
Last modified on 2013-03-22 15:01:37
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 54H05
Classification msc 03E15
Synonym set derivative
Related topic DerivedSet
Defines Cantor-Bendixson rank