Editing Derived set (mathematics) (section)

==Cantor–Bendixson rank==

For [[ordinal number]]s <math>\alpha,</math> the <math>\alpha</math>-th '''Cantor–[[Ivar Otto Bendixson|Bendixson]] derivative''' of a topological space is defined by repeatedly applying the derived set operation using [[transfinite recursion]] as follows:
*<math>\displaystyle X^0 = X</math>
*<math>\displaystyle X^{\alpha+1} = \left(X^\alpha\right)'</math>
*<math>\displaystyle X^\lambda = \bigcap_{\alpha < \lambda} X^\alpha</math> for [[limit ordinal]]s <math>\lambda.</math> 
The transfinite sequence of Cantor–Bendixson derivatives of <math>X</math> is [[decreasing]] and must eventually be constant. The smallest ordinal <math>\alpha</math> such that <math>X^{\alpha+1} = X^\alpha</math> is called the '''{{visible anchor|Cantor–Bendixson rank}}''' of <math>X.</math> 

This investigation into the derivation process was one of the motivations for introducing [[ordinal numbers]] by [[Georg Cantor]].