Definition:Cantor-Bendixson Derivative

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Definition

Let $\left({X, \tau}\right)$ be a topological space.

Let $S \subseteq X$.


Then for all ordinals $\beta$, the $\beta$th Cantor-Bendixson derivative of $S$ is defined by Transfinite Recursion thus:

$S^{\left({\beta}\right)} = \begin{cases} S & : \beta = 0 \\ \left({ S^{\left({\alpha}\right)} }\right)' & : \beta = \alpha^+ \\ \displaystyle \bigcap_{\alpha \mathop < \lambda} S^{\left({\alpha}\right)} & : \beta = \lambda \end{cases}$

where:

$\left({ S^{\left({\alpha}\right)} }\right)'$ is the derived set of $S^{\left({\alpha}\right)}$
$\lambda$ is a limit ordinal.


Source of Name

This entry was named for Georg Cantor and Ivar Otto Bendixson.


Sources