Definition:Cantor-Bendixson Derivative

Definition

Let $\struct {X, \tau}$ be a topological space.

Let $S \subseteq X$.

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

$S^{\paren \beta} = \begin {cases} S & : \beta = 0 \\

\paren {S^{\paren \alpha} }' & : \beta = \alpha^+ \\ \ds \bigcap_{\alpha \mathop < \lambda} S^{\paren \alpha} & : \beta = \lambda \end{cases}$

where:

$\paren {S^{\paren \alpha} }'$ is the derived set of $S^{\paren \alpha}$

$\lambda$ is a limit ordinal.