User:Dfeuer/Cantor-Bendixson Derivative

Definition
Let $(X, \tau)$ be a topological space.

Let $S \subseteq X$.

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


 * $S^{(0)} = S$
 * $S^{(\alpha^+)} = \left({ S^{(\alpha)} }\right)'$
 * $\displaystyle S^{(\lambda)} = \bigcap_{\alpha < \lambda} S^{(\alpha)}$ if $\lambda$ is a limit ordinal.