\begin{definition}[Definition:Cantor-Bendixson Derivative]
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 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.
{{NamedforDef|Georg Cantor|name2 = Ivar Otto Bendixson|cat = Cantor|cat2 = Bendixson}}
\end{definition}