Definition:Cantor-Bendixson Rank

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

Let $S \subseteq X$.

For each ordinal $\alpha$, let $S^{(\alpha)}$ be the $\alpha$th Cantor-Bendixson derivative of $S$.

Then the Cantor-Bendixson rank of $S$ is the least ordinal $\alpha$ such that $S^{(\alpha^+)} = S^{(\alpha)}$, if such an ordinal exists.