# Definition:Cantor-Bendixson Rank

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## Definition

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

Let $S \subseteq X$.

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

Then the **Cantor-Bendixson rank** of $S$ is the least ordinal $\alpha$ such that:

- $S^{\left({\alpha^+}\right)} = S^{\left({\alpha}\right)}$

if such an ordinal exists.