# Definition:Cantor-Bendixson Rank

## 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.