# Definition:Cantor-Bendixson Rank

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## Definition

Let $\struct {X, \tau}$ be a topological space.

Let $S \subseteq X$.

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

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

- $S^{\paren {\alpha^+} } = S^{\paren \alpha}$

if such an ordinal exists.

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## Source of Name

This entry was named for Georg Cantor and Ivar Otto Bendixson.

## Sources

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