Let $A$ be a subset of a topological space. Its Cantor-Bendixson derivative $A'$ is defined as the set of accumulation points of $A$ . In other words$$ A' = \{ x\in A \mid x\in \overline{A\setminus \{x\}} \}.$$ Through transfinite induction, the Cantor-Bendixson derivative can be defined to any order $\alpha$ , where $\alpha$ is an arbitrary ordinal. Let $A^{(0)} = A$ . If $\alpha$ is a successor ordinal, then $A^{(\alpha)} = \left(A^{(\alpha-1)}\right)'$ . If $\lambda$ is a limit ordinal, then $A^{(\lambda)} = \bigcap_{\alpha<\lambda} A^{(\alpha)}$ . The Cantor-Bendixson rank of the set $A$ is the least ordinal $\alpha$ such that $A^{(\alpha)} = A^{(\alpha+1)}$ . Note that $A' = A$ implies that $A$ is a perfect set.

Some basic properties of the Cantor-Bendixson derivative include

1. $(A\cup B)' = A'\cup B'$ ,
2. $(\bigcup_{i\in I} A_i)' \spse \bigcup_{i\in I} A_i'$ ,
3. $(\bigcap_{i\in I} A_i)' \sse \bigcap_{i\in I} A_i'$ ,
4. $(A\setminus B)' \spse A' \setminus B'$ ,
5. $A\sse B \impl A' \sse B'$ ,
6. $\overline{A} = A \cup A'$ ,
7. $\overline{A'} = A'$ .

Finally, from the definition of the Cantor-Bendixson rank and the above properties, if $A$ has Cantor-Bendixson rank $\alpha$ , the sets$$ A^{(1)} \supset A^{(2)} \supset \cdots \supset A^{(\alpha)}$$ form a strictly decreasing chain of closed sets.