# Cantor-Bendixson derivative

Let $A$ be a subset of a topological space $X$. Its Cantor-Bendixson derivative $A^{\prime}$ is defined as the set of accumulation points of $A$. In other words

 $A^{\prime}=\{x\in X\mid x\in\overline{A\setminus\{x\}}\}.$

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 $A^{(\alpha)}=\left(A^{(\alpha-1)}\right)^{\prime}$. 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 $A^{\prime}=A$ implies that $A$ is a perfect set.

Some basic properties of the Cantor-Bendixson derivative include

1. 1.

$(A\cup B)^{\prime}=A^{\prime}\cup B^{\prime}$,

2. 2.

$(\bigcup_{i\in I}A_{i})^{\prime}\supseteq\bigcup_{i\in I}A_{i}^{\prime}$,

3. 3.

$(\bigcap_{i\in I}A_{i})^{\prime}\subseteq\bigcap_{i\in I}A_{i}^{\prime}$,

4. 4.

$(A\setminus B)^{\prime}\supseteq A^{\prime}\setminus B^{\prime}$,

5. 5.

$A\subseteq B\Rightarrow A^{\prime}\subseteq B^{\prime}$,

6. 6.

$\overline{A}=A\cup A^{\prime}$,

7. 7.

$\overline{A^{\prime}}=A^{\prime}$.

The last property requires some justification. Obviously, $A^{\prime}\subseteq\overline{A^{\prime}}$. Suppose $a\in\overline{A^{\prime}}$, then every neighborhood of $a$ contains some points of $A^{\prime}$ distinct from $a$. But by definition of $A^{\prime}$, each such neighborhood must also contain some points of $A$. This implies that $a$ is an accumulation point of $A$, that is $a\in A^{\prime}$. Therefore $\overline{A^{\prime}}\subseteq A^{\prime}$ and we have $\overline{A^{\prime}}=A^{\prime}$.

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.

Title Cantor-Bendixson derivative CantorBendixsonDerivative 2013-03-22 15:01:37 2013-03-22 15:01:37 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 54H05 msc 03E15 set derivative DerivedSet Cantor-Bendixson rank