CantorBendixson derivative
Let $A$ be a subset of a topological space^{} $X$. Its CantorBendixson 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 CantorBendixson 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 $$. The CantorBendixson 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 CantorBendixson derivative include

1.
${(A\cup B)}^{\prime}={A}^{\prime}\cup {B}^{\prime}$,

2.
${({\bigcup}_{i\in I}{A}_{i})}^{\prime}\supseteq {\bigcup}_{i\in I}{A}_{i}^{\prime}$,

3.
${({\bigcap}_{i\in I}{A}_{i})}^{\prime}\subseteq {\bigcap}_{i\in I}{A}_{i}^{\prime}$,

4.
${(A\setminus B)}^{\prime}\supseteq {A}^{\prime}\setminus {B}^{\prime}$,

5.
$A\subseteq B\Rightarrow {A}^{\prime}\subseteq {B}^{\prime}$,

6.
$\overline{A}=A\cup {A}^{\prime}$,

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 CantorBendixson rank and the above properties, if $A$ has CantorBendixson rank $\alpha $, the sets
$${A}^{(1)}\supset {A}^{(2)}\supset \mathrm{\cdots}\supset {A}^{(\alpha )}$$ 
form a strictly decreasing chain of closed sets^{}.
Title  CantorBendixson derivative 

Canonical name  CantorBendixsonDerivative 
Date of creation  20130322 15:01:37 
Last modified on  20130322 15:01:37 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54H05 
Classification  msc 03E15 
Synonym  set derivative 
Related topic  DerivedSet 
Defines  CantorBendixson rank 