PlanetMath: Cantor-Bendixson derivative

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Cantor-Bendixson derivative

(Definition)

Let be a subset of a topological space. Its Cantor-Bendixson derivative is defined as the set of accumulation points of . In other words

$\displaystyle A' = \{ x\in A \mid x\in \overline{A\setminus \{x\}} \}.$

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

is the least ordinal $\alpha$ such that $A^{(\alpha)} = A^{(\alpha+1)}$ . Note that

implies that

is a perfect set.

Some basic properties of the Cantor-Bendixson derivative include

$(A\cup B)' = A'\cup B'$ ,
$(\bigcup_{i\in I} A_i)' \supseteq \bigcup_{i\in I} A_i'$ ,
$(\bigcap_{i\in I} A_i)' \subseteq \bigcap_{i\in I} A_i'$ ,
$(A\setminus B)' \supseteq A' \setminus B'$ ,
$A\subseteq B \Rightarrow A' \subseteq B'$ ,
$\overline{A} = A \cup A'$ ,
$\overline{A'} = A'$ .

The last property requires some justification. Obviously, $A'\subseteq \overline{A'}$ . Suppose $a\in \overline{A'}$ , then every neighborhood of

contains some points of

distinct from

. But by definition of

, each such neighborhood must also contain some points of

. This implies that

is an accumulation point of

, that is $a\in A'$ . Therefore $\overline{A'}\subseteq A'$ and we have $\overline{A'}=A'$ .

Finally, from the definition of the Cantor-Bendixson rank and the above properties, if has Cantor-Bendixson rank $\alpha$ , the sets

$\displaystyle A^{(1)} \supset A^{(2)} \supset \cdots \supset A^{(\alpha)}$

form a strictly decreasing chain of closed sets.

"Cantor-Bendixson derivative" is owned by CWoo. [ owner history (1) ]

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See Also: derived set

Other names:

set derivative

Also defines:

Cantor-Bendixson rank

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Cross-references: closed sets, chain, strictly decreasing, points, contains, neighborhood, properties, perfect set, implies, limit ordinal, successor ordinal, ordinal, transfinite induction, accumulation points, topological space, subset

This is version 4 of Cantor-Bendixson derivative, born on 2005-02-10, modified 2005-02-10.
Object id is 6736, canonical name is CantorBendixsonDerivative.
Accessed 3403 times total.

Classification:

AMS MSC:	54H05 (General topology :: Connections with other structures, applications :: Descriptive set theory )
	03E15 (Mathematical logic and foundations :: Set theory :: Descriptive set theory)

Pending Errata and Addenda

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