Cantor-Bendixson derivative
Let be a subset of a topological space . Its Cantor-Bendixson derivative is defined as the set of accumulation points of . In other words
Through transfinite induction, the Cantor-Bendixson derivative can be defined to any order , where is an arbitrary ordinal. Let . If is a successor ordinal, then . If is a limit ordinal, then . The Cantor-Bendixson rank of the set is the least ordinal such that . Note that implies that is a perfect set.
Some basic properties of the Cantor-Bendixson derivative include
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The last property requires some justification. Obviously, . Suppose , 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 . Therefore and we have .
Finally, from the definition of the Cantor-Bendixson rank and the above properties, if has Cantor-Bendixson rank , the sets
form a strictly decreasing chain of closed sets.
Title | Cantor-Bendixson derivative |
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Canonical name | CantorBendixsonDerivative |
Date of creation | 2013-03-22 15:01:37 |
Last modified on | 2013-03-22 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 | Cantor-Bendixson rank |