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
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
|Date of creation||2013-03-22 15:01:37|
|Last modified on||2013-03-22 15:01:37|
|Last modified by||CWoo (3771)|