The collection of all limit points of a set in a topological space (cf. Limit point of a set). A set that coincides with its derived set is called perfect.
This process can be iterated.
In general one defines, for an ordinal number , the -th derived set of , , as follows: , is the derived set of , and if is a limit ordinal then .
One then shows that there is a first ordinal number such that . If , then is called scattered; if , then is called the perfect kernel of .
In this way one can prove the Cantor–Bendixson theorem: If is a subspace of the real line, then , with a countable set, a perfect set and .
For this reason is sometimes called the Cantor–Bendixson height of . Perfect spaces are sometimes called dense-in-itself.
Derived set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derived_set&oldid=11226