Derived set

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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.

How to Cite This Entry:
Derived set. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article