Derived set
From Encyclopedia of Mathematics
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.
Comments
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: http://encyclopediaofmath.org/index.php?title=Derived_set&oldid=11226
Derived set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derived_set&oldid=11226
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article