Derived set
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.
Derived set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derived_set&oldid=33537