In mathematics, more specifically in point-set topology, the derived set of a subset
of a topological space is the set of all limit points of
The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.
The derived set of a subset
is endowed with its usual Euclidean topology then the derived set of the half-open interval
with the topology (open sets) consisting of the empty set and any subset of
are subsets of the topological space
then the derived set has the following properties:[2] A subset
of a topological space is closed precisely when
[3] The derived set of a subset of a space
with the trivial topology, the set
is a T1 space, the derived set of every subset of
are separated precisely when they are disjoint and each is disjoint from the other's derived set
[6] A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset.
[7] A space is a T1 space if every subset consisting of a single point is closed.
[8] In a T1 space, the derived set of a set consisting of a single element is empty (Example 2 above is not a T1 space).
It follows that in T1 spaces, the derived set of any finite set is empty and furthermore,
In other words, the derived set is not changed by adding to or removing from the given set a finite number of points.
contains no isolated points) is called dense-in-itself.
is called a perfect set.
[11] Equivalently, a perfect set is a closed dense-in-itself set, or, put another way, a closed set with no isolated points.
Perfect sets are particularly important in applications of the Baire category theorem.
The Cantor–Bendixson theorem states that any Polish space can be written as the union of a countable set and a perfect set.
Because any Gδ subset of a Polish space is again a Polish space, the theorem also shows that any Gδ subset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology.
Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology.
will define a topology on the space in which
is the derived set operator, that is,
-th Cantor–Bendixson derivative of a topological space is defined by repeatedly applying the derived set operation using transfinite recursion as follows: The transfinite sequence of Cantor–Bendixson derivatives of
is decreasing and must eventually be constant.
This investigation into the derivation process was one of the motivations for introducing ordinal numbers by Georg Cantor.