Derived set (mathematics)


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. It is usually denoted by.
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.

Examples

  1. Consider ℝ with the usual topology. If is the half-open interval .
  2. Consider ℝ with the topology consisting of the empty set and any subset of ℝ that contains 1. If =, then = ℝ -.

    Properties

If and are subsets of the topological space, then the derived set has the following properties:
A subset of a topological space is closed precisely when, that is, when contains all its limit points. For any subset, the set is closed and is the closure of .
Two subsets and are separated precisely when they are disjoint and each is disjoint from the other's derived set. This condition is often, using closures, written as
and is known as the Hausdorff-Lennes Separation Condition.
A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image of any subset of the first space is the image of the derived set of that subset.
A space is a T1 space if every subset consisting of a single point is closed. In a T1 space, the derived set of a set consisting of a single element is empty. It follows that in T1 spaces, the derived set of any finite set is empty and furthermore,
for any subset and any point of the space. In other words, the derived set is not changed by adding to or removing from the given set a finite number of points. It can also be shown that in a T1 space, for any subset.
A set with is called dense-in-itself and can contain no isolated points. A set with is called perfect. 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.

Topology in terms of derived sets

Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology. A set of points X can be equipped with an operator S ↦ S* mapping subsets of X to subsets of X, such that for any set S and any point a:
Calling a set closed if will define a topology on the space in which is the derived set operator, that is,.

Cantor–Bendixson rank

For ordinal numbers α, the α-th Cantor–Bendixson derivative of a topological space is defined by transfinite induction as follows, where is the set of all limit points of :
The transfinite sequence of Cantor–Bendixson derivatives of X must eventually be constant. The smallest ordinal α such that Xα+1 = Xα is called the Cantor–Bendixson rank of X.