Derived set (mathematics)

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In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the set of all limit points of S. It is usually denoted by S'.

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.

Properties[edit]

A subset S of a topological space is closed precisely when S' \subseteq S, when S contains all its limit points. Two subsets S and T are separated precisely when they are disjoint and each is disjoint from the other's derived set (though the derived sets don't need to be disjoint from each other).

The set S is defined to be a perfect set if S = S'. Equivalently, a perfect set is a closed set with no isolated points. Perfect sets are particularly important in applications of the Baire category theorem.

Two topological spaces are homeomorphic if and only if there is a bijection from one to the other such that the derived set of the image of any subset is the image of the derived set of that subset.

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[edit]

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 * mapping subsets of X to subsets of X, such that for any set S and any point a:

  1. \empty^* = \empty
  2. S^{**} \subseteq S^*
  3. a \in S^* \implies a \in (S \setminus \{a\})^*
  4. (S \cup T)^* \subseteq S^* \cup T^*
  5. S \subseteq T \implies S^* \subseteq T^*

Note that given 5, 3 is equivalent to 3' below, and that 4 and 5 together are equivalent to 4' below, so we have the following equivalent axioms:

  1. \empty^* = \empty
  2. S^{**} \subseteq S^*
  • 3'.   S^* = (S \setminus \{a\})^*
  • 4'.    \, (S \cup T)^* = S^* \cup T^*

Calling a set S closed if S^* \subseteq S will define a topology on the space in which * is the derived set operator, that is, S^* = S' \,\!. If we also require that the derived set of a set consisting of a single element be empty, the resulting space will be a T1 space. In fact, 2 and 3' can fail in a space that is not T1.

Cantor–Bendixson rank[edit]

For ordinal numbers α, the α-th Cantor–Bendixson derivative of a topological space is defined by transfinite induction as follows:

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.

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