Derived set (mathematics)

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.

Examples

1. Consider ℝ with the usual topology. If A is the half-open interval [0,1) then the derived set A' is the closed interval [0,1].
2. Consider ℝ with the topology (open sets) consisting of the empty set and any subset of ℝ that contains 1. If A = {1}, then A' = ℝ - {1}.

Properties

If A and B are subsets of the topological space $(X,{\mathcal {F}})$, then the derived set has the following properties:

• $\emptyset '=\emptyset$
• $a\in A'\implies a\in (A\setminus \{a\})'$
• $(A\cup B)'=A'\cup B'$
• $A\subseteq B\implies A'\subseteq B'$

A subset S of a topological space is closed precisely when S ' ⊆ S, that is, when S contains all its limit points. For any subset S, the set S ∪ S' is closed and is the closure of S (= S).

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). This condition is often, using closures, written as

$\left(S\cap {\bar {T}}\right)\cup \left({\bar {S}}\cap T\right)=\emptyset ,$

and is known as the Hausdorff-Lennes Separation Condition.

A set S with S = S' is called perfect. Equivalently, a perfect set is a closed set with no isolated points. Perfect sets are particularly important in applications of the Baire category theorem.

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.

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 G_δ subset of a Polish space is again a Polish space, the theorem also shows that any G_δ 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:

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

Calling a set S closed if $S^{*}\subseteq S$ will define a topology on the space in which S ↦ S^* 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

For ordinal numbers α, the α-th Cantor–Bendixson derivative of a topological space is defined by transfinite induction as follows, where $X'$ is the set of all limit points of $X$:

• $\displaystyle X^{0}=X$
• $\displaystyle X^{\alpha +1}=(X^{\alpha })'$
• $\displaystyle X^{\lambda }=\bigcap _{\alpha <\lambda }X^{\alpha }$ for limit ordinals λ.

The transfinite sequence of Cantor–Bendixson derivatives of X must eventually be constant. The smallest ordinal α such that X^α+1 = X^α is called the Cantor–Bendixson rank of X.

References

• Baker, Crump W. (1991), Introduction to Topology, Wm C. Brown Publishers, ISBN 0-697-05972-3
• Pervin, William J. (1964), Foundations of General Topology, Academic Press

Further reading

• Kechris, Alexander S. (1995). Classical Descriptive Set Theory (Graduate Texts in Mathematics 156 ed.). Springer. ISBN 978-0-387-94374-9.
• Sierpiński, Wacław F.; translated by Krieger, C. Cecilia (1952). General Topology. University of Toronto Press.

External links

• PlanetMath's article on the Cantor–Bendixson derivative