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

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].
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}.[1]

Properties

If A and B are subsets of the topological space \( (X,{\mathcal {F}}) \), then the derived set has the following properties:[2]

\( {\displaystyle \emptyset '=\emptyset } \)
\( {\displaystyle a\in A'\implies a\in (A\setminus \{a\})'} \)
\( ( A ∪ B ) ′ = A ′ ∪ B ′ {\displaystyle (A\cup B)'=A'\cup B'} {\displaystyle (A\cup B)'=A'\cup B'} \)
\( A ⊆ B ⟹ A ′ ⊆ B ′ {\displaystyle A\subseteq B\implies A'\subseteq B'} {\displaystyle A\subseteq B\implies A'\subseteq B'} \)

A subset S of a topological space is closed precisely when S ' ⊆ S,[1] 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).[3]

The derived set of a subset of a space X need not be closed in general. For example, if \( {\displaystyle X=\{a,b\}} \) with the trivial topology, the set \( {\displaystyle S=\{a\}} \) has derived set \( {\displaystyle S'=\{b\}} \), which is not closed in X. But the derived set of a closed set is always closed. (Proof: Assuming S is a closed subset of X, i.e., \( S'\subseteq S \), take the derived set on both sides to get \( {\displaystyle S''\subseteq S'} \), i.e., S' is closed in X.) In addition, if X is a T1 space, the derived set of every subset of X is closed in X. [4][5]

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

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

and is known as the Hausdorff-Lennes Separation Condition.[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,

\( {\displaystyle (S-\{p\})'=S'=(S\cup \{p\})',} \)

for any subset S and any point p 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.[9] It can also be shown that in a T1 space, (S ')' ⊆ S' for any subset S.[10]

A set S with S ⊆ S ' is called dense-in-itself and can contain no isolated points. A set S with S = S' is called perfect.[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.
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:

\( \emptyset ^{*}=\emptyset \)
\( {\displaystyle S^{**}\subseteq S^{*}\cup S} \)
\( a\in S^{*}\implies a\in (S\setminus \{a\})^{*} \)
\( (S\cup T)^{*}\subseteq S^{*}\cup T^{*} \)
\( S\subseteq T\implies S^{*}\subseteq T^{*} \)

Calling a set S closed if S* ⊆ S will define a topology on the space in which S ↦ S* is the derived set operator, that is, S* = S'.
Cantor–Bendixson rank

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

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

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

Adherent point
Condensation point
Isolated point
Limit point – Point to which functions converge in topology

Notes

Baker 1991, p. 41
Pervin 1964, p.38
Baker 1991, p. 42
Engelking 1989, p. 47
https://math.stackexchange.com/a/940849/52912
Pervin 1964, p. 51
Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, p. 4, ISBN 0-486-65676-4
Pervin 1964, p. 70
Kuratowski 1966, p.77
Kuratowski 1966, p.76

Pervin 1964, p. 62

References

Baker, Crump W. (1991), Introduction to Topology, Wm C. Brown Publishers, ISBN 0-697-05972-3
Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
Kuratowski, K. (1966), Topology, 1, Academic Press, ISBN 0-12-429201-1
Pervin, William J. (1964), Foundations of General Topology, Academic Press