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.

Definition

The derived set of a subset $S$ of a topological space $X,$ denoted by $S',$ is the set of all points $x\in X$ that are limit points of $S,$ that is, points $x$ such that every neighbourhood of $x$ contains a point of $S$ other than $x$ itself.

Examples

If $\mathbb {R}$ is endowed with its usual Euclidean topology then the derived set of the half-open interval $[0,1)$ is the closed interval $[0,1].$

Consider $\mathbb {R}$ with the topology (open sets) consisting of the empty set and any subset of $\mathbb {R}$ that contains 1. The derived set of $A:=\{1\}$ is $A'=\mathbb {R} \setminus \{1\}.$ ^[1]

Properties

Let $X$ denote a topological space in what follows.

If $A$ and $B$ are subsets of $X,$ the derived set has the following properties:^[2]

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

A set $S\subseteq X$ is closed precisely when $S'\subseteq S,$ ^[1] that is, when $S$ contains all its limit points. For any $S\subseteq X,$ the set $S\cup S'$ is closed and is the closure of $S$ (that is, the set ${\overline {S}}$ ).^[3]

Closedness of derived sets

The derived set of a set need not be closed in general. For example, if $X=\{a,b\}$ with the indiscrete topology, the set $S=\{a\}$ has derived set $S'=\{b\},$ which is not closed in $X.$ But the derived set of a closed set is always closed.^{[proof 1]}

For a point $x\in X,$ the derived set of the singleton $\{x\}$ is the set $\{x\}'={\overline {\{x\}}}\setminus \{x\},$ consisting of the points in the closure of $\{x\}$ and different from $x.$ A space $X$ is called a T_D space^[4] if the derived set of every singleton in $X$ is closed; that is, if ${\overline {\{x\}}}\setminus \{x\}$ is closed for every $x\in X;$ in other words, if every point $x$ is isolated in ${\overline {\{x\}}}.$ A space $X$ has the property that $S'$ is closed for all sets $S\subseteq X$ if and only if it is a T_D space.^[5]

Every T_D space is a T₀ space.^[6]

Every T₁ space is a T_D space,^[6] since every singleton is closed, hence $\{x\}'={\overline {\{x\}}}\setminus \{x\}=\varnothing ,$ which is closed. Consequently, in a T₁ space, the derived set of any set is closed.^[7]^[8]

The relation between these properties can be summarized as

T_{1}\implies T_{D}\implies T_{0}.

The implications are not reversible. For example, the Sierpiński space is T_D and not T₁. And the right order topology on $\mathbb {R}$ is T₀ and not T_D.

More properties

Two subsets $S$ and $T$ are separated precisely when they are disjoint and each is disjoint from the other's derived set ${\textstyle S'\cap T=\varnothing =T'\cap S.}$ ^[9]

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.^[10]

In a T₁ space, the derived set of any finite set is empty and furthermore, $(S-\{p\})'=S'=(S\cup \{p\})',$ 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.^[11]

A set $S$ with $S\subseteq S'$ (that is, $S$ contains no isolated points) is called dense-in-itself. A set $S$ with $S=S'$ is called a perfect set.^[12] 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 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 $S\mapsto S^{*}$ mapping subsets of $X$ to subsets of $X,$ such that for any set $S$ and any point $a$ :

$\varnothing ^{*}=\varnothing$
$S^{**}\subseteq S^{*}\cup S$
$a\in S^{*}$ 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 $S^{*}\subseteq S$ will define a topology on the space in which $S\mapsto S^{*}$ is the derived set operator, that is, $S^{*}=S'.$

Cantor–Bendixson rank

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

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

The transfinite sequence of Cantor–Bendixson derivatives of $X$ is decreasing and must eventually be constant. The smallest ordinal $\alpha$ such that $X^{\alpha +1}=X^{\alpha }$ is called the Cantor–Bendixson rank of $X.$

This investigation into the derivation process was one of the motivations for introducing ordinal numbers by Georg Cantor.

Notes

^ ^a ^b Baker 1991, p. 41
^ Pervin 1964, p.38
^ Baker 1991, p. 42
^ Aull, C. E.; Thron, W. J. (1962). "Separation axioms between T0 and T1" (PDF). Nederl. Akad. Wetensch. Proc. Ser. A. 65: 26–37. Zbl 0108.35402.Definition 3.1
^ Aull & Thron 1962, Theorem 5.1.
^ ^a ^b Goubault-Larrecq, Jean. "TD spaces". Non-Hausdorff Topology and Domain Theory.
^ Engelking 1989, p. 47
^ "Proving the derived set E' is closed".
^ Pervin 1964, p. 51
^ Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, p. 4, ISBN 0-486-65676-4
^ Kuratowski 1966, p.77
^ Pervin 1964, p. 62

Proofs

^ Proof: Assuming $S$ is a closed subset of $X,$ which shows that $S'\subseteq S,$ take the derived set on both sides to get $S''\subseteq S';$ that is, $S'$ is closed in $X.$

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, vol. 1, Academic Press, ISBN 0-12-429201-1
Pervin, William J. (1964), Foundations of General Topology, Academic Press

External links

PlanetMath's article on the Cantor–Bendixson derivative

[Baker41-1] Baker 1991, p. 41

[2] Pervin 1964, p.38

[3] Baker 1991, p. 42

[aull-thron-5] Aull, C. E.; Thron, W. J. (1962). "Separation axioms between T0 and T1" (PDF). Nederl. Akad. Wetensch. Proc. Ser. A. 65: 26–37. Zbl 0108.35402.Definition 3.1

[FOOTNOTEAullThron1962Theorem_5.1-6] Aull & Thron 1962, Theorem 5.1.

[jgl-7] Goubault-Larrecq, Jean. "TD spaces". Non-Hausdorff Topology and Domain Theory.

[8] Engelking 1989, p. 47

[9] "Proving the derived set E' is closed".

[10] Pervin 1964, p. 51

[11] Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, p. 4, ISBN 0-486-65676-4

[12] Kuratowski 1966, p.77

[13] Pervin 1964, p. 62

[4] Proof: Assuming $S$ is a closed subset of $X,$ which shows that $S'\subseteq S,$ take the derived set on both sides to get $S''\subseteq S';$ that is, $S'$ is closed in $X.$

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[proof 1]

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