Derived set (mathematics)

In mathematics, more specifically in

, the derived set of a subset ${\displaystyle S}$ of a of ${\displaystyle S.}$ It is usually denoted by ${\displaystyle S'.}$

The concept was first introduced by

.

Definition

The derived set of a subset ${\displaystyle S}$ of a topological space ${\displaystyle X,}$ denoted by ${\displaystyle S',}$ is the set of all points ${\displaystyle x\in X}$ that are

of ${\displaystyle S,}$ that is, points ${\displaystyle x}$ such that every neighbourhood of ${\displaystyle x}$ contains a point of ${\displaystyle S}$ other than ${\displaystyle x}$ itself.

Examples

If ${\displaystyle \mathbb {R} }$ is endowed with its usual

${\displaystyle [0,1)}$ is the ${\displaystyle [0,1].}$

Consider ${\displaystyle \mathbb {R} }$ with the

and any subset of ${\displaystyle \mathbb {R} }$ that contains 1. The derived set of ${\displaystyle A:=\{1\}}$ is ${\displaystyle A'=\mathbb {R} \setminus \{1\}.}$[1]

Properties

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

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

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

The derived set of a subset of a space ${\displaystyle X}$ 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 ${\displaystyle X.}$ But the derived set of a closed set is always closed.^{[proof 1]} In addition, if ${\displaystyle X}$ is a T₁ space, the derived set of every subset of ${\displaystyle X}$ is closed in ${\displaystyle X.}$^[4][5]

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

for any subset ${\displaystyle S}$ and any point ${\displaystyle p}$ 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, ${\displaystyle \left(S'\right)'\subseteq S'}$ for any subset ${\displaystyle S.}$^[10]

A set ${\displaystyle S}$ with ${\displaystyle S\subseteq S'}$ (that is, ${\displaystyle S}$ contains no isolated points) is called dense-in-itself. A set ${\displaystyle S}$ with ${\displaystyle S=S'}$ is called a perfect set.^[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

.

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 ${\displaystyle X}$ can be equipped with an operator ${\displaystyle S\mapsto S^{*}}$ mapping subsets of ${\displaystyle X}$ to subsets of ${\displaystyle X,}$ such that for any set ${\displaystyle S}$ and any point ${\displaystyle a}$:

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

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

Cantor–Bendixson rank

For ordinal numbers ${\displaystyle \alpha ,}$ the ${\displaystyle \alpha }$-th Cantor–

as follows:

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

The transfinite sequence of Cantor–Bendixson derivatives of ${\displaystyle X}$ is

decreasing

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

This investigation into the derivation process was one of the motivations for introducing

ordinal numbers by Georg Cantor

.

See also

• Adherent point – Point that belongs to the closure of some given subset of a topological space
• Condensation point – a stronger analog of limit pointPages displaying wikidata descriptions as a fallback
• Isolated point – Point of a subset S around which there are no other points of S
• Limit point
   – Cluster point in a topological spacePages displaying short descriptions of redirect targets

Notes

1. ^ ^{a b} Baker 1991, p. 41
2. ^ Pervin 1964, p.38
3. ^ Baker 1991, p. 42
4. ^ Engelking 1989, p. 47
5. ^ "General topology - Proving the derived set $E'$ is closed".
6. ^ Pervin 1964, p. 51
7. ISBN 0-486-65676-4
8. ^ Pervin 1964, p. 70
9. ^ Kuratowski 1966, p.77
10. ^ Kuratowski 1966, p.76
11. ^ Pervin 1964, p. 62

Proofs

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

References

• Baker, Crump W. (1991), Introduction to Topology, Wm C. Brown Publishers,
  ISBN 0-697-05972-3
• ISBN 3-88538-006-4
  .
• ISBN 0-12-429201-1
• Pervin, William J. (1964), Foundations of General Topology, Academic Press

Further reading

• 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