Editing Derived set (mathematics)

{{Short description|Set of all limit points of a set}}
In mathematics, more specifically in [[point-set topology]], the '''derived set''' of a subset <math>S</math> of a [[topological space]] is the set of all [[limit point]]s of <math>S.</math> It is usually denoted by <math>S'.</math> 

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 [[Set (mathematics)|subset]] <math>S</math> of a [[topological space]] <math>X,</math> denoted by <math>S',</math> is the set of all points <math>x \in X</math> that are [[limit point]]s of <math>S,</math> that is, points <math>x</math> such that every [[neighbourhood (mathematics)|neighbourhood]] of <math>x</math> contains a point of <math>S</math> other than <math>x</math> itself.

==Examples==

If <math>\Reals</math> is endowed with its usual [[Euclidean topology]] then the derived set of the [[half-open interval]] <math>[0, 1)</math> is the [[closed interval]] <math>[0, 1].</math> 

Consider <math>\Reals</math> with the [[Topology (structure)|topology]] (open sets) consisting of the [[empty set]] and any subset of <math>\Reals</math> that contains 1. The derived set of <math>A := \{1\}</math> is <math>A' = \Reals \setminus \{1\}.</math><ref name=Baker41>{{harvnb|Baker|1991|loc = p. 41}}</ref> 

==Properties==

Let <math>X</math> denote a topological space in what follows.

If <math>A</math> and <math>B</math> are subsets of <math>X,</math> the derived set has the following properties:<ref>{{harvnb|Pervin|1964|loc=p.38}}</ref>
* <math>\varnothing' = \varnothing</math>
* <math>a \in A'</math> implies <math>a \in (A \setminus \{a\})'</math>
* <math>(A \cup B)' = A' \cup B'</math>
* <math>A \subseteq B</math> implies <math>A' \subseteq B'</math>

A set <math>S\subseteq X</math> is [[closed set|closed]] precisely when <math>S' \subseteq S,</math><ref name=Baker41 /> that is, when <math>S</math> contains all its limit points. For any <math>S\subseteq X,</math> the set <math>S \cup S'</math> is closed and is the [[Closure (topology)|closure]] of <math>S</math> (that is, the set <math>\overline{S}</math>).<ref>{{harvnb|Baker|1991|loc=p. 42}}</ref>

===Closedness of derived sets===

The derived set of a set need not be closed in general.  For example, if <math>X = \{a, b\}</math> with the [[indiscrete topology]], the set <math>S = \{a\}</math> has derived set <math>S' = \{b\},</math> which is not closed in <math>X.</math>  But the derived set of a closed set is always closed.<ref group=proof>''Proof:'' Assuming <math>S</math> is a closed subset of <math>X,</math> which shows that <math>S' \subseteq S,</math> take the derived set on both sides to get <math>S'' \subseteq S';</math> that is, <math>S'</math> is closed in <math>X.</math></ref>

For a point <math>x\in X,</math> the derived set of the singleton <math>\{x\}</math> is the set <math>\{x\}'=\overline{\{x\}}\setminus\{x\},</math> consisting of the points in the closure of <math>\{x\}</math> and different from <math>x.</math>
A space <math>X</math> is called a '''T<sub>D</sub> space'''<ref name="aull-thron">{{cite journal |last1=Aull |first1=C. E. |last2=Thron |first2=W. J. |title=Separation axioms between T0 and T1 |journal=Nederl. Akad. Wetensch. Proc. Ser. A |date=1962 |volume=65 |pages=26–37 |doi=10.1016/S1385-7258(62)50003-6 |url=https://core.ac.uk/download/pdf/82702431.pdf |zbl=0108.35402}}Definition 3.1</ref> if the derived set of every singleton in <math>X</math> is closed; that is, if <math>\overline{\{x\}}\setminus\{x\}</math> is closed for every <math>x\in X;</math> in other words, if every point <math>x</math> is isolated in <math>\overline{\{x\}}.</math>  A space <math>X</math> has the property that <math>S'</math> is closed for all sets <math>S\subseteq X</math> if and only if it is a T<sub>D</sub> space.{{sfn|Aull|Thron|1962|loc=Theorem 5.1}}

Every T<sub>D</sub> space is a [[T0 space|T<sub>0</sub> space]].<ref name="jgl">{{cite web|last1=Goubault-Larrecq |first1=Jean |title=TD spaces |url=https://projects.lsv.ens-cachan.fr/topology/?page_id=2626 |website=Non-Hausdorff Topology and Domain Theory}}</ref>

Every [[T1 space|T<sub>1</sub> space]] is a T<sub>D</sub> space,<ref name="jgl"/> since every singleton is closed, hence
<math>\{x\}'=\overline{\{x\}}\setminus\{x\}=\varnothing,</math> which is closed.
Consequently, in a T<sub>1</sub> space, the derived set of any set is closed.<ref>{{harvnb|Engelking|1989|loc=p. 47}}</ref><ref>{{Cite web|url=https://math.stackexchange.com/a/940849/52912|title=Proving the derived set E' is closed}}</ref>

The relation between these properties can be summarized as
:<math>T_1\implies T_D\implies T_0.</math>

The implications are not reversible.  For example, the [[Sierpiński space]] is T<sub>D</sub> and not T<sub>1</sub>.
And the [[right order topology]] on <math>\R</math> is T<sub>0</sub> and not T<sub>D</sub>.

===More properties===

Two subsets <math>S</math> and <math>T</math> are [[Separated sets|separated]] precisely when they are [[Disjoint sets|disjoint]] and each is disjoint from the other's derived set <math display=inline>S' \cap T = \varnothing = T' \cap S.</math><ref>{{harvnb|Pervin|1964|loc=p. 51}}</ref>

A [[bijection]] between two topological spaces is a [[homeomorphism]] if and only if the derived set of the [[image (mathematics)|image]] (in the second space) of any subset of the first space is the image of the derived set of that subset.<ref>{{citation|first1=John G.|last1=Hocking|first2=Gail S.|last2=Young|title=Topology|year=1988|orig-date=1961|publisher=Dover|isbn=0-486-65676-4|page=[https://archive.org/details/topology00hock_0/page/4 4]|url=https://archive.org/details/topology00hock_0/page/4}}</ref>

In a T<sub>1</sub> space, the derived set of any finite set is empty and furthermore,
<math display=block>(S - \{p\})' = S' = (S \cup \{p\})',</math>
for any subset <math>S</math> and any point <math>p</math> 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.<ref>{{harvnb|Kuratowski|1966|loc=p.77}}</ref>

A set <math>S</math> with <math>S \subseteq S'</math> (that is, <math>S</math> contains no [[isolated point]]s) is called [[dense-in-itself]]. A set <math>S</math> with <math>S = S'</math> is called a [[perfect set]].<ref>{{harvnb|Pervin|1964|loc=p. 62}}</ref>  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-delta set|G<sub>δ</sub>]] subset of a Polish space is again a Polish space, the theorem also shows that any G<sub>δ</sub> 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 <math>X</math> can be equipped with an operator <math>S \mapsto S^*</math> mapping subsets of <math>X</math> to subsets of <math>X,</math> such that for any set <math>S</math> and any point <math>a</math>:

# <math>\varnothing^* = \varnothing</math>
# <math>S^{**} \subseteq S^*\cup S</math>
# <math>a \in S^*</math> implies <math>a \in (S \setminus \{a\})^*</math>
# <math>(S \cup T)^* \subseteq S^* \cup T^*</math>
# <math>S \subseteq T</math> implies <math>S^* \subseteq T^*.</math>
<!--
The following is wrong, see discussion page, section "S** subset S*"
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:

# <math>\varnothing^* = \varnothing</math>
# <math>S^{**} \subseteq S^*</math>
*3'. &nbsp; <math>S^* = (S \setminus \{a\})^*</math>
*4'. &nbsp; <math> \, (S \cup T)^* = S^* \cup T^*</math>
-->

Calling a set <math>S</math> {{em|closed}} if <math>S^* \subseteq S</math> will define a topology on the space in which <math>S \mapsto S^*</math> is the derived set operator, that is, <math>S^* = S'.</math>

==Cantor–Bendixson rank==

For [[ordinal number]]s <math>\alpha,</math> the <math>\alpha</math>-th '''Cantor–[[Ivar Otto Bendixson|Bendixson]] derivative''' of a topological space is defined by repeatedly applying the derived set operation using [[transfinite recursion]] as follows:
*<math>\displaystyle X^0 = X</math>
*<math>\displaystyle X^{\alpha+1} = \left(X^\alpha\right)'</math>
*<math>\displaystyle X^\lambda = \bigcap_{\alpha < \lambda} X^\alpha</math> for [[limit ordinal]]s <math>\lambda.</math> 
The transfinite sequence of Cantor–Bendixson derivatives of <math>X</math> is [[decreasing]] and must eventually be constant. The smallest ordinal <math>\alpha</math> such that <math>X^{\alpha+1} = X^\alpha</math> is called the '''{{visible anchor|Cantor–Bendixson rank}}''' of <math>X.</math> 

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

==See also==

* {{annotated link|Adherent point}}
* {{annotated link|Condensation point}}
* {{annotated link|Isolated point}}
* {{annotated link|Limit point}}

==Notes==

{{reflist}}

'''Proofs'''

{{reflist|group=proof}}

==References==

* {{citation|first=Crump W.|last=Baker|title=Introduction to Topology|year=1991|publisher=Wm C. Brown Publishers|isbn=0-697-05972-3}}
* {{cite book|last=Engelking|first=Ryszard| authorlink=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989| isbn=3-88538-006-4}}
* {{citation|first=K.|last=Kuratowski|authorlink = Kazimierz Kuratowski|title=Topology|volume=1|year=1966|publisher=Academic Press|isbn=0-12-429201-1}}
* {{citation|first=William J.|last=Pervin|title=Foundations of General Topology|year=1964|publisher=Academic Press}}

==Further reading==

* {{cite book|author = Kechris, Alexander S. |authorlink = Alexander Kechris| title = Classical Descriptive Set Theory |url = https://archive.org/details/classicaldescrip0000kech |url-access = registration | edition = [[Graduate Texts in Mathematics]] 156 | publisher = Springer | year = 1995 | isbn =978-0-387-94374-9}}
* [[Wacław Sierpiński|Sierpiński, Wacław F.]]; translated by [[Cecilia Krieger|Krieger, C. Cecilia]] (1952). ''General Topology''. [[University of Toronto]] Press.

==External links==
* [http://planetmath.org/cantorbendixsonderivative PlanetMath's article on the Cantor–Bendixson derivative]

[[Category:General topology]]