derived set (mathematics)

The derived set of a subset $S$ of a topological space $X,$ denoted by $S\text{'},$ is the set of all points $x\; \backslash in\; X$ 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.

If $\backslash Reals$ 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 $\backslash Reals$ with the topology (open sets) consisting of the empty set and any subset of $\backslash Reals$ that contains 1. The derived set of $A\; :=\; \backslash \{1\backslash \}$ is $A\text{'}\; =\; \backslash Reals\; \backslash setminus\; \backslash \{1\backslash \}.${{harvnb|Baker|1991|loc = p. 41}}

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

If $A$ and $B$ are subsets of $X,$ the derived set has the following properties:{{harvnb|Pervin|1964|loc=p.38}}

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

A set $S\backslash subseteq\; X$ is closed precisely when $S\text{'}\; \backslash subseteq\; S,$ that is, when $S$ contains all its limit points. For any $S\backslash subseteq\; X,$ the set $S\; \backslash cup\; S\text{'}$ is closed and is the closure of $S$ (that is, the set $\backslash overline\{S\}$).{{harvnb|Baker|1991|loc=p. 42}}

The derived set of a set need not be closed in general. For example, if $X\; =\; \backslash \{a,\; b\backslash \}$ with the indiscrete topology, the set $S\; =\; \backslash \{a\backslash \}$ has derived set $S\text{'}\; =\; \backslash \{b\backslash \},$ 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 $X,$ which shows that $S\text{'}\; \backslash subseteq\; S,$ take the derived set on both sides to get $S\text{'}\text{'}\; \backslash subseteq\; S\text{'};$ that is, $S\text{'}$ is closed in $X.$

For a point $x\backslash in\; X,$ the derived set of the singleton $\backslash \{x\backslash \}$ is the set $\backslash \{x\backslash \}\text{'}=\backslash overline\{\backslash \{x\backslash \}\}\backslash setminus\backslash \{x\backslash \},$ consisting of the points in the closure of $\backslash \{x\backslash \}$ and different from $x.$

A space $X$ is called a T_D space{{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 if the derived set of every singleton in $X$ is closed; that is, if $\backslash overline\{\backslash \{x\backslash \}\}\backslash setminus\backslash \{x\backslash \}$ is closed for every $x\backslash in\; X;$ in other words, if every point $x$ is isolated in $\backslash overline\{\backslash \{x\backslash \}\}.$ A space $X$ has the property that $S\text{'}$ is closed for all sets $S\backslash subseteq\; X$ if and only if it is a T_D space.{{sfn|Aull|Thron|1962|loc=Theorem 5.1}}

Every T_D space is a T₀ space.{{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}}

Every T₁ space is a T_D space, since every singleton is closed, hence

$\backslash \{x\backslash \}\text{'}=\backslash overline\{\backslash \{x\backslash \}\}\backslash setminus\backslash \{x\backslash \}=\backslash varnothing,$ which is closed.

Consequently, in a T₁ space, the derived set of any set is closed.{{harvnb|Engelking|1989|loc=p. 47}}{{Cite web|url=https://math.stackexchange.com/a/940849/52912|title=Proving the derived set E' is closed}}

The relation between these properties can be summarized as

:$T\_1\backslash implies\; T\_D\backslash implies\; T\_0.$

The implications are not reversible. For example, the Sierpiński space is T_D and not T₁.

And the right order topology on $\backslash R$ is T₀ and not T_D.

Two subsets $S$ and $T$ are separated precisely when they are disjoint and each is disjoint from the other's derived set $S\text{'}\; \backslash cap\; T\; =\; \backslash varnothing\; =\; T\text{'}\; \backslash cap\; S.${{harvnb|Pervin|1964|loc=p. 51}}

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.{{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}}

In a T₁ space, the derived set of any finite set is empty and furthermore,

$$(S\; -\; \backslash \{p\backslash \})\text{'}\; =\; S\text{'}\; =\; (S\; \backslash cup\; \backslash \{p\backslash \})\text{'},$$

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.{{harvnb|Kuratowski|1966|loc=p.77}}

A set $S$ with $S\; \backslash subseteq\; S\text{'}$ (that is, $S$ contains no isolated points) is called dense-in-itself. A set $S$ with $S\; =\; S\text{'}$ is called a perfect set.{{harvnb|Pervin|1964|loc=p. 62}} 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.

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 $S\; \backslash mapsto\; S^*$ mapping subsets of $X$ to subsets of $X,$ such that for any set $S$ and any point $a$:

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

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

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

• $\backslash displaystyle\; X^0\; =\; X$
• $\backslash displaystyle\; X^\{\backslash alpha+1\}\; =\; \backslash left(X^\backslash alpha\backslash right)\text{'}$
• for limit ordinals $\backslash lambda.$

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

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

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

{{reflist}}

Proofs

{{reflist|group=proof}}

• {{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}}

• {{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}}
• Sierpiński, Wacław F.; translated by Krieger, C. Cecilia (1952). General Topology. University of Toronto Press.

• [http://planetmath.org/cantorbendixsonderivative PlanetMath's article on the Cantor–Bendixson derivative]

Category:General topology