In mathematics, more specifically in

(open sets) consisting of the empty set and any subset of $\backslash mathbb$ that contains 1. The derived set of $A\; :=\; \backslash $ is $A\text{'}\; =\; \backslash mathbb\; \backslash setminus\; \backslash .$
point-set topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...

, the derived set of a subset $S$ of a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

is the set of all limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...

s of $S.$ It is usually denoted by $S\text{'}.$
The concept was first introduced by Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...

in 1872 and he developed set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

in large part to study derived sets on the real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...

.
Examples

If $\backslash mathbb$ is endowed with its usual Euclidean topology then the derived set of the half-open interval $;\; href="/html/ALL/l/,\_1)$_is_the_closed_interval_$[0,1$_ Consider_$\backslash mathbb$_with_the_Topology_(structure).html" "title=",1.html" ;"title=", 1) is the closed interval $[0,1">,\; 1)$ is the closed interval $[0,1$ Consider $\backslash mathbb$ with the Topology (structure)">topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...Properties

If $A$ and $B$ are subsets of the topological space $\backslash left(X,\; \backslash mathcal\backslash right),$ then the derived set has the following properties: * $\backslash varnothing\text{'}\; =\; \backslash varnothing$ * $a\; \backslash in\; A\text{'}\; \backslash implies\; a\; \backslash in\; (A\; \backslash setminus\; \backslash )\text{'}$ * $(A\; \backslash cup\; B)\text{'}\; =\; A\text{'}\; \backslash cup\; B\text{'}$ * $A\; \backslash subseteq\; B\; \backslash implies\; A\text{'}\; \backslash subseteq\; B\text{'}$ A subset $S$ of a topological space is closed precisely when $S\text{'}\; \backslash subseteq\; S,$ that is, when $S$ contains all its limit points. For any subset $S,$ the set $S\; \backslash cup\; S\text{'}$ is closed and is the closure of $S$ (i.e. the set $\backslash overline$). The derived set of a subset of a space $X$ need not be closed in general. For example, if $X\; =\; \backslash $ with thetrivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...

, the set $S\; =\; \backslash $ has derived set $S\text{'}\; =\; \backslash ,$ 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 $S\text{'}\text{'}\; \backslash subseteq\; S\text{'},$ i.e., $S\text{'}$ is closed in $X.$) In addition, if $X$ is a Tbijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

between two topological spaces is a homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

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.
A space is a Tdense-in-itself
In general topology, a subset A of a topological space is said to be dense-in-itself or crowded
if A has no isolated point.
Equivalently, A is dense-in-itself if every point of A is a limit point of A.
Thus A is dense-in-itself if and only if A\su ...

and can contain no isolated point
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...

s. A set $S$ with $S\; =\; S\text{'}$ is called perfect set, perfect. 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 Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...

.
The Cantor–Bendixson theorem In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a p ...

states that any Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named bec ...

can be written as the union of a countable set and a perfect set. Because any Ginduced topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...

.
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 intopology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

. 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 $a$:
# $\backslash varnothing^*\; =\; \backslash varnothing$
# $S^\; \backslash subseteq\; S^*\backslash cup\; S$
# $a\; \backslash in\; S^*$ implies $a\; \backslash in\; (S\; \backslash setminus\; \backslash )^*$
# $(S\; \backslash cup\; T)^*\; \backslash subseteq\; S^*\; \backslash cup\; T^*$
# $S\; \backslash subseteq\; T$ implies $S^*\; \backslash subseteq\; T^*.$
Calling a set $S$ if $S^\; \backslash subseteq\; S$ will define a topology on the space in which $S\; \backslash mapsto\; X^*$ is the derived set operator, that is, $S^\; =\; S\text{'}.$
Cantor–Bendixson rank

Forordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

s $\backslash alpha,$ the $\backslash alpha$-th Cantor–Bendixson derivative of a topological space is defined by repeatedly applying the derived set operation using transfinite recursion
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for a ...

as follows:
*$\backslash displaystyle\; X^0\; =\; X$
*$\backslash displaystyle\; X^\; =\; \backslash left(\; X^\; \backslash right)\text{'}$
*$\backslash displaystyle\; X^\; =\; \backslash bigcap\_\; X^$ for limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...

s $\backslash lambda.$
The transfinite sequence of Cantor–Bendixson derivatives of $X$ must eventually be constant. The smallest ordinal $\backslash alpha$ such that $X^\; =\; X^$ is called the Cantor–Bendixson rank of $X.$
This investigations into the derivation process was one of the motivations for introducing ordinal numbers
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

by Georg Cantor.
See also

* * * *Notes

References

* * * *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
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard s ...

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
The University of Toronto (UToronto or U of T) is a public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution ...

Press.
External links

PlanetMath's article on the Cantor–Bendixson derivative

General topology