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In mathematics, more specifically in
point-set topology In mathematics, general topology (or point set 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 differ ...
, the derived set of a subset S of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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 contains a point of S other than x itself. A ...
s of S. It is usually denoted by S'. The concept was first introduced by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
in 1872 and he developed
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
in large part to study derived sets on the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
.


Definition

The derived set of a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
S of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
X, denoted by S', is the set of all points x \in X that are
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 contains a point of S other than x itself. A ...
s of S, that is, points x such that every
neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
of x contains a point of S other than x itself.


Examples

If \Reals is endowed with its usual
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot ...
then the derived set of the
half-open interval In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
, 1) is the closed interval [0, 1">closed_interval.html" ;"title=", 1) is the closed interval">, 1) is the closed interval [0, 1 Consider \Reals with the Topology (structure)">topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
(open sets) consisting of the empty set and any subset of \Reals that contains 1. The derived set of A := \ is A' = \Reals \setminus \.


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: * \varnothing' = \varnothing * a \in A' implies a \in (A \setminus \)' * (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, 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).


Closedness of derived sets

The derived set of a set need not be closed in general. For example, if X = \ with the
indiscrete 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 conseque ...
, the set S = \ has derived set S' = \, 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' \subseteq S, take the derived set on both sides to get S'' \subseteq S'; that is, S' is closed in X. For a point x\in X, the derived set of the singleton \ is the set \'=\overline\setminus\, consisting of the points in the closure of \ and different from x. A space X is called a TD spaceDefinition 3.1 if the derived set of every singleton in X is closed; that is, if \overline\setminus\ is closed for every x\in X; in other words, if every point x is isolated in \overline. A space X has the property that S' is closed for all sets S\subseteq X if and only if it is a TD space. Every TD space is a T0 space. Every T1 space is a TD space, since every singleton is closed, hence \'=\overline\setminus\=\varnothing, which is closed. Consequently, in a T1 space, the derived set of any set is closed. 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 In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The ...
is TD and not T1. And the right order topology on \R is T0 and not TD.


More properties

Two subsets S and T are separated precisely when they are disjoint and each is disjoint from the other's derived set S' \cap T = \varnothing = T' \cap S. A
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
between two topological spaces is a
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
if and only if the derived set of the
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
(in the second space) of any subset of the first space is the image of the derived set of that subset. In a T1 space, the derived set of any finite set is empty and furthermore, (S - \)' = S' = (S \cup \)', 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. A set S with S \subseteq S' (that is, S contains no
isolated point In mathematics, a point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton i ...
s) is called
dense-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 ...
. A set S with S = S' is called a
perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all limit points of S, also known as the derived set of S. (Some ...
. 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 th ...
. The
Cantor–Bendixson theorem In the mathematical field of 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 n ...
states that any
Polish space In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...
can be written as the union of a
countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
and a perfect set. Because any Gδ subset of a Polish space is again a Polish space, the theorem also shows that any Gδ subset of a Polish space is the union of a countable set and a set that is perfect with respect to the
induced topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology, o ...
.


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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
. 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 \)^* # (S \cup T)^* \subseteq S^* \cup T^* # S \subseteq T implies S^* \subseteq T^*. Calling a set S 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 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 leas ...
s \alpha, the \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: *\displaystyle X^0 = X *\displaystyle X^ = \left(X^\alpha\right)' *\displaystyle X^\lambda = \bigcap_ X^\alpha 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 a ...
s \lambda. The transfinite sequence of Cantor–Bendixson derivatives of X is decreasing and must eventually be constant. The smallest ordinal \alpha such that X^ = X^\alpha is called the of X. This investigation 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 leas ...
by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
.


See also

* * * *


Notes

Proofs


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) () 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 size (with va ...
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 university, public research university whose main campus is located on the grounds that surround Queen's Park (Toronto), Queen's Park in Toronto, Ontario, Canada. It was founded by ...
Press.


External links


PlanetMath's article on the Cantor–Bendixson derivative
General topology