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In mathematics, more specifically in
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
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
It is usually denoted by
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
is endowed with its
usual Euclidean topology then the derived set of the half-open interval
_is_the_closed_interval_
_
Consider_
_with_the_Topology_(structure).html" "title=",1.html" ;"title=", 1) is the closed interval
is the closed interval
Consider
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 ...
(open sets) consisting of the empty set and any subset of
that contains 1. The derived set of
is
Properties
If
and
are subsets of the topological space
then the derived set has the following properties:
*
*
*
*
A subset
of a topological space is
closed precisely when
that is, when
contains all its limit points. For any subset
the set
is closed and is the
closure of
(i.e. the set
).
The derived set of a subset of a space
need not be closed in general. For example, if
with the
trivial 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
has derived set
which is not closed in
But the derived set of a closed set is always closed. (''Proof:'' Assuming
is a closed subset of
which shows that
take the derived set on both sides to get
i.e.,
is closed in
) In addition, if
is a
T1 space, the derived set of every subset of
is closed in
Two subsets
and
are
separated precisely when they are
disjoint and each is disjoint from the other's derived set (though the derived sets don't need to be disjoint from each other). This condition is often, using closures, written as
:
and is known as the .
A
bijection
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
T1 space if every subset consisting of a single point is closed. In a T
1 space, the derived set of a set consisting of a single element is empty (Example 2 above is not a T
1 space). It follows that in T
1 spaces, the derived set of any finite set is empty and furthermore,
:
for any subset
and any point
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. It can also be shown that in a T
1 space,
for any subset
A set
with
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 ...
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
with
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
Gδ 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
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 in
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 ...
. A set of points
can be equipped with an operator
mapping subsets of
to subsets of
such that for any set
and any point
:
#
#
#
implies
#
#
implies
Calling a set
if
will define a topology on the space in which
is the derived set operator, that is,
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 least n ...
s
the
-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:
*
*
*
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
The transfinite sequence of Cantor–Bendixson derivatives of
must eventually be constant. The smallest ordinal
such that
is called the Cantor–Bendixson rank of
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