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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 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 po ...
is the set of all limit points of S. It is usually denoted by S'. 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 In elementary mathematics, a number line is a picture of a graduated straight 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 number to a poin ...
.


Examples

If \mathbb is endowed with its usual Euclidean topology then the derived set of the half-open interval ,_1)_is_the_closed_interval_[0,1_ Consider_\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 \mathbb with the Topology (structure)">topology (open sets) consisting of the empty set and any subset of \mathbb that contains 1. The derived set of A := \ is A' = \mathbb \setminus \.


Properties

If A and B are subsets of the topological space \left(X, \mathcal\right), then 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 subset S of a topological space is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
precisely when S' \subseteq S, that is, when S contains all its limit points. For any subset S, the set S \cup S' is closed and is the closure of S (i.e. the set \overline). The derived set of a subset of a space X need not be closed in general. For example, if X = \ with the trivial topology, 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', i.e., S' is closed in X.) In addition, if X is a T1 space, the derived set of every subset of X is closed in X. Two subsets S and T 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 :\left( S \cap \bar \right) \cup \left( \bar \cap T \right) = \varnothing, 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 ...
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 isomorph ...
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 T1 space, the derived set of a set consisting of a single element is empty (Example 2 above is not a T1 space). It follows that in T1 spaces, the derived set of any finite set is empty and furthermore, :\left( S - \ \right)' = S' = \left( S \cup \ \right)', 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. It can also be shown that in a T1 space, \left( S' \right)' \subseteq S' for any subset S. A set S with S \subseteq 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\ ...
and can contain no isolated points. A set S with S = S' is called 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 per ...
states that any Polish space 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 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.


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 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 X^* 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 least ...
s \alpha, the \alpha-th Cantor–Bendixson derivative of a topological space is defined by repeatedly applying the derived set operation using transfinite recursion as follows: *\displaystyle X^0 = X *\displaystyle X^ = \left( X^ \right)' *\displaystyle X^ = \bigcap_ X^ for limit ordinals \lambda. The transfinite sequence of Cantor–Bendixson derivatives of X must eventually be constant. The smallest ordinal \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 ...
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 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.


External links


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