 Derived set (mathematics)

In mathematics, more specifically in pointset topology, the derived set of a subset S of a topological space 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.
Contents
Properties
A subset S of a topological space is closed precisely when . 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).
The set S is defined to be a perfect set if S = S′. Equivalently, a perfect set is a closed set with no isolated points. Perfect sets are particularly important in applications of the Baire category theorem.
Two topological spaces are homeomorphic if and only if there is a bijection from one to the other such that the derived set of the image of any subset is the image of the derived set of that subset.
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.
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 ^{*} mapping subsets of X to subsets of X, such that for any set S and any point a:
Note that given 5, 3 is equivalent to 3' below, and that 4 and 5 together are equivalent to 4' below, so we have the following equivalent axioms:
 3'.
 4'.
Calling a set S closed if will define a topology on the space in which ^{*} is the derived set operator, that is, . If we also require that the derived set of a set consisting of a single element be empty, the resulting space will be a T_{1} space.
Cantor–Bendixson rank
For ordinal numbers α, the αth Cantor–Bendixson derivative of a topological space is defined by transfinite induction as follows:
 X^{0} = X
 X^{α+1} = (X^{α})'
 X^{λ} = X^{α} for limit ordinals λ.
The transfinite sequence of Cantor–Bendixson derivatives of X must eventually be constant. The smallest ordinal α such that X^{α+1} = X^{α} is called the Cantor–Bendixson rank of X.
External links
References
 Kechris, A. (1995). Classical Descriptive Set Theory (Graduate Texts in Mathematics 156 ed.). Springer. ISBN 0387943749 ISBN 3540943749.
 Sierpiński, Wacław F.; translated by Krieger, C. Cecilia (1952). General Topology. University of Toronto Press.
Categories:
Wikimedia Foundation. 2010.
См. также в других словарях:
Derived set — A derived set may refer to: Derived set (mathematics), a construction in point set topology Derived row, a concept in musical set theory This disambiguation page lists articles associated with the same title. If an internal … Wikipedia
Set (mathematics) — This article gives an introduction to what mathematicians call intuitive or naive set theory; for a more detailed account see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory. The intersection of two sets is… … Wikipedia
Set theory — This article is about the branch of mathematics. For musical set theory, see Set theory (music). A Venn diagram illustrating the intersection of two sets. Set theory is the branch of mathematics that studies sets, which are collections of objects … Wikipedia
Set (music) — Six element set of rhythmic values used in Variazioni canoniche by Luigi Nono[1] A set (pitch set, pitch class set, set class, set form, pitch collection) in music theory, as in mat … Wikipedia
Mathematics and art — have a long historical relationship. The ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid,[1] the Parthenon … Wikipedia
mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… … Universalium
Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity. Computer scientist Manindra Agrawal of the… … Universalium
set theory — the branch of mathematics that deals with relations between sets. [1940 45] * * * Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its… … Universalium
Derived category — In mathematics, the derived category D(C) of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C. The construction proceeds on the… … Wikipedia
mathematics, foundations of — Scientific inquiry into the nature of mathematical theories and the scope of mathematical methods. It began with Euclid s Elements as an inquiry into the logical and philosophical basis of mathematics in essence, whether the axioms of any system… … Universalium