Deriving a set
In mathematics, the derivative of a set is the set of all accumulation points in this set. It is assumed that a distance concept or, more generally, a topology is defined on the set . A synonymous term is the derivative of the set. I.e., the amount of such signs are for their dissipation , or, for the first derivative .
Higher amount derivatives
Higher quantity derivatives are defined inductively: The -th derivative is the derivative of the -th derivative . The closed envelope of is also referred to as the zeroth derivative of . More generally, for each isolated ordinal the th derivation through and, for each limit number by defined.
properties
The derivation of a set can be empty. The following rules apply in a T 1 space:
A lot is perfect precisely when . The dense core of a set is the intersection of its derivatives.
Spaces with a countable basis
Let be the set of condensation points of . In a topological space with a countable basis:
- First set of Lindelof : ,
- Cantor-Bendixson theorem, I : Every closed set can be represented as a union of a perfect and a at most countable set. This representation is clear in Polish areas .
It follows as a consequence:
- Every closed set is either at most countable or has the thickness of the continuum.
One possible evidence used
- Cantor-Bendixson theorem, II : In spaces with a countable basis, the sequence of its derivatives always ends with a perfect set for every subset, i.e. H. for every set there is an ordinal number such that .
The smallest such ordinal number is called the Cantor-Bendixson degree of the set.
Cantor-Bendixson's second theorem is a generalization of the first. Consider the topology induced by X on M. If the Cantor-Benidixson degree is the set in this space, then is
- .
The sets consist only of isolated points and are at most countable. The amount
as a union of at most countable many at most countable sets itself is at most countable. The crowd is perfect because of .
Individual evidence
- ↑ For the history of the introduction of the term set derivation by G. Cantor see ordinal number: History of discovery .
- ^ Willi Rinow : Textbook of Topology (= university books for mathematics . Vol. 79). VEB Deutscher Verlag der Wissenschaften, Berlin 1975.
- ↑ a b Kazimierz Kuratowski : Topology. Volume 1. New edition, revised and augmumented. Academic Press et al., New York NY et al. 1966, ISBN 0-12-429201-1 , § 9., § 24.IV.
- ^ Josef Naas , Hermann Ludwig Schmid : Mathematical dictionary. With the inclusion of theoretical physics. 2 volumes. 3rd edition, unchanged reprint. Akademie-Verlag et al., Berlin et al. 1979, ISBN 3-519-02400-4 (vol. 1).
- ↑ a b Pawel Sergejewitsch Alexandrow : Textbook of set theory. 6th, revised edition. Harri Deutsch, Frankfurt am Main et al. 1994, ISBN 3-8171-1365-X .
- ↑ Oliver Deiser: Real Numbers. The classical continuum and the natural sequences (= Springer textbook ). Springer, Berlin et al. 2007, ISBN 978-3-540-45387-1 .