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 . ${\displaystyle M}$ ${\displaystyle M}$ ^{${\displaystyle {\ }^{\prime }}$} ${\displaystyle M^{\rm {d}}}$ ${\displaystyle M^{(1)}}$

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. ${\displaystyle n}$ ${\displaystyle M^{(n)}}$ ${\displaystyle (n-1)}$ ${\displaystyle M^{(n-1)}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \xi }$ ${\displaystyle \xi }$ ${\displaystyle M^{(\xi )}}$ ${\displaystyle M^{(\xi )}=(M^{(\xi -1)})^{(1)}}$ ${\displaystyle \xi }$ ${\displaystyle M^{(\xi )}=\textstyle {\bigcap _{\eta <\xi }}M^{(\eta )}}$

properties

The derivation of a set can be empty. The following rules apply in a T ₁ space:

• ${\displaystyle (M\cup N)^{(1)}=M^{(1)}\cup N^{(1)}}$
• ${\displaystyle M^{(1)}\setminus N^{(1)}\subseteq (M\setminus N)^{(1)}}$
• ${\displaystyle (M^{(1)})^{(1)}\subseteq M^{(1)}}$
• ${\displaystyle \textstyle \left(\bigcap _{t\in T}M_{t}\right)^{(1)}\subseteq \bigcap _{t\in T}M_{t}^{(1)}}$
• ${\displaystyle \textstyle \bigcup _{t\in T}M_{t}^{(1)}\subseteq \left(\bigcup _{t\in T}M_{t}\right)^{(1)}}$
• ${\displaystyle M\subseteq N\ \Rightarrow \ M^{(1)}\subseteq N^{(1)}}$
• ${\displaystyle {\overline {M}}=M\cup M^{(1)}}$
• ${\displaystyle {\overline {M}}^{(1)}={\overline {M^{(1)}}}=M^{(1)}}$
• ${\displaystyle \eta <\xi \Rightarrow M^{(\eta )}\supseteq M^{(\xi )}.}$

A lot is perfect precisely when . The dense core of a set is the intersection of its derivatives. ${\displaystyle M}$ ${\displaystyle M^{(1)}=M}$

Spaces with a countable basis

Let be the set of condensation points of . In a topological space with a countable basis: ${\displaystyle \operatorname {cp} (M)}$ ${\displaystyle M}$

• First set of Lindelof : , ${\displaystyle \operatorname {card} (M\setminus \operatorname {cp} (M))<\aleph _{1}}$
• 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 . ${\displaystyle M}$ ${\displaystyle \xi <\Omega }$ ${\displaystyle M^{(\xi )}=M^{(\xi +1)}}$

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 ${\displaystyle \beta }$ ${\displaystyle M}$

${\displaystyle M=\bigcup _{\alpha <\beta }(M^{(\alpha )}\setminus M^{(\alpha +1)})\cup M^{(\beta )}}$.

The sets consist only of isolated points and are at most countable. The amount ${\displaystyle M^{(\alpha )}\setminus M^{(\alpha +1)}}$

${\displaystyle \bigcup _{\alpha <\beta }(M^{(\alpha )}\setminus M^{(\alpha +1)})}$

as a union of at most countable many at most countable sets itself is at most countable. The crowd is perfect because of . ${\displaystyle M^{(\beta )}}$ ${\displaystyle M^{(\beta )}=M^{(\beta +1)}}$

Individual evidence

1. ↑ For the history of the introduction of the term set derivation by G. Cantor see ordinal number: History of discovery .
2. ^ Willi Rinow : Textbook of Topology (= university books for mathematics . Vol. 79). VEB Deutscher Verlag der Wissenschaften, Berlin 1975.
3. ↑ ^{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.
4. ^ 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).
5. ↑ ^{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 .
6. ↑ Oliver Deiser: Real Numbers. The classical continuum and the natural sequences (= Springer textbook ). Springer, Berlin et al. 2007, ISBN 978-3-540-45387-1 .