Confinality

In order theory and set theory , the property confinal (also: cofinal , English cofinal ) is used for topological subnetworks , including per-finite numbers . The term derived cofinality (also: Kofinalität , Eng. Cofinality ) denotes a particular attribute of partially ordered subsets, namely a cardinal number .

The term was introduced by Felix Hausdorff .

Definitions

• Let be a set partially ordered by and . The set is called confinal (cofinal) in (or, if there are several partial orders on , confinal in ), if there is a with for each . ${\displaystyle \lambda }$ ${\displaystyle \leq }$ ${\displaystyle X\subset \lambda }$ ${\displaystyle X}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle (\lambda ,\leq )}$ ${\displaystyle \mu \in \lambda }$ ${\displaystyle \xi \in X}$ ${\displaystyle \mu \leq \xi }$

• The confinality of is denoted by and is defined as the smallest cardinality of a confinal subset, i.e. H. ${\displaystyle \lambda }$ ${\displaystyle \operatorname {cf} (\lambda )}$

${\displaystyle \operatorname {cf} (\lambda )\;:=\min _{X\subset \lambda {\text{ konfinal}}}|X|}$.

• For an ordinal number and thus also for every cardinal number , the following conceptualization is available: ${\displaystyle \lambda }$ ${\displaystyle \lambda }$

If so, it is called singular. ${\displaystyle \operatorname {cf} (\lambda )<\lambda }$ ${\displaystyle \lambda }$

If so, it is called regular. ${\displaystyle \operatorname {cf} (\lambda )=\lambda }$ ${\displaystyle \lambda }$

Concept formation in the sense of Hausdorff

In Hausdorff's Fundamentals of Set Theory , one finds a more general conceptual formation for the confinality, which, in the case of a linearly ordered set , corresponds to the above. This more general term can be represented as follows:

• If a non-empty partially ordered set and a non-empty subset lying within it , then one says that with is confinal if there is no element which is really larger than each element . ${\displaystyle (S,\preccurlyeq )}$ ${\displaystyle T\subseteq S}$ ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle s\in S}$ ${\displaystyle t\in T}$

Inferences

• The relation

${\displaystyle X{\overset {\underset {\mathrm {cof} }{}}{\subset }}\lambda \quad :\Longleftrightarrow \quad X\subset \lambda \;\;\wedge \;\;X}$ is cofinal in ${\displaystyle \lambda }$

is transitive and reflexive , i.e. a quasi-order .

Transitivity: Is and , then is first . Secondly, for every one with . Well , then there is a with , therefore also a with . Taken together follows . ${\displaystyle X{\overset {\underset {\mathrm {cof} }{}}{\subset }}\lambda }$ ${\displaystyle Y{\overset {\underset {\mathrm {cof} }{}}{\subset }}X}$ ${\displaystyle Y\subset X\subset \lambda }$ ${\displaystyle \xi \in X}$ ${\displaystyle \eta \in Y}$ ${\displaystyle \xi \leq \eta }$ ${\displaystyle \mu \in \lambda }$ ${\displaystyle \xi \in X}$ ${\displaystyle \mu \leq \xi }$ ${\displaystyle \eta \in Y}$ ${\displaystyle \mu \leq \xi \leq \eta }$ ${\displaystyle Y{\overset {\underset {\mathrm {cof} }{}}{\subset }}\lambda }$

The reflexivity is trivial.

• The confinality is if and only if the partially ordered set is empty. ${\displaystyle 0}$
• The confinality is exactly when the order has a maximum , for example when it is a successor ordinal number. ${\displaystyle 1}$
• For non-empty partially ordered sets without maximum elements , the confinality is at least countable , i.e. (see Aleph function ), and at most the cardinality of the set itself, because every partially ordered set lies confinally in itself. ${\displaystyle \aleph _{0}}$
• For total parent applies , ie is regular. ${\displaystyle \lambda }$ ${\displaystyle \operatorname {cf} (\operatorname {cf} (\lambda ))=\operatorname {cf} (\lambda )}$ ${\displaystyle \operatorname {cf} (\lambda )}$
• For a Limes number (understood as a Von Neumann ordinal number) a subset is confinal if and only if its union is equal . ${\displaystyle \lambda }$ ${\displaystyle X}$ ${\displaystyle \textstyle \bigcup X}$ ${\displaystyle \lambda }$
• If an infinite set has regular cardinality , one needs at least -many sets with cardinality less than , in order to represent the union of these sets. ${\displaystyle K}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle K}$
• For a limit number a subset is confinal if and only if it converges as a network , provided with the natural order, in the order topology of against . ${\displaystyle \lambda }$ ${\displaystyle \lambda +1}$ ${\displaystyle \lambda }$

Examples

• The affinity of with the natural order is , because the natural numbers form a countable confinal subset. ${\displaystyle \mathbb {R} }$ ${\displaystyle \aleph _{0}}$
• ${\displaystyle \aleph _{0}}$ is regular.
• If you restrict a network to a confinal subset while adopting the order, you get a subnet (however, not every subnet has to have this shape).
• The cardinal number is singular. It is true because it is a confinal subset. ${\displaystyle \aleph _{\omega }}$ ${\displaystyle \operatorname {cf} (\aleph _{\omega })=\aleph _{0}}$ ${\displaystyle \{\aleph _{i}\mid i\in \mathbb {N} \}}$
• If there is a successor ordinal and the axiom of choice applies , it is always regular. The question of whether there are other and therefore uncountable, regular Limes cardinal numbers is the core of the large cardinal number axioms , i.e. H. of the axioms about the existence of large cardinal numbers. ${\displaystyle \alpha }$ ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \aleph _{0}}$

literature

• Ulf Friedrichsdorf, Alexander Prestel: Set theory for the mathematician (= Vieweg study. 58 Basic course in mathematics. ). Vieweg, Braunschweig a. a. 1985, ISBN 3-528-07258-X .
• Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded. Springer, Berlin a. a. 2003, ISBN 3-540-44085-2 .
• PS Alexandroff : Textbook of set theory . Translated from Russian by Manfred Peschel, Wolfgang Richter and Horst Antelmann. Publishing house Harri Deutsch , Thun and Frankfurt am Main 1994, ISBN 3-8171-1365-X .
• Egbert Harzheim : Ordered Sets (=  Advances in Mathematics . Volume 7 ). Springer Verlag , New York 2005, ISBN 0-387-24219-8 ( MR2127991 ).
• Felix Hausdorff: Fundamentals of set theory . Reprinted, New York, 1965. Chelsea Publishing Company , New York, NY 1965.
• Erich Kamke : Set theory (=  Göschen Collection . 999 / 999a). 7th edition. Walter de Gruyter , Berlin, New York 1971.

Individual evidence

1. ↑ Felix Hausdorff: Fundamentals of set theory. Reprinted, New York, 1965, p. 140.
2. ↑ Erich Kamke: Set theory. 1971, pp. 167-168.
3. ↑ With regard to the present order relation . ${\displaystyle \preccurlyeq }$