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 .
- The confinality of is denoted by and is defined as the smallest cardinality of a confinal subset, i.e. H.
- .
- For an ordinal number and thus also for every cardinal number , the following conceptualization is available:
- If so, it is called singular.
- If so, it is called regular.
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 .
Inferences
- The relation
- is cofinal in
- 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 .
- The reflexivity is trivial.
- The confinality is if and only if the partially ordered set is empty.
- The confinality is exactly when the order has a maximum , for example when it is a successor ordinal number.
- 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.
- For total parent applies , ie is regular.
- For a Limes number (understood as a Von Neumann ordinal number) a subset is confinal if and only if its union is equal .
- 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.
- 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 .
Examples
- The affinity of with the natural order is , because the natural numbers form a countable confinal subset.
- 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.
- 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.
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
- ↑ Felix Hausdorff: Fundamentals of set theory. Reprinted, New York, 1965, p. 140.
- ↑ Erich Kamke: Set theory. 1971, pp. 167-168.
- ↑ With regard to the present order relation .