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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 .


  • 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 .


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 .


  • 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.


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 .