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

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

## 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 .${\ displaystyle \ preccurlyeq}$