Editing Cofinality (section)

{{Short description|Size of subsets in order theory}}
{{Distinguish|cofiniteness}}
In [[mathematics]], especially in [[order theory]], the '''cofinality''' cf(''A'') of a [[partially ordered set]] ''A'' is the least of the [[cardinality|cardinalities]] of the [[cofinal (mathematics)|cofinal]] subsets of ''A''. Formally,<ref>{{cite arXiv |last=Shelah |first=Saharon |date=26 November 2002 |title=Logical Dreams |eprint=math/0211398}}</ref>
:<math>\operatorname{cf}(A) = \inf \{|B| : B \subseteq A, (\forall x \in A) (\exists y \in B) (x \leq y)\}</math>

This definition of cofinality relies on the [[axiom of choice]], as it uses the fact that every non-empty set of [[cardinal number]]s has a least member. The cofinality of a partially ordered set ''A'' can alternatively be defined as the least [[ordinal number|ordinal]] ''x'' such that there is a function from ''x'' to ''A'' with cofinal [[Image (mathematics)|image]]. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.

Cofinality can be similarly defined for a [[directed set]] and is used to generalize the notion of a [[subsequence]] in a [[Net (mathematics)|net]].