Cofinality

Let A be a partially ordered set. A subset B of A is said to be cofinal if for every a in A there is a b in B such that a ≤ b. The cofinality of A is the smallest cardinality of a cofinal subset. Note that the cofinality always exists, since the cardinal numbers are well ordered. Cofinality is only an interesting concept if there is no maximal element in A; otherwise the cofinality is 1.

If A admits a totally ordered cofinal subset B, then we can find a subset of B which is well-ordered and cofinal in B (and hence in A). Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered and order-isomorphic to its own cardinality.

For any infinite well-orderable cardinal number κ, an equivalent and useful definition is cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ; more precisely

That the set above is nonempty comes from the fact that

i.e. the disjoint union of κ singleton sets. This implies immediately that cf(κ) ≤ κ. A cardinal κ such that cf(κ) = κ is called regular; otherwise it is called singular.

The fact that a countable union of countable sets is countable implies that the cofinality of the cardinality of the continuum must be uncountable, and hence we have

the ordinal number ω being the first infinite ordinal; this is because

.

so that the cofinality of is ω. Many more interesting results relating cardinal numbers and cofinality follow from a useful theorem of König (e.g., κ < κ^cf(κ) and κ < cf(2^κ) for any infinite cardinal κ).

Cofinality can also be similarly defined for a directed set and it is used to generalize the notion of a subsequence in a net.