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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.
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