Cofinality

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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 ab. 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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