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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