# 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.