Cofinality
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In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a ≤ b. The cofinality of A is the least 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 greatest element in A since otherwise the cofinality is 1.
Cofinality can also be similarly defined for a directed set and it is used to generalize the notion of a subsequence in a net.
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
- <math>\mathrm{cf}(\kappa) = \inf \{ \mathrm{card}(I)\ |\ \kappa = \sum_{i \in I} \lambda_i\ \mathrm{and}\ (\forall i)(\lambda_i < \kappa)\}<math>
That the set above is nonempty comes from the fact that
- <math>\kappa = \bigcup_{i \in \kappa} \{i\}<math>
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
- <math>2^{\aleph_0}\neq\aleph_\omega,<math>
the ordinal number ω being the first infinite ordinal; this is because
- <math> \aleph_\omega = \bigcup_{n < \omega} \aleph_n <math>.
so that the cofinality of <math>\aleph_\omega<math> is ω. Many more interesting results relating cardinal numbers and cofinality follow from a useful theorem of König (e.g., κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ).