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.