In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.
Formally,[1] This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member.
The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image.
This second definition makes sense without the axiom of choice.
If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.
Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net.
admits a totally ordered cofinal subset, then we can find a subset
have the countable cardinality of the cofinality of
that is the order type of a cofinal subset of
-indexed strictly increasing sequence with limit
ranges over the natural numbers) tends to
but, more generally, any countable limit ordinal has cofinality
An uncountable limit ordinal may have either cofinality
The cofinality of any successor ordinal is 1.
The cofinality of any nonzero limit ordinal is an infinite regular cardinal.
Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular.
Assuming the axiom of choice,
are initial ordinals that are not regular.
So the cofinality operation is idempotent.
is an infinite cardinal number, then
is the least cardinal such that there is an unbounded function from
is also the cardinality of the smallest set of strictly smaller cardinals whose sum is
cf ( κ ) = min
That the set above is nonempty comes from the fact that
The cofinality of any totally ordered set is regular, so
Using König's theorem, one can prove
The last inequality implies that the cofinality of the cardinality of the continuum must be uncountable.
(Compare to the continuum hypothesis, which states
) Generalizing this argument, one can prove that for a limit ordinal
On the other hand, if the axiom of choice holds, then for a successor or zero ordinal