Examples
- The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subset).
- In particular, the cofinality of any nonzero finite ordinal, or indeed any finite directed set, is 1, since such sets have a greatest element.
- Every cofinal subset of a partially ordered set must contain all maximal elements of that set. Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.
- In particular, let A be a set of size n, and consider the set of subsets of A containing no more than m elements. This is partially ordered under inclusion and the subsets with m elements are maximal. Thus the cofinality of this poset is n choose m.
- A subset of the natural numbers N is cofinal in N if and only if it is infinite, and therefore the cofinality of ℵ0 is ℵ0. Thus ℵ0 is a regular cardinal.
- The cofinality of the real numbers with their usual ordering is ℵ0, since N is cofinal in R. The usual ordering of R is not order isomorphic to c, the cardinality of the real numbers, which has cofinality strictly greater than ℵ0. This demonstrates that the cofinality depends on the order; different orders on the same set may have different cofinality.
Read more about this topic: Cofinality
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