Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets.
An ordinal is called regular if it is cofinal with any smaller ordinal; otherwise it is singular.
For example, the second theorem above fails for the Tychonoff plank if we restrict ourselves to cofinal subnets.
For example, the even and odd natural numbers form disjoint cofinal subsets of the set of all natural numbers.
Cofinal subsets are very important in the theory of directed sets and cofinal subnet is the appropriate generalization of subsequence.
Cofinal subsets are very important in the theory of directed sets and cofinal subnet is the appropriate generalization of subsequence.
Every cyclically ordered group can be expressed as a quotient, where is a linearly ordered group and is a cyclic cofinal subgroup of.
The cofinality of an ordinal \ alpha is the smallest ordinal \ delta that is the order type of a cofinal subset of \ alpha.
A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it.
In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.