Cofinal (mathematics)

In mathematics, a subset B of a partially ordered set A is cofinal if for every a in A there is b in B such that a ≤ b.

Also, a sequence or net of elements of A will be called cofinal if its image is cofinal in A.

Concerning the cardinality of cofinal subsets, see cofinality.

Cofinal set of subsets

A particular but important case is given if A is a subset of the power set P(E) of some set E, ordered by inclusion (⊃).

Thus, B ⊂ A ⊂ P(E) will be called cofinal, iff for any a ∈ A there is b ∈ B such that b ⊂ a.

For example, if E is a group, A could be the set of normal subgroups of finite index. Then, cofinal subsets of A (or sequences, or nets) are used to define Cauchy sequences and the completion of the group.

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