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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