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. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ...

Also, a sequence or net of elements of A will be called cofinal if its image is cofinal in A. This is a page about mathematics. ... In mathematics the term net has at least two meanings. ...

Concerning the cardinality of cofinal subsets, see cofinality. In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a ≤ b. ...

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 (⊃). In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ...

Thus, B ⊂ A ⊂ P(E) will be called cofinal, iff for any a ∈ A there is b ∈ B such that b ⊂ a. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P...

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. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written: . Another way... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ...

References

• Lang, Serge (1997). Algebra (3rd ed., reprint w/ corr.). Addison-Wesley. ISBN 0-201-55540-9.