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Cofinal
Posted:
Oct 25, 2011 6:00 AM


Within a (partial) order A set A is cofinal to B when A subset B and for all b in B, there's some a in A with b <= a.
Defining the lower set of A, down A = { x  some a in A with x <= a },
the definition can be rewritten as A subset B subset down A
which is equivalent to A subset B and down B = down A.
Has the notion of cofinal been extended beyond subsets? Can two sets A and B for which down A = down B or equivalently A subset down B and B subset down A that is for all a in A, some b in B with a <= b and for all b in B, some a in A with b <= a be consider cofinal.
For example, are the sets A = { 1  1/2n  n in N } and B = { 1  1/(2n1)  n in N } considered cofinal?
Within complete orders, cofinal would be equivalent to having the same supremum.



