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Cofinal
Posted:
Oct 25, 2011 6:00 AM
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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/(2n-1) | n in N }
considered cofinal?
Within complete orders, cofinal would be equivalent
to having the same supremum.
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