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Topic: Cofinal
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William Elliot

Posts: 248
Registered: 10/7/08
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 }
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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