Cofinal

Discussion in 'Math Research' started by William Elliot, Oct 25, 2011.

  1. 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.
     
    William Elliot, Oct 25, 2011
    #1
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