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Find a closed interval topology

  1. Aug 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Let X be an ordered set where every closed interval is compact. Prove that X has the least upper bound property.


    2. Relevant equations
    X having the least upper bound property means that every nonempty subset that is bounded from above has a least upper bound, in other words, an upper bound where any number less that it is not an upper bound.
    Compact means that every open cover has a finite subcover.


    3. The attempt at a solution
    Let A be a subset of X that is bounded from above. I know I should try to find a closed interval from this, but I'm not sure where to get it from.
     
  2. jcsd
  3. Aug 23, 2010 #2
    Re: topology

    Hello,
    I'll try to give an intuitive explanation.

    "Let A be a subset of X that is bounded from above."
    If A has a greatest element, we're done. If A does not, we can focus on an interval B of the form [...[ with [tex]A \cap B = B[/tex]. Note that B has no greatest element thus there is an open cover that has no finite subcover (is there?).
    Define [tex]C = \overline{B}[/tex].
    • If C = B, then B is closed and thus compact (CONTRADICTION! (?))
    • So C has at least an element more than B. What is this element?
     
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