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Is [a,b] ever an open set in the order topology?

  1. Sep 19, 2010 #1
    My textbook is indicating to me that sometimes {x \in X : a <= x <= b} is an open set. How can this happen?

    My only guess is that if X has a smallest and largest element, called a and b, then sure. Otherwise?
     
  2. jcsd
  3. Sep 19, 2010 #2

    Office_Shredder

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    Let's look at the natural numbers for inspiration

    [3,5]=(2,6)

    [4,9]=(3,10)

    etc. Hopefully the generalization becomes clear
     
  4. Sep 19, 2010 #3
    So that would result from our space being the natural numbers and using the order topology. But why is that interval open? I suppose that for any element x in [3,5], we can find an open set, U, around x such that U \subset X.

    For 3 we would choose the open set (2,4)?
     
  5. Sep 19, 2010 #4

    Office_Shredder

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    I'm confused. The definition of an open set in the order topology is that it's a union of open intervals. What do you mean here?
     
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