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Why not a topology?

  1. Sep 19, 2007 #1
    1. The problem statement, all variables and given/known data
    X is the space of all real numbers

    topology A={empty set} U {R} U {(-infinity,x];x in R}

    3. The attempt at a solution
    Is it because (-infinity, x] is not an open set usuing the usual metric on R but is using a metric allowed as it was not specified in the question.

    If not then is it because an infinite union of (-infinity,x] is not in A?

    i.e take x=1/n then an infinite union of (-infinity,-1/n] when n goes to infinity should be (-infinity,0) which is not in A.
  2. jcsd
  3. Sep 19, 2007 #2

    matt grime

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    What does the usual topology have to do with anything. You're asked to show that this is not a topology so find one of the axioms it fails to satisfy.

    that doesn't make sense. The union of what index? Do you mean the union

    [tex]\cup_{x \in \mathbb{R}} (-\infty,x][/tex]?

    Because that is just R.

    There we go. Now you're talking, though what you really mean is

    [tex]\cup_{n \in \mathbb{N}} (-\infty,-1/n]=(-\infty,0)[/tex].

    There are no limits of anything involved.
    Last edited: Sep 19, 2007
  4. Sep 19, 2007 #3
    Finally a (the first) complement from Matt Grime.

    I figured that one up while typing the question.
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