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Confusion with open set and a subset of R2

  1. Sep 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Using the definition of an open set, prove that the subset [itex]S = \left({a}_{1}, {b}_{1}\right)\times \left({a}_{2}, {b}_{2}\right)[/itex] of the Euclidean space [itex]{R}^{2}[/itex] is open.

    2. Relevant equations



    3. The attempt at a solution

    I so far don't have much of an attempt, as I am merely trying to figure out what the question is asking. If I interpret the (a1, b1) x (a2, b2) as two vectors on a plane then S would be a vector pointing outwards from the plane? Which doesn't seem like an open set in R2 to me.

    I am not looking for a solution, but looking for a clarification of the question in mind.

    Thanks!
     
  2. jcsd
  3. Sep 30, 2009 #2
    (a1, a2) is an open interval on the real number line, just like in highschool or calculus.

    here, the X refers to the cartesian product. so (a1, b1) X (a2, b2) is the set that consists of tuples1 (x1, x2) where x1 is in (a1, b1) and x2 is in (a2, b2). There is an unfortunate overlap of notation between vector/tuple notation and the notation for intervals.

    In symbols and for general sets:

    [tex]S_{1} \times S_{2} = \left{\{ (a, b) : a \in S_{1}, b \in S_{2} \right}\} [/tex]


    1 - tuples are essentially vectors but with no necessarily assumed linear structure
     
    Last edited: Sep 30, 2009
  4. Sep 30, 2009 #3
    Thank you very much!
     
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