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Prove set is open using open ball

  1. Sep 18, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that the set O = {(y1, y2) : y1 - y2 > 0} is an open subset of R2 in the Euclidean metric.


    2. Relevant equations



    3. The attempt at a solution
    Try as I might I cannot figure out what to take for my r. Any suggestions?
     
  2. jcsd
  3. Sep 18, 2011 #2

    Dick

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    Graph the set! It's bounded by the line y1=y2, right? Find the distance from an arbitrary point to that line.
     
  4. Sep 18, 2011 #3
    I did graph it. It is the graph of y2 > y1. So on the coordinate plane it would be the dashed line y=x shaded above. I just guessed that my r might be (y1 +y2)/2 but when I try all my inequalities I can't get to x2>x1 (assuming that (x1, x2) is my other point in the ball that I need to show is in O)
     
  5. Sep 18, 2011 #4

    Dick

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  6. Sep 18, 2011 #5
    Wait, but if (y1, y2) is in the open set, the y2-y1>0 which implies that y1-y2<0 so then wouldn't (y1-y2)/sqrt 2 be less than 0? I always have the hardest time with these open set questions
     
  7. Sep 18, 2011 #6

    Dick

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    Your original post said y1-y2>0. Seems to have gotten scrambled somewhere along the line. I would have a hard time proving x2<x1 just using basic inequalities. I would just draw the picture. Not good enough for your course??
     
  8. Sep 18, 2011 #7
    Oh yes, the original post is incorrect. The set is y2-y1>0 or y2>y1. Sorry. No, a picture proof will not due. I am actually doing this as part of a larger problem. Let S be in R2 be the set defined by S = {(a1,a2) : 0 <= a2 <= a1}. Prove that S is a closed subset of R2 in the Euclidean metric. So, I was going about by proving that the complement is open. Well, the complement is the union of two open sets (which means the complement will be open). The first open set is x<0 which I was able to prove easily. The second set is y>x. Which I cannot figure out. Hence, the post. Grrr, the questions are so aggravating. once you think that you understand it, they throw out a question that stumps me.
     
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