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Math Help - Two balls such that the bigger one is contained inside the smaller

  1. #1
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    Two balls such that the bigger one is contained inside the smaller

    I've been told to find a metric space and two balls in it such that the ball with the smaller radius contains but is not equal to the ball with the larger radius

    So I understand this means find (X,d) such that  B_d(a,R)= \{x\in X | d(a,x)<R\}\subset B_d(a,r)=\{  x\in X | d(a,x)<r\} with R>r and R,r\in\mathbb{R} (I realize they do not have to be centered at the same point, but I do not believe assuming so affects the generality if I can find an example like this)


    I can't get anywhere. I only know of the euclidean metric, the taxicab metric, the square metric, and the discrete metric, and my professor said that I'd have to consider subsets and restrictions of the distance function is some way....


    I don't need a complete answer, just a push in the right direction, thanks
    Last edited by artvandalay11; September 20th 2009 at 01:08 PM.
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    The situation described is impossible!! With the same center and the same metric, the ball with smaller radius is contained always in the ball of bigger radius! I suppose one of two, the metric or the center has to be different. If it is the metric, just consider the discrete case. If it is the center, unfortunately, I don't have anything immediate in mind. I can't think a little about it, but it sounds strange to me because I am used to Banach or Frechet spaces, with each point being an accumulation one.
    Last edited by Enrique2; September 20th 2009 at 01:58 PM.
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  3. #3
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    "Find a metric space and two balls in it such that the ball with smaller radius contains and is not equal to the ball with larger radius. "

    That quote was taken directly from the website with my homework posted, I'm not sure he would make us try something that's impossible, but yeah it's not like I have any insight
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    If the big ball with different center was contained inside a small ball, then we could just move the big ball to the center of the small one no? which is why I don't think it matters that I am assuming centers are the same
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    Ok! the exact quote suggest that the centers must be different, and the metric the same. I was confused with your attempt. I have no example in mind, we need something that near the boundary of the ball, outside there is "nothing", then the ball centering in one point near the boundary only arrives "near" the center, but I have not in mind eany example, sorry!
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  6. #6
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    alright thanks, I am pretty sure I can take it from here, sorry for the confusion
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  7. #7
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    Quote Originally Posted by artvandalay11 View Post
    If the big ball with different center was contained inside a small ball, then we could just move the big ball to the center of the small one no? which is why I don't think it matters that I am assuming centers are the same
    I believe you are like me, thinking in normed spaces or something like this, with any point being accumulating.

    Perhaps three points!, defining d(x,y)=1, d(x,z)=1, d(y,z)=2 gives the answer! The closed ball centered at x with radius 1 contains all the space, but the ball centered at y with radius 3/2 does not contain z!!
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