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Math Help - Closure of open ball

  1. #1
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    Closure of open ball

    How do you show that the closure of an open ball is a closed ball?
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  2. #2
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    Let A be the the closure of the open ball B

    Let x be in  A^c

    We want to show  A^c is open (because a set is closed if its complement is open)

    Suppose  A^c is not open, then for all r > 0, the open ball  B_r(x) intersects A. But then x is in the closure of A which equals A. Contradiction
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  3. #3
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    Quote Originally Posted by vinnie100 View Post
    How do you show that the closure of an open ball is a closed ball?
    Quote Originally Posted by southprkfan1 View Post
    Let A be the the closure of the open ball B
    Let x be in  A^c We want to show  A^c is open (because a set is closed if its complement is open)
    Suppose  A^c is not open, then for all r > 0, the open ball  B_r(x) intersects A. But then x is in the closure of A which equals A. Contradiction

    southprkfan1, does that show that the closure is a closed ball?
    Oh, is does show that the complement of a closed set is open. BUT?

    Here is a hint: In a metric space the closure of a set is the set of points a distance of zero from the set.
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  4. #4
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    oh i was convinced, but i can see plato's point... hmmmm
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  5. #5
    MHF Contributor Drexel28's Avatar
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    I'm not sure exacdtly what you're asking but if it's to prove that B_{\delta}[x]=\overline{B_{\delta}(x)} ([] means closed and () means open) then this isn't always true. Consider the three point discrete space \{0,1,2\}. Then, B_{1}[1]=\{0,1,2\} but \overline{B_{1}[1]}=\{1\} since the discrete space has no limit points.
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  6. #6
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    yep that is exactly what I was asking to prove...but its false now I see! sorry for the confusion! thanks for all your help everyone
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