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Uncountability of neighborhoods in R^n

  1. Dec 3, 2008 #1
    Quick question: Is it true that all neighborhoods in R^n form an uncountable set? It seems obvious to me that the answer is yes, but there is no proof in my analysis book and I can't think of one.
     
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  3. Dec 3, 2008 #2

    Hurkyl

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    Some bases for the topology on R^n are uncountable, and some bases are countably infinite. It depends on which one you have chosen.
     
  4. Dec 3, 2008 #3
    How about just a neighborhood of the real line centered at x with radius r>0.
     
  5. Dec 3, 2008 #4
    Try proving the the set of all balls around any single point is uncountable, and think about it's relationship to your problem.
     
  6. Dec 3, 2008 #5

    morphism

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    Wait, what exactly are you asking? Do you want to prove that the set {y : |x-y|<r} is uncountable? Or do you want to prove the collection of all such neighborhoods is uncountable?
     
  7. Dec 4, 2008 #6
    Yeah sorry I wasn't that clear. I just want to know if the set {x in R : |x-y|<r} is uncountable.
     
  8. Dec 4, 2008 #7
    Well, that is just your open interval (y - r, y + r), which is indeed uncountable. (A bijection from the real numbers to this interval is [tex]f(x) = y + r \tanh(x)[/tex].)
     
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