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Open Balls and Neighborhoods

  1. Sep 11, 2010 #1
    Are open balls and neighborhoods the exact same thing? If not, could you please shed some light on this for me?
     
  2. jcsd
  3. Sep 11, 2010 #2

    Office_Shredder

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    A neighborhood is just an open set. An open ball requires being in a metric space. The word neighborhood is usually used as opposed to just open set because you want to give the impression that the open set is supposed to be a small one, similar to saying let [tex] \epsilon>0[/tex] vs saying let [tex]M>0[/tex]. They both say the exact same thing but one of them indicates we're interested in picking small numbers and one large numbers. It's not a formal definition but just to give the reader some intuition
     
  4. Sep 11, 2010 #3

    Fredrik

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    There are at least three inequivalent definitions of "neigborhood of x":

    1. An open ball around x.
    2. An open set that contains x.
    3. A set that has an open subset that contains x.
     
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