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Open balls and limit points

  1. Nov 11, 2011 #1
    While learning about limit points the use of an open ball has been of high discussion. Why can you not use a closed ball to define a limit point?

    If someone could give me some intuition as to why I think I may get it.

    Thanks.
     
  2. jcsd
  3. Nov 11, 2011 #2
    Are you working in metric space?? In that case, you can define limit points using open balls.

    Just define x to be a limit point of A if every ball around x intersects A\{x}.
     
  4. Nov 11, 2011 #3
    yes, I am working in a metric space.

    So can you use your definition for open and closed balls?
     
  5. Nov 11, 2011 #4
    No, only for open balls. Think about what could go wrong for closed balls.
     
  6. Nov 11, 2011 #5
    I've been trying to but I can't think of an example where having boundaries on the ball would cause a problem. That's really why I asked the original question.

    Can you give me one?
     
  7. Nov 11, 2011 #6
    What if the ball has radius 0?
     
  8. Nov 11, 2011 #7
    got it, thanks!
     
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