Open balls and limit points

  • Thread starter srfriggen
  • Start date
  • #1
301
4
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.
 

Answers and Replies

  • #2
22,089
3,293
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}.
 
  • #3
301
4
yes, I am working in a metric space.

So can you use your definition for open and closed balls?
 
  • #4
22,089
3,293
yes, I am working in a metric space.

So can you use your definition for open and closed balls?
No, only for open balls. Think about what could go wrong for closed balls.
 
  • #5
301
4
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?
 
  • #6
22,089
3,293
What if the ball has radius 0?
 
  • #7
301
4
got it, thanks!
 

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