Open balls and limit points

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

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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}.

yes, I am working in a metric space.

So can you use your definition for open and closed balls?

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.

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?

What if the ball has radius 0?

got it, thanks!