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.

 

yes, I am working in a metric space.

So can you use your definition for open and 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?

 

got it, thanks!