Why Are Open Balls Essential for Defining Limit Points in Mathematics?

[srfriggen]

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.

 

[micromass]

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}.

 

[srfriggen]

yes, I am working in a metric space.

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

 

[micromass]

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.

 

[srfriggen]

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?

 

[micromass]

What if the ball has radius 0?

 

[srfriggen]

got it, thanks!