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

AI Thread Summary
Open balls are essential for defining limit points in metric spaces because they ensure that every neighborhood around a point intersects the set minus that point. Closed balls can lead to ambiguities, particularly when considering points on the boundary, which may not accurately reflect the limit point's behavior. Using closed balls could result in a situation where a point is incorrectly classified as a limit point due to the inclusion of its boundary. An example illustrating this is when a ball has a radius of zero, which would not allow for any intersection with the set. Thus, open balls provide a clearer and more precise definition of limit points.
srfriggen
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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.
 
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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?
 
srfriggen said:
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!
 

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