Open balls and limit points

  • Thread starter srfriggen
  • Start date
  • #1
srfriggen
306
5
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
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,178
3,316
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
srfriggen
306
5
yes, I am working in a metric space.

So can you use your definition for open and closed balls?
 
  • #4
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,178
3,316
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
srfriggen
306
5
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
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,178
3,316
What if the ball has radius 0?
 
  • #7
srfriggen
306
5
got it, thanks!
 

Suggested for: Open balls and limit points

Replies
1
Views
368
  • Last Post
Replies
2
Views
664
Replies
14
Views
696
  • Last Post
Replies
5
Views
688
Replies
7
Views
447
Replies
1
Views
604
  • Last Post
Replies
0
Views
1K
Replies
7
Views
882
Replies
1
Views
428
Top