Limit points and exterio points Quick quesiton

In summary, a limit point is a point in a sequence or set that is approached by other points in the sequence or set, while an exterior point is a point that is not part of the sequence or set. Limit points and exterior points are important in analysis as they help us understand the behavior of a sequence or set and determine if it has a limit or if it is open or closed. A limit point and exterior point cannot be the same point, and they can be identified by looking for points that are approached by other points in the sequence or set, or those that are not part of the sequence or set.
  • #1
Buri
273
0
Is the following true?

X\lim(A) = ext(A), where A is a subset of a metric space X.

I think I've found a proof, but I don't feel very secure about it. So could someone just tell me if this is a correct assertion? What I'm trying to prove is that lim(A) is closed.

Thanks
 
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  • #2
Take A being a one-point set. What is lim(A)? What is ext(A)?
 
  • #3
I hate isolated points! lol Thanks
 

FAQ: Limit points and exterio points Quick quesiton

1. What is a limit point?

A limit point is a point in a sequence or set that is approached by other points in the sequence or set. It is not necessary for the limit point to be a member of the sequence or set.

2. How is a limit point different from an exterior point?

A limit point is a point that is approached by other points in a sequence or set, while an exterior point is a point that is not part of the sequence or set. In other words, a limit point is a point within the boundary of the sequence or set, while an exterior point is outside the boundary.

3. What is the importance of limit points and exterior points in analysis?

Limit points and exterior points are important in analysis because they help us understand the behavior of a sequence or set. They can help us determine if a sequence or set has a limit or if it is open or closed.

4. Can a limit point also be an exterior point?

No, a limit point and exterior point cannot be the same point. A limit point is an accumulation point within the boundary of a sequence or set, while an exterior point is outside the boundary. Therefore, a point cannot be both within and outside the boundary at the same time.

5. How do we identify limit points and exterior points?

To identify limit points, we look for points that are approached by other points in the sequence or set. For exterior points, we look for points that are not part of the sequence or set. We can also use mathematical notation and definitions to help identify these points.

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