# Interesting problem? boundaries/limit points etc

• Buri
In summary, the conversation is discussing finding an example of a nonempty set A subset of R such that A = br(A) = Lim(A) = Cl(A). It is mentioned that A cannot be dense in R and that int(A) must be empty. The idea of A being a set of discrete points is suggested, but there are concerns about the limit points of such a set. The possibility of A being a convergent infinite sequence or the Cantor set is also mentioned. The main issue seems to be understanding the behavior of limit points in a discrete set.
Buri
I've been working on this problem and would like some help or any hints.

Give an example of a nonempty set A subset of R such that A = br(A) = Lim(A) = Cl(A), where br(A) denotes the boundary points of A, Lim(A) denotes the limit points of A and Cl(A) denotes the closure of A.

I've tried finding conditions that this A seems to have to satisfy and this is what I have so far. It appears that A cannot be dense in R. Otherwise, br(A) = R which won't equal A. And since Cl(A) = int(A) U br(A) then I should have that int(A) is empty. So it seems like A is going to be a set of discrete points. But at the same time, the fact that it isn't dense sort of confuses me because I must have that every point is a boundary point and hence any ball around it has points of A and R\A, so its rather weird.

Any help or ideas?

EDIT:

Hmm I just thought of something, maybe I could let A to be an convergent infinite sequence...once the terms begin to become arbitrarily close to each other...I'll have to look at this...

EDIT:

Wait does the Cantor set satisfy these properties?

Any ideas??

What's so weird about a discrete point being in the ball centered at it?

What should be weirding you out is the limit points of such a discrete set.

ZioX said:
What's so weird about a discrete point being in the ball centered at it?.

I didn't say that...

ZioX said:
What should be weirding you out is the limit points of such a discrete set.

Hmm I think that's my problem

## 1. What is the definition of an interesting problem?

An interesting problem is a question or challenge that requires critical thinking, creativity, and often multiple approaches to solve. It may also have real-world applications or implications.

## 2. What are boundaries in relation to problem-solving?

Boundaries in problem-solving refer to the limitations or constraints that must be considered when attempting to solve a problem. These can include physical, technical, or ethical boundaries.

## 3. How do limit points play a role in problem-solving?

Limit points are important in problem-solving because they can help determine the scope and direction of a problem. They can also indicate potential solutions or areas that need further investigation.

## 4. Can boundaries and limit points change during the problem-solving process?

Yes, boundaries and limit points can change as new information is discovered or as the problem is approached from different perspectives. It is important to continuously reassess and adjust these boundaries and limit points as needed during the problem-solving process.

## 5. How can understanding boundaries and limit points improve problem-solving?

By understanding boundaries and limit points, a scientist can identify potential challenges and limitations in their problem-solving approach. This can help them develop more effective and efficient solutions and avoid dead ends or ethical dilemmas.

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