An interesting problem (closure, limit points)

In summary, the conversation discusses finding a nonempty set A that is a subset of R and satisfies the conditions 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. The conversation explores different ideas and possibilities, and eventually concludes that the Cantor set satisfies these properties.
  • #1
Buri
273
0
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??
 
Last edited:
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  • #2
I reckon the Cantor set will do.
 
  • #3
Yes it does. I realized this last night after looonngg hours of thinking lol And I've written up my proof that it does. Thanks a lot!
 

Related to An interesting problem (closure, limit points)

1. What is a closure in mathematics?

A closure in mathematics refers to the set of all points that are either a part of the original set or are limit points of the original set. It is a fundamental concept in topology and is used to describe the behavior of a set of points near its boundary.

2. How is closure related to limit points?

Closure and limit points are closely related in mathematics. A closure contains all of the limit points of a set, and a set is closed if and only if it contains all of its limit points. In other words, the closure of a set is the smallest closed set that contains all of the points in the original set.

3. How do you determine if a point is a limit point of a set?

A point is a limit point of a set if every neighborhood of the point contains at least one point from the set other than the point itself. In other words, no matter how small the neighborhood is, there will always be points from the set within that neighborhood.

4. Can a set have multiple closures?

No, a set can only have one closure. The closure of a set is unique and is the smallest closed set that contains all of the points in the original set.

5. How is closure used in real-world applications?

Closure is a fundamental concept in topology and is used in various fields such as physics, engineering, and computer science. It is used to describe the behavior of a system near its boundary or to determine if a system is stable or not. It is also used in data analysis, where it is used to identify outliers or missing data points.

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