Exterior and limit points: Quick Question

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Homework Help Overview

The discussion revolves around the relationship between limit points and exterior points in the context of a subset A of a metric space X. The original poster is questioning the validity of the assertion that the exterior of A equals the limit points of A, seeking clarification on whether this assertion is correct and how it relates to proving that limit points form a closed set.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to establish a proof regarding the closure of limit points and questions the truth of their assertion. Some participants challenge this assertion, suggesting that the exterior of A is not simply the complement of the limit points, and they hint at the role of isolated points in this context.

Discussion Status

The discussion is ongoing, with some participants expressing uncertainty and reconsidering their understanding of the concepts involved. Guidance has been offered regarding the nature of isolated points and their implications for the original assertion.

Contextual Notes

The original poster is working within the constraints of a homework problem, which may limit the information they can provide or the depth of their exploration into the topic.

Buri
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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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No, this is false. The exterior of A can be expressed as X minus something, but that "something" is not the set of limit points of A. (As a hint, think about isolated points.)
 
Ahh really? Hmm I'm going to ahve to think about it more then...damnit
 
In R, consider A= [0, 1]\cup \{2\}.
 
Yup, those damn isolated points. Thanks!
 

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