Open sets in R^2: Are open sets contained in their closures?

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Homework Help Overview

The discussion revolves around the properties of open sets in R^2, specifically examining whether an open set A contained in the closure of another open set B implies that A is also contained in B.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to reason through the relationship between open sets and their closures, suggesting that if a point in A can be approximated by points in B, then A might be contained in B. However, this reasoning is questioned by others who seek counterexamples.

Discussion Status

Participants are exploring different interpretations of the problem. Some express skepticism about the original poster's intuition, while others are actively seeking counterexamples to challenge the assertion. There is no explicit consensus yet, but the discussion is generating productive questioning.

Contextual Notes

Participants mention specific examples, such as B being R^2 minus the origin, to illustrate their points and question the assumptions involved in the original statement.

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[SOLVED] open sets in R^2

Homework Statement


If A, B are open in R^2, and A is contained in the closure(B), then is it true that A is contained in B?

Homework Equations


The Attempt at a Solution


It seems intuitively true.

If a is in A, then there is a sequence b_n in B that converges to a by definition of closure. But I don't know what to do with that?
 
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It doesn't seem intuitively true to me. Take B=R^2-{(0,0)}. Give me a counterexample.
 
Last edited:
Dick said:
It doesn't seem intuitively true to me. Take B=R^2-{(0,0)}. Give me a counterexample.

any open set containing the origin
 
Yep.
 

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