Open Sets of R^n: Countable Union of Open Rectangles/Balls?

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Discussion Overview

The discussion revolves around the characterization of open sets in R^n, specifically whether any open set can be expressed as a countable union of open rectangles or balls. Participants explore the implications of this idea and its connection to known theorems, while expressing varying degrees of understanding and skepticism.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant notes that open sets in the real line are countable unions of disjoint open intervals and questions if this extends to R^n.
  • Another participant suggests that the countable union of open rectangles can be used to describe open sets in R^n, questioning the plausibility of this for open balls.
  • A different participant challenges the idea of transitioning from R to R^n without difficulty, seeking clarification.
  • One participant mentions finding a proof that relies on the density of the rationals in the reals, indicating a different approach to the problem.

Areas of Agreement / Disagreement

Participants express differing views on the transition from the real line to R^n, with some asserting it is straightforward while others remain skeptical. The discussion does not reach a consensus on the characterization of open sets in R^n.

Contextual Notes

Participants reference the Heine-Borel theorem and the density of rationals, but the implications of these concepts in the context of the discussion remain unresolved.

qspeechc
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Hi

Ok, so I know open sets of the real line are the countable union of disjoint open intervals (or open balls). Does this in any way extent to R^n? Say, any open set in R^n is the countable union of open rectangles or balls? I ask because I was reading some proof, and at a crucial step they use the fact that open sets in R^n can be expressed as the countable union of open rectangles, and I have no idea where this comes from! It doesn't even seem plausible to me, if one considers the open ball in the plane-- how would you describe that as the union of countably many open rectangles?
I know every open set is the union of open balls, and maybe the Heine-Borel theorem comes in somewhere...but I'm just lost.
Any help?
Thanks.
 
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qspeechc said:
I ask because I was reading some proof, and at a crucial step they use the fact that open sets in R^n can be expressed as the countable union of open rectangles, and I have no idea where this comes from! It doesn't even seem plausible to me, if one considers the open ball in the plane-- how would you describe that as the union of countably many open rectangles?

Hi qspeechc! :smile:

What's uncountable about it?

n times countable is still countable. :wink:

For your open ball, just keep halving the size of the rectangles … that'll do the job, won't it? :smile:
 
Errr...I'm not sure I follow you...? Are you saying I can go from the case of the real line to R^n with no difficulty?
 
qspeechc said:
Errr...I'm not sure I follow you...? Are you saying I can go from the case of the real line to R^n with no difficulty?

Yes. :smile:
 
Hhmm, actually I found a proof, and it has nothing to do with the case of the real line, lol. It has to do with the fact that the rationals are dense in the reals.
 

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