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

[qspeechc]

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.

 

[tiny-tim]

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!

What's uncountable about it?

n times countable is still countable.

For your open ball, just keep halving the size of the rectangles … that'll do the job, won't it?

 

[qspeechc]

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?

 

[tiny-tim]

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.

 

[qspeechc]

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.