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Open Sets of R^n

  1. May 18, 2009 #1
    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.
     
  2. jcsd
  3. May 18, 2009 #2

    tiny-tim

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    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:
     
  4. May 18, 2009 #3
    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?
     
  5. May 18, 2009 #4

    tiny-tim

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    Yes. :smile:
     
  6. May 19, 2009 #5
    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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