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Math Help - [SOLVED] Rationals and countable intersection of open sets

  1. #1
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    [SOLVED] Rationals and countable intersection of open sets

    Is it possible to use Baire's theorem to prove that the set of rational numbers isn't a countable intersection of open sets?
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  2. #2
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    I found the answer in this wikipedia article. Here it is:

    Quote Originally Posted by Wikipedia
    The rational numbers Q are not a Gδ set. If we were able to write Q as the intersection of open sets An, each An would have to be dense in R since Q is dense in R. However, the construction above gave the irrational numbers as a countable intersection of open dense subsets. Taking the intersection of both of these sets gives the empty set as a countable intersection of open dense sets in R, a violation of the Baire category theorem.
    Last edited by JoachimAgrell; February 9th 2010 at 04:28 AM.
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