I found the answer in this wikipedia article. Here it is:
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; Feb 9th 2010 at 05:28 AM.