Thread: [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?

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.