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.