Question: are the rationals a borel set?
Answer: Prove that the set of rational numbers Q is a Borel set in R. Solution: For every x ∈ R, the set {x} is the complement of an open set, and hence Borel. Since there are only countably many rational numbers1, we may express Q as the countable union of Borel sets: Q = ∪x∈Q{x}. Therefore Q is a Borel set.