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How would one go about proving that the rationals are not the countable intersection of open sets?
As the topic, prove that set $S$ of rational numbers in the interval (0,1), cannot be expressed as the intersection of a countable collection of open sets.
This follows immediately from the Baire category theorem. Suppose that $\Bbb Q\cap(0,1)=\bigcap_{n\in\Bbb N}U_n$, where each $U_n$ is open. For each $q\in\Bbb Q\cap(0,1)$ let $V_q=(0,1)\setminus\{q\}$. Then $$\{U_n:n\in\Bbb N\}\cup\{V_q:q\in\Bbb Q\cap(0,1)\}$$ is a countable family of dense open subsets of $(0,1)$ whose intersection is empty. Since $(0,1)$ is locally compact, however, the Baire category theorem ensures that the intersection of a countable family of dense open sets is dense in $(0,1)$.
