 Čech-complete spaces are Baire spaces

Write the dense space $X$ as ${\bigcap }_n{G}_n$ in compact Hausdorff $K$. Every $G_n$ is dense and open. A dense open subset $U_n$ of $X$ has the form $X\cap O_n$ with $O_n$ dense and open in $K$. The compact Baire category theorem makes ${\bigcap }_n(G_n\cap O_n)={\bigcap }_n{U}_n$ dense in $K$, hence in $X$. Therefore every Čech-complete space is a Baire space. The rationals are not a Baire space, as deleting their points one at a time gives dense open sets with empty intersection.