= Čech-complete spaces are Baire spaces
{c}
Write the dense space $X$ as $\bigcap_nG_n$ 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 $\bigcap_n(G_n\cap O_n)=\bigcap_nU_n$ 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.
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