Čech-complete space

A Tychonoff space is Čech-complete if it is a G-delta set in a Hausdorff compactification. The G-delta compactification invariance makes this independent of the chosen compactification, and equivalent to being a G-delta set in its Stone-Čech compactification.

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.

For the comparison map $\pi :\beta X\to K$, the fibers of compactification comparison maps identity ${\pi }^{-1}(X)=X$ pulls a G-delta set presentation of $X$ back from $K$. Conversely if $\beta X\setminus X={\bigcup }_n{F}_n$ with $F_n$ closed compact, then $K\setminus X={\bigcup }_n\pi (F_n)$. Each image is closed and misses $X$. Thus $X$ is a G-delta set in one Hausdorff compactification exactly when it is in every one.