G-delta compactification invariance
= G-delta compactification invariance
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 $\beta X\setminus X=\bigcup_nF_n$ 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.