= Fibers of compactification comparison maps
{title2=$\pi^{-1}(X)=X$}
If $\pi:\beta X\to K$ is the canonical map from the <Stone-Čech compactification> onto a <Hausdorff compactification>, its fibers over the original points are singletons. If $\pi(z)=x\in X$ and $z\ne x$, choose an open neighborhood $U$ of $x$ in $\beta X$ whose closure excludes $z$. Choose open $V\subseteq K$ with $x\in V$ and $V\cap X\subseteq U$. The open neighborhood $\pi^{-1}V$ of $z$ has dense trace $V\cap X$, placing $z$ in $\overline U$, a contradiction. This controls when images of closed remainder sets stay in the remainder.
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