Hausdorff compactification ($j:X\hookrightarrow K$)

A Hausdorff compactification is a compact Hausdorff space $K$ with a topological embedding $j:X\to K$ whose image is dense. It retains the original topology on $X$ while adjoining boundary points. For a locally compact noncompact Hausdorff space, its one-point compactification is an example. For every Tychonoff space, the Stone-Čech compactification extends all continuous maps into compact Hausdorff targets.

If a locally compact Hausdorff space is densely embedded in a Hausdorff space, its image is open. Around a point choose an ambient open set whose trace lies in a compact neighborhood $K$. Density puts that ambient open set inside the ambient closure of its trace, which lies in $K$ because compact subsets of a Hausdorff space are closed. Thus the ambient open set lies in the image.

For a Hausdorff compactification $h:\mathbb{R}\to Y$, let $C^{\pm }$ be the intersections of the closures of the positive and negative tails, respectively. Each is nonempty and connected by the closed-filter-base theorem. Their union equals the remainder: an image point has a bounded-interval neighbourhood excluding sufficiently long tails, whereas every neighbourhood of a remainder point meets tails by density and compactness of bounded intervals. If the remainder is finite, each $C^{\pm }$ must be a singleton, so there can be only one or two remainder points. In particular there is no three-point compactification of the real line.

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.

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.