 Hausdorff compactification

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.