= Hausdorff compactification
{title2=$j:X\hookrightarrow K$}
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.
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