compactification

Let $X$ be a topological space. A (Hausdorff) compactification of $X$ is a pair $(K,h)$ where $K$ is a Hausdorff topological space and $h:X\to K$ is a continuous function such that

• $K$ is compact

• $h$ is a homeomorphism between $X$ and $h(X)$

• ${\overline{h(X)}}^K=K$ where ${\overline{A}}^K$ denotes closure in $K$ for any subset $A$ of $K$

$h$ is often considered to be the inclusion map, so that $X\subseteq K$ with ${\overline{X}}^K=K$.