Let $X$ be a topological space. A (Hausdorff) compactification of $X$ is a pair $(K,h)$ where $K$ is a Hausdorff topological space and $h:X\rightarrow K$ 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 $\overline{X}^K=K$