# 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 $h:X\rightarrow K$ 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 $\overline{X}^{K}=K$.

Title compactification Compactification 2013-03-22 12:15:42 2013-03-22 12:15:42 Evandar (27) Evandar (27) 8 Evandar (27) Definition msc 54D35 Hausdorff compactification Compact AlexandrovOnePointCompactification