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

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$K$ is compact^{}

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

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${\overline{h(X)}}^{K}=K$ 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 

Canonical name  Compactification 
Date of creation  20130322 12:15:42 
Last modified on  20130322 12:15:42 
Owner  Evandar (27) 
Last modified by  Evandar (27) 
Numerical id  8 
Author  Evandar (27) 
Entry type  Definition 
Classification  msc 54D35 
Synonym  Hausdorff compactification 
Related topic  Compact 
Related topic  AlexandrovOnePointCompactification 