compactification Compactification 2002-02-02 10:45:26 2003-09-04 21:14:08 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \begin{itemize} \item $K$ is compact \item $h$ is a homeomorphism between $X$ and $h(X)$ \item $\overline{h(X)}^K=K$ where $\overline{A}^K$ denotes closure in $K$ for any subset $A$ of $K$ \end{itemize} $h$ is often considered to be the inclusion map, so that $X\subseteq K$ with $\overline{X}^K=K$.