compact Compact 2001-10-25 12:06:08 2004-03-28 17:51:52 Definition compact set compact subset \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} %\usepackage{graphicx} %\usepackage{xypic} \PMlinkescapeword{term} A topological space $X$ is {\em compact} if, for every collection $\{U_i\}_{i \in I}$ of open sets in $X$ whose union is $X$, there exists a finite subcollection $\{U_{i_j}\}_{j=1}^n$ whose union is also $X$. A subset $Y$ of a topological space $X$ is said to be compact if $Y$ with its subspace topology is a compact topological space. \textbf{Note:} Some authors require that a compact topological space be Hausdorff as well, and use the term quasi-compact to refer to a non-Hausdorff compact space. The modern convention seems to be to use compact in the sense given here, but the old definition is still occasionally encountered (particularly in the French school).