normal NormalTopologicalSpace 2002-01-22 12:36:21 2007-05-23 11:06:14 Definition topology % 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 \PMlinkescapeword{normal} A topological space $X$ is said to be \emph{normal} if $X$ is $T_1$ (i.e. singletons are closed), and for all disjoint closed sets $D,F\subseteq X$ there exist disjoint open sets $U$ and $V$ such that $C\subseteq U$ and $D\subseteq V$ (i.e, $X$ is also $T_4$). Some authors do not require the $T_1$ axiom as part of this definition.