residual Residual 2002-09-27 19:50:15 2005-10-22 23:42:24 Definition comeager % 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 A subspace~$A$ of a topological space~$X$ is called {\em residual} (or {\em comeager}) if and only if it is second category and its complement $X\setminus A$ is first category. Equivalently, a set is residual if and only if it contains a countable intersection of \PMlinkname{open}{OpenSet} dense sets.