Generic 2003-06-11 17:17:59 2003-07-25 11:53:54 Definition generically % 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} \usepackage{mathrsfs} % 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 \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Per}{\operatorname{Per}} \PMlinkescapeword{satisfies} A property that holds for all $x$ in some residual subset of a Baire space $X$ is said to be \emph{generic} in $X$, or to \emph{hold generically} in $X$. In the study of generic properties, it is common to state ``generically, $P(x)$'', where $P(x)$ is some proposition about $x\in X$. The useful fact about generic properties is that, given countably many generic properties $P_n$, all of them hold simultaneously in a residual set, i.e. we have that, generically, $P_n(x)$ holds for each $n$.