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| Title of object: |
generic |
| Canonical Name: |
Generic |
| Type: |
Definition |
| Created on: |
2003-06-11 17:17:59 |
| Modified on: |
2003-06-11 17:34:31 |
| Classification: |
msc:54E52 |
| Defines: |
generically |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathrsfs}
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%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
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% define commands here
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Per}{\operatorname{Per}} |
Content:
\PMlinkescapeword{satisfies}
If $X$ is a Baire space, we say that a subset $S$ of $X$ is \emph{generic} (in $X$) if $X-S$ is meagre, or equivalently, if $S$ contains a countable intersection of open and dense sets. From the Baire property, we know that a countable intersection of generic sets is generic as well.
\textbf{Remark} If a property holds for a generic subset of $X$, it is said that the property is generic in $X$, or that it \emph{holds generically} in $X$. In the study of generic properties, it is common to state ``generically, $P(x)$'', where $P$ is some proposition about $x\in X$. The useful fact about generic properties is that, if $P$ and $Q$ are two generic properties, then the property ``P and Q'' also holds generically. |
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