ComparisonOfFilters 2004-10-06 02:21:23 2004-10-07 15:08:42 Definition filter finer coarser strictly finer strictly coarser comparable % 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 Let $\mathbb{F}_1$ and $\mathbb{F}_2$ be two filters on the same set. The following terminology is commonly used to describe the relation of $\mathbb{F}_1$ to $\mathbb{F}_2$: $\mathbb{F}_2$ is said to be \emph{finer} than $\mathbb{F}_1$ if $\mathbb{F}_1 \subseteq \mathbb{F}_2$. $\mathbb{F}_2$ is said to be \emph{coarser} than $\mathbb{F}_1$ if $\mathbb{F}_1 \supseteq \mathbb{F}_2$. $\mathbb{F}_2$ is said to be \emph{strictly finer} than $\mathbb{F}_1$ if $\mathbb{F}_1 \subset \mathbb{F}_2$. $\mathbb{F}_2$ is said to be \emph{strictly coarser} than $\mathbb{F}_1$ if $\mathbb{F}_1 \supset \mathbb{F}_2$. $\mathbb{F}_1$ and $\mathbb{F}_2$ are said to be \emph{comparable} if either $\mathbb{F}_1 \subseteq \mathbb{F}_2$ or $\mathbb{F}_1 \supseteq \mathbb{F}_2$.