SectionFilter 2007-02-12 16:40:33 2007-02-12 16:40:33 Definition filter \usepackage{amssymb,amscd} \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} \usepackage{pst-plot} \usepackage{psfrag} % define commands here Let $X$ be a set and $(x_i)_{i\in D}$ a non-empty net in $X$. For each $j\in D$, define $S(j):=\lbrace x_i\mid i\le j\rbrace$. Then the set $$S:=\lbrace S(j)\mid j\in D\rbrace$$ is a filter basis: $S$ is non-empty because $(x_i)\neq \varnothing$, and for any $j,k\in D$, there is a $\ell$ such that $j\le \ell$ and $k\le \ell$, so that $S(\ell) \subseteq S(j)\cap S(k)$. Let $\mathcal{A}$ be the family of all filters containing $S$. $\mathcal{A}$ is non-empty since the filter generated by $S$ is in $\mathcal{A}$. Order $\mathcal{A}$ by inclusion so that $\mathcal{A}$ is a poset. Any chain $\mathcal{F}_1\subseteq \mathcal{F}_2\subseteq\cdots $ has an upper bound, namely, $$\mathcal{F}:=\bigcup_{i=1}^{\infty} \mathcal{F}_i.$$ By Zorn's lemma, $\mathcal{A}$ has a maximal element $\mathcal{X}$. \textbf{Definition}. $\mathcal{X}$ defined above is called the \emph{section filter} of the net $(x_i)$ in $X$. \textbf{Remark}. A section filter is obviously a filter. The name ``section'' comes from the elements $S(j)$ of $S$, which are sometimes known as ``sections'' of the net $(x_i)$.