FilterBasis 2004-10-06 00:19:35 2008-01-02 01:33:53 Definition filter filter subbasis equivalent % 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 \emph{filter subbasis} for a set $S$ is a collection of subsets of $S$ which has the finite intersection property. A \emph{filter basis} $B$ for a set $S$ is a non-empty collection of subsets of $S$ which does not contain the empty set such that, for every $u \in B$ and every $v \in B$, there exists a $w \in B$ such that $w \subset u \cap v$. Given a filter basis $B$ for a set $S$, the set of all supersets of elements of $B$ forms a filter on the set $S$. This filter is known as the filter generated by the basis. Given a filter subbasis $B$ for a set $S$, the set of all supersets of finite intersections of elements of $B$ is a filter. This filter is known as the filter generated by the subbasis. Two filter bases are said to be \emph{equivalent} if they generate the same filter. Likewise, two filter subbases are said to be equivalent if they generate the same filter. \textbf{Note:} Not every author requires that filters do not contain the empty set. Because every filter is a filter basis then accordingly some authors allow that a filter base can contain the empty set.