filter Filter 2002-01-05 08:41:07 2008-05-01 10:23:23 Definition principal filter nonprincipal filter non-principal filter free filter fixed filter neighbourhood filter principal element convergent filter topology set theory \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{fixed} \PMlinkescapeword{non-principal} \PMlinkescapeword{principal} \PMlinkescapeword{free} \newcommand{\F}{\mathbb{F}} Let $X$ be a set. A filter on $X$ is a set $\F$ of subsets of $X$ such that \begin{itemize} \item $X\in\F$ \item The intersection of any two elements of $\F$ is an element of $\F$. \item $\emptyset\notin\F$ (some authors do not include this axiom in the definition of filter) \item If $F\in\F$ and $F\subset G\subset X$ then $G\in\F$. \end{itemize} The first two axioms can be replaced by one: \begin{itemize} \item Any finite intersection of elements of $\F$ is an element of $\F$. \end{itemize} with the usual understanding that the intersection of an empty family of subsets of $X$ is the whole set $X$. A filter $\F$ is said to be \emph{fixed} or \emph{principal} if there is $F\in \F$ such that no proper subset of $F$ belongs to $\F$. In this case, $\F$ consists of all subsets of $X$ containing $F$, and $F$ is called a \emph{principal element} of $\F$. If $\F$ is not principal, it is said to be \emph{non-principal} or \emph{free}. If $x$ is any point (or any subset) of any topological space $X$, the set $\mathcal{N}_x$ of neighbourhoods of $x$ in $X$ is a filter, called the \emph{neighbourhood filter} of $x$. If $\F$ is any filter on the space $X$, $\F$ is said to \emph{converge} to $x$, and we write $\F\to x$, if $\mathcal{N}_x\subset\F$. If every neighbourhood of $x$ meets every set of $\F$, then $x$ is called an \emph{accumulation point} or \emph{cluster point} of $\F$. \textbf{Remarks: } The notion of filter (due to H. Cartan) has a simplifying effect on various proofs in analysis and topology. Tychonoff's theorem would be one example. Also, the two kinds of limit that one sees in elementary real analysis -- the limit of a sequence at infinity, and the limit of a function at a point -- are both special cases of the limit of a filter: the Fr\'echet filter and the neighbourhood filter respectively. The notion of a Cauchy sequence can be extended with no difficulty to any uniform space (but not just a topological space), getting what is called a Cauchy filter; any convergent filter on a uniform space is a Cauchy filter, and if the converse holds then we say that the uniform space is \emph{complete}.