Let $X$ be a set. A filter on $X$ is a set $\F$ of subsets of $X$ such that

• $X\in\F$
• The intersection of any two elements of $\F$ is an element of $\F$ .
• $\emptyset\notin\F$ (some authors do not include this axiom in the definition of filter)
• If $F\in\F$ and $F\subset G\subset X$ then $G\in\F$ .

• Any finite intersection of elements of $\F$ is an element of $\F$ .

A filter $\F$ is said to be fixed or 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 principal element of $\F$ . If $\F$ is not principal, it is said to be non-principal or 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 neighbourhood filter of $x$ . If $\F$ is any filter on the space $X$ , $\F$ is said to 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 accumulation point or cluster point of $\F$ .

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 Fréchet 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 complete.