# filter

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

• $X\in\mathbb{F}$

• The intersection of any two elements of $\mathbb{F}$ is an element of $\mathbb{F}$.

• $\emptyset\notin\mathbb{F}$ (some authors do not include this axiom in the definition of filter)

• If $F\in\mathbb{F}$ and $F\subset G\subset X$ then $G\in\mathbb{F}$.

The first two axioms can be replaced by one:

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

with the usual understanding that the intersection of an empty family of subsets of $X$ is the whole set $X$.

A filter $\mathbb{F}$ is said to be fixed or principal if there is $F\in\mathbb{F}$ such that no proper subset of $F$ belongs to $\mathbb{F}$. In this case, $\mathbb{F}$ consists of all subsets of $X$ containing $F$, and $F$ is called a principal element of $\mathbb{F}$. If $\mathbb{F}$ 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 $\mathbb{F}$ is any filter on the space $X$, $\mathbb{F}$ is said to converge to $x$, and we write $\mathbb{F}\to x$, if $\mathcal{N}_{x}\subset\mathbb{F}$. If every neighbourhood of $x$ meets every set of $\mathbb{F}$, then $x$ is called an accumulation point or cluster point of $\mathbb{F}$.

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.

 Title filter Canonical name Filter Date of creation 2013-03-22 12:09:06 Last modified on 2013-03-22 12:09:06 Owner Koro (127) Last modified by Koro (127) Numerical id 19 Author Koro (127) Entry type Definition Classification msc 03E99 Classification msc 54A99 Related topic Ultrafilter Related topic KappaComplete Related topic KappaComplete2 Related topic Net Related topic LimitAlongAFilter Related topic UpperSet Related topic OrderIdeal Defines principal filter Defines nonprincipal filter Defines non-principal filter Defines free filter Defines fixed filter Defines neighbourhood filter Defines principal element Defines convergent filter