filter

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 $𝔽$.

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

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

The first two axioms can be replaced by one:

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

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 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 ${𝒩}_x$ 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 ${𝒩}_x\subset 𝔽$. 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.

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