filter
Let $X$ be a set. A filter on $X$ is a set $\mathrm{\pi \x9d\x94\xbd}$ of subsets of $X$ such that

β’
$X\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$

β’
The intersection^{} of any two elements of $\mathrm{\pi \x9d\x94\xbd}$ is an element of $\mathrm{\pi \x9d\x94\xbd}$.

β’
$\mathrm{\beta \x88\x85}\beta \x88\x89\mathrm{\pi \x9d\x94\xbd}$ (some authors do not include this axiom in the definition of filter)

β’
If $F\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$ and $F\beta \x8a\x82G\beta \x8a\x82X$ then $G\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$.
The first two axioms can be replaced by one:

β’
Any finite intersection of elements of $\mathrm{\pi \x9d\x94\xbd}$ is an element of $\mathrm{\pi \x9d\x94\xbd}$.
with the usual understanding that the intersection of an empty family of subsets of $X$ is the whole set $X$.
A filter $\mathrm{\pi \x9d\x94\xbd}$ is said to be fixed or principal if there is $F\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$ such that no proper subset^{} of $F$ belongs to $\mathrm{\pi \x9d\x94\xbd}$. In this case, $\mathrm{\pi \x9d\x94\xbd}$ consists of all subsets of $X$ containing $F$, and $F$ is called a principal element of $\mathrm{\pi \x9d\x94\xbd}$. If $\mathrm{\pi \x9d\x94\xbd}$ is not principal, it is said to be nonprincipal or free.
If $x$ is any point (or any subset) of any topological space^{} $X$, the set ${\mathrm{\pi \x9d\x92\copyright}}_{x}$ of neighbourhoods of $x$ in $X$ is a filter, called the neighbourhood filter of $x$. If $\mathrm{\pi \x9d\x94\xbd}$ is any filter on the space $X$, $\mathrm{\pi \x9d\x94\xbd}$ is said to converge^{} to $x$, and we write $\mathrm{\pi \x9d\x94\xbd}\beta \x86\x92x$, if ${\mathrm{\pi \x9d\x92\copyright}}_{x}\beta \x8a\x82\mathrm{\pi \x9d\x94\xbd}$. If every neighbourhood of $x$ meets every set of $\mathrm{\pi \x9d\x94\xbd}$, then $x$ is called an accumulation point^{} or cluster point of $\mathrm{\pi \x9d\x94\xbd}$.
Remarks: The notion of filter (due to H. Cartan) has a simplifying effect on various proofs in analysis^{} and topology. Tychonoff^{}βs theorem would be one example. Also, the two kinds of limit that one sees in elementary real analysis β the limit of a sequence at infinity^{}, and the limit of a function at a point β are both special cases of the limit of a filter: the FrΓ©chet 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  20130322 12:09:06 
Last modified on  20130322 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  nonprincipal filter 
Defines  free filter 
Defines  fixed filter 
Defines  neighbourhood filter 
Defines  principal element 
Defines  convergent filter 