Let be a set. A filter on is a set of subsets of such that
The intersection of any two elements of is an element of .
(some authors do not include this axiom in the definition of filter)
If and then .
The first two axioms can be replaced by one:
Any finite intersection of elements of is an element of .
with the usual understanding that the intersection of an empty family of subsets of is the whole set .
A filter is said to be fixed or principal if there is such that no proper subset of belongs to . In this case, consists of all subsets of containing , and is called a principal element of . If is not principal, it is said to be non-principal or free.
If is any point (or any subset) of any topological space , the set of neighbourhoods of in is a filter, called the neighbourhood filter of . If is any filter on the space , is said to converge to , and we write , if . If every neighbourhood of meets every set of , then is called an accumulation point or cluster point of .
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.
|Date of creation||2013-03-22 12:09:06|
|Last modified on||2013-03-22 12:09:06|
|Last modified by||Koro (127)|