Editing Filter (mathematics) (section)

==In topology==
{{Main|Filters in topology}}
In [[general topology]] and analysis, filters are used to define convergence in a manner similar to the role of [[sequence]]s in a [[metric space]].  They unify the concept of a [[Limit (mathematics)|limit]] across the wide variety of arbitrary [[topological space]]s.

To understand the need for filters, begin with the equivalent concept of a [[Net (mathematics)|net]].   A [[sequence]] is usually indexed by the [[natural numbers]]&nbsp;{{Math|{{mathbb|N}}}}, which are a [[totally ordered set]]. Nets generalize the notion of a sequence by replacing {{Math|{{mathbb|N}}}} with an arbitrary [[directed set]]. In certain categories of topological spaces, such as [[first-countable space]]s, sequences characterize most topological properties, but this is not true in general.  However, nets&nbsp;— as well as filters&nbsp;— always do characterize those topological properties.

Filters do not involve any set external to the topological space&nbsp;{{Mvar|X}}, whereas sequences and nets rely on other directed sets.  For this reason, the collection of all filters on {{Mvar|X}} is always a [[Set (mathematics)|set]], whereas the collection of all {{Mvar|X}}-valued nets is a [[proper class]].

=== Neighborhood bases ===
Any point&nbsp;{{Mvar|x}} in the topological space&nbsp;{{Mvar|X}} defines a [[Neighbourhood system|neighborhood filter or system]]&nbsp;{{Math|{{mathcal|N}}<sub>''x''</sub>}}: namely, the family of all sets containing {{Mvar|x}} in their [[Interior (topology)|interior]].  A set&nbsp;{{Math|{{mathcal|N}}}} of neighborhoods of {{Mvar|x}} is a [[neighbourhood base|neighborhood base]] at {{Mvar|x}} if {{Math|{{mathcal|N}}}} generates {{Math|{{mathcal|N}}<sub>''x''</sub>}}.  Equivalently, {{Math|''S'' &subseteq; ''X''}} is a neighborhood of {{Mvar|x}} if and only if there exists {{Math|''N'' &isin; {{mathcal|N}}}} such that {{Math|''N'' &subseteq; ''S''}}.

==== Convergent filters and cluster points ====
A prefilter&nbsp;{{Mvar|B}} [[Convergent prefilter|converges]] to a point&nbsp;{{Mvar|x}}, written {{Math|''B'' &rarr; ''x''}}, if and only if {{Mvar|B}} generates a filter&nbsp;{{Mvar|F}} that contains the neighborhood filter {{Math|{{mathcal|N}}<sub>''x''</sub>}}&nbsp;— explicitly, for every neighborhood&nbsp;{{Mvar|U}} of {{Mvar|x}}, there is some {{Math|''V'' &isin; ''B''}} such that {{Math|''V'' &subseteq; ''U''}}.  Less explicitly, {{Math|''B'' &rarr; ''x''}} if and only if {{Mvar|B}} refines {{Math|{{mathcal|N}}<sub>''x''</sub>}}, and any neighborhood base at {{Mvar|x}} can replace {{Math|{{mathcal|N}}<sub>''x''</sub>}} in this condition.  Clearly, every [[neighbourhood base|neighborhood base]] at {{Mvar|x}} converges to {{Mvar|x}}.

A filter&nbsp;{{Mvar|F}} (which generates itself) converges to {{Mvar|x}} if {{Math|{{mathcal|N}}<sub>''x''</sub> &subseteq; ''F''}}.  The above can also be reversed to characterize the neighborhood filter {{Math|{{mathcal|N}}<sub>''x''</sub>}}: {{Math|{{mathcal|N}}<sub>''x''</sub>}} is the finest filter coarser than each filter converging to {{Mvar|x}}.

If {{Math|''B'' &rarr; ''x''}}, then {{Mvar|x}} is called a [[Limit of a filter|limit]] (point) of {{Mvar|B}}.  The prefilter {{Mvar|B}} is said to cluster at {{Mvar|x}} (or have {{Mvar|x}} as a [[Cluster point of a filter|cluster point]]) if and only if each element of {{Mvar|B}} has non-empty intersection with each neighborhood of {{Mvar|x}}.  Every limit point is a cluster point but the converse is not true in general. However, every cluster point of an {{em|ultra}}filter is a limit point.