**Filters in Topology**

In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space.

In topology and related areas of mathematics, a filter is a generalization of a net. Both nets and filters provide very general contexts to unify the various notions of limit to arbitrary topological spaces.

A sequence is usually indexed by the natural numbers, which are a totally ordered set. Thus, limits in first-countable spaces can be described by sequences. However, if the space is not first-countable, nets or filters must be used. Nets generalize the notion of a sequence by requiring the index set simply be a directed set. Filters can be thought of as sets built from multiple nets. Therefore, both the limit of a filter and the limit of a net are conceptually the same as the limit of a sequence.

Read more about this topic: Filter (mathematics), Filter On A Set

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### Famous quotes containing the word filters:

“Raise a million *filters* and the rain will not be clean, until the longing for it be refined in deep confession. And still we hear, If only this nation had a soul, or, Let us change the way we trade, or, Let us be proud of our region.”

—Leonard Cohen (b. 1934)