# Cauchy filter

A filter on a uniform space such that for any entourage of the uniform structure of there exists a set which is -small and belongs to . In other words, a Cauchy filter is a filter which contains arbitrarily small sets in a uniform space . The concept is a generalization of the concept of a Cauchy sequence in metric spaces.

Every convergent filter is a Cauchy filter. Every filter which is finer than a Cauchy filter is also a Cauchy filter. The image of a Cauchy filterbase under a uniformly-continuous mapping is again a Cauchy filterbase. A uniform space in which every Cauchy filter is convergent is a complete space.

#### References

[1] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) pp. Chapt. II: Uniform structures (Translated from French) |

#### Comments

A Cauchy filterbase (or Cauchy -filterbase) is a filterbase in a metric space such that for every there is some for which (cf. [a1]).

A filterbase in a space is a family of subsets of with the properties: 1) for all ; and 2) for all there is a such that (see also Filter).

#### References

[a1] | J. Dugundji, "Topology" , Allyn & Bacon (1978) |

**How to Cite This Entry:**

Cauchy filter.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cauchy_filter&oldid=12827