Cauchy filter

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

Every convergent filter (cf. Limit) 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 $d$-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 Directed set).

References

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