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Cauchy filter

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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 $\mathfrak{A} = \{ A_\alpha : \alpha \in \mathcal{A} \}$ in a metric space $(X,d)$ such that for every $\epsilon > 0$ there is some $\alpha \in \mathcal{A}$ for which $\text{diam}\,A_\alpha < \epsilon$ (cf. [a1]).

A filterbase in a space $X$ is a family $\{ A_\alpha : \alpha \in \mathcal{A} \}$ of subsets of $X$ with the properties: 1) $A_\alpha \neq \emptyset$ for all $\alpha \in \mathcal{A}$; and 2) for all $\alpha,\beta \in \mathcal{A}$ there is a $\gamma \in \mathcal{A}$ such that $A_\gamma \subseteq A_\alpha \cap A_\beta$ (see also Directed set).

References

[a1] J. Dugundji, "Topology" , Allyn & Bacon (1978)
How to Cite This Entry:
Cauchy filter. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Cauchy_filter&oldid=38771
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article