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A filter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c0208601.png" /> on a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c0208602.png" /> such that for any entourage <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c0208603.png" /> of the uniform structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c0208604.png" /> there exists a set which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c0208605.png" />-small and belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c0208606.png" />. In other words, a Cauchy filter is a filter which contains arbitrarily small sets in a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c0208607.png" />. The concept is a generalization of the concept of a Cauchy sequence in metric spaces.
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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 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.
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  pp. Chapt. II: Uniform structures  (Translated from French)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  pp. Chapt. II: Uniform structures  (Translated from French)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
A Cauchy filterbase (or Cauchy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c0208609.png" />-filterbase) is a filterbase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086010.png" /> in a [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086011.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086012.png" /> there is some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086013.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086014.png" /> (cf. [[#References|[a1]]]).
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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. [[#References|[a1]]]).
  
A filterbase in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086015.png" /> is a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086016.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086017.png" /> with the properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086019.png" />; and 2) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086020.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020860/c02086022.png" /> (see also [[Filter|Filter]]).
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  "Topology" , Allyn &amp; Bacon  (1978)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  "Topology" , Allyn &amp; Bacon  (1978)</TD></TR>
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</table>
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Latest revision as of 21:05, 2 May 2016

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://encyclopediaofmath.org/index.php?title=Cauchy_filter&oldid=12827
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article