View source for 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====
<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>



====Comments====
A Cauchy filterbase (or Cauchy $d$-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]]]).

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 [[Directed set]]).

====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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