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) |
Cauchy filter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_filter&oldid=38766