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=12827