What is cauchy filter?

Filter (mathematics) - Filter On A Set - Filters in Topology - Cauchy Filters
... To say that a filter base B on X is Cauchy means that for each real number ε>0, there is a B0 ∈ B such that the metric diameter of B0 is less than ε ... (xn) is a Cauchy sequence if and only if the filter base {{xN,xN+1...} N ∈ {1,2,3...} } is Cauchy ... More generally, given a uniform space X, a filter F on X is called Cauchy filter if for every entourage U there is an A ∈ F with (x,y) ∈ U for all x,y ∈ A ...

Uniform Space - Completeness
... Instead of working with Cauchy sequences, one works with Cauchy filters (or Cauchy nets) ... A Cauchy filter F on a uniform space X is a filter F such that for every entourage U, there exists A∈F with A×A ⊆ U ... In other words, a filter is Cauchy if it contains "arbitrarily small" sets ...

Cauchy Space
... In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense ... Cauchy spaces were introduced by H ... Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces ...