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. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. The category of Cauchy spaces and Cauchy continuous maps is cartesian closed, and contains the category of proximity spaces.

A Cauchy space is a set X and a collection C of proper filters in the power set P(X) such that

An element of C is called a Cauchy filter, and a map f between Cauchy spaces (X,C) and (Y,D) is Cauchy continuous if f(C)⊆D; that is, each the image of each Cauchy filter in X is Cauchy in Y.

Any Cauchy space is also a convergence space, where a filter F converges to x if F∩U(x) is Cauchy. In particular, a Cauchy space carries a natural topology.

The natural notion of morphism between Cauchy spaces is that of a Cauchy-continuous function, a concept that had earlier been studied for uniform spaces.

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