Concept

Cauchy space

Résumé
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 of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces. Definition Throughout, X is a set, \wp(X) denotes the power set of X, and all filters are assumed to be proper/non-degenerate (i.e. a filter may not contain the empty set). A Cauchy space is a pair (X, C) consisting of a set X together a family C \subseteq \wp(\wp(X)) of (proper) filters on X having all of the following properties:

For each x \in X, the discrete ultrafilter at x, denoted by U(x),

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