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. Definition. Throughout, formula_1 is a set, formula_2 denotes the power set of formula_3 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 formula_4 consisting of a set formula_1 together a family formula_6 of (proper) filters on formula_1 having all of the following properties: An element of formula_21 is called a Cauchy filter, and a map formula_22 between Cauchy spaces formula_4 and formula_24 is Cauchy continuous if formula_25; that is, the image of each Cauchy filter in formula_1 is a Cauchy filter base in formula_27 Properties and definitions. Any Cauchy space is also a convergence space, where a filter formula_14 converges to formula_29 if formula_30 is Cauchy . In particular, a Cauchy space carries a natural topology. Category of Cauchy spaces. 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.