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.
denotes the power set of
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
is called a Cauchy filter, and a map
; that is, the image of each Cauchy filter in
is a Cauchy filter base in
Any Cauchy space is also a convergence space, where a filter
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.