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 filter in the power set P such that
  1. for each x in X, the ultrafilter at x, U, is in C.
  2. if F is in C, G is a proper filter, and F is a subset of G, then G is in C.
  3. if F and G are in C and each member of F intersects each member of G, then F ∩ G is in C.
An element of C is called a Cauchy filter, and a map f between Cauchy spaces and is Cauchy continuous if f ⊆ D; that is, the image of each Cauchy filter in X is a Cauchy filter base in Y.

Properties and definitions

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

Examples

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.