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 Complete [metric space|completeness] in topological spaces. The category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces.

Definition

Let be a set, the power set of, and assume all filters are proper.
A Cauchy space is a pair consisting of a set together with a family of filters on having all of the following properties:

For each the discrete ultrafilter at denoted by is in
If is a proper filter, and is a subset of then
If and if each member of intersects each member of then

An element of is called a Cauchy filter, and a map between Cauchy spaces and is Cauchy continuous if ; that is, the image of each Cauchy filter in is a Cauchy filter base in

Properties and definitions

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

Examples

Any uniform space is a Cauchy space; see Cauchy filter for definitions.
A lattice-ordered group carries a natural Cauchy structure.
Any directed set may be made into a Cauchy space by declaring a filter to be Cauchy if, given any element there is an element such that is either a singleton or a subset of the tail Then given any other Cauchy space the Cauchy-continuous functions from to are the same as the Cauchy nets in indexed by If is complete, then such a function may be extended to the completion of which may be written the value of the extension at will be the limit of the net. In the case where is the set of natural numbers, then receives the same Cauchy structure as the metric space

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.