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,  is a set,  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  consisting of a set  together a family  of (proper) 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 (hence any metric space, topological vector space, or topological group) 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 (so that a Cauchy net indexed by  is the same as a Cauchy sequence), 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.

See also

References

 Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989.
  

General topology