A Cauchy space is a generalisation of a metric space with a bare minimum of structure for the concepts of Cauchy sequence, Cauchy-continuous map, and Cauchy completion to make sense. Topologically (that is, up to continuous maps), any Cauchy space is a convergence space, but not much more than that. Besides Cauchy sequences, we can also speak more generally of Cauchy nets and Cauchy filters in a Cauchy space; in fact, the usual definition is in terms of Cauchy filters.
That is, the set of Cauchy filters is a local filter of proper filters that contains all principal ultrafilters (sort of a tongue twister).
The morphisms of Cauchy spaces are the Cauchy-continuous functions; a function between Cauchy spaces is Cauchy-continuous if is a (base of a) Cauchy filter whenever is. In this way, Cauchy spaces form a concrete category .
Any metric space is a Cauchy space: is a Cauchy filter iff it has elements of arbitrarily small diameter. This reconstructs the usual definitions of Cauchy sequence and Cauchy-continuous map for metric spaces. (In particular, a map between metric spaces is Cauchy-continuous iff it maps every Cauchy sequence to a Cauchy sequence; the result for general nets follows since a metric space is sequential.) The forgetful functor from (metric spaces and short maps) to is faithful but not full.
More generally, any uniform space is a Cauchy space: is a Cauchy filter if, given any entourage , for some . This reconstructs the usual definitions of Cauchy net and Cauchy-continuous map for uniform spaces. (In general, we need nets rather than just sequences here.) The forgetful functor from (uniform spaces and uniformly continuous maps) to is faithful but still not full.
Assuming the ultrafilter principle (and excluded middle, which might follow from the ultrafilter principle for all that I know), we may take any collection of free ultrafilters and define a proper filter to be Cauchy if (hence iff) every free ultrafilter that refines belongs to .
Every Cauchy space is a convergence space; if the intersection of with the principal ultrafilter is Cauchy. Note that any convergent proper filter must be Cauchy. Conversely, if every Cauchy filter is convergent, then the Cauchy space is called complete.
The set of Cauchy filters on a Cauchy space has a natural Cauchy structure which is complete and (as a convergence space) preregular; we identify the indistinguishable Cauchy filters to get a Hausdorff space, the Hausdorff completion of the original Cauchy space. The complete Hausdorff Cauchy spaces thus form a reflective subcategory of . This completion agrees with the completion of a metric or uniform space; that is, Cauchy completion, even of a metric space, is an operation on its Cauchy structure only.
Conversely, every Hausdorff convergence space becomes a complete Hausdorff Cauchy space upon declaring that the Cauchy filters are precisely the convergent proper filters. The convergence structure on this Cauchy space matches the original convergence structure.
A Cauchy space is precompact (or totally bounded) if every proper filter is contained in a Cauchy filter. Equivalently (assuming the ultrafilter principle), is precompact iff every ultrafilter is Cauchy. A Cauchy space is compact (as a convergence space) iff it is both complete and precompact. Conversely, it is precompact iff its completion is compact.