The precise definition depends on what sort of space is, up to the full generality of a Cauchy space.
In a metric space, the diameter? of a subset is the supremum of the distances for (which is a lower real number in general). However, we need not think precisely about these diameters; it is enough to characterise those sets with diameter at most .
In a gauge space, instead of a single number to estimate diameter, we use together with a gauging distance .
A Cauchy filter on a gauge space is a proper filter with, for each gauging distance and each strictly positive number , a set with, for each , .
(It is actually sufficient to consider a base of gauging distances, as well as enough sufficiently small .)
A Cauchy filter on a topological group is a proper filter with, for each neighbourhood of the identity, a set with, for each , (or equivalently, for each , for some , ).
(It is sufficient to consider a neighbourhood base? at the identity. There is no difference between left and right even for nonabelian groups.)
A Cauchy filter on a uniform space is a proper filter with, for each entourage , a set with, for each , (that is, ).
(It is sufficient to consider a base of the uniformity.)
If you want to define uniformities in terms of uniform covers:
A Cauchy filter on a uniform space is a proper filter with, for each uniform cover , a set with .
(It is sufficient to consider a base of uniform covers.)
All of the above have non-symmetric versions: quasimetric spaces, quasigauge spaces, topological monoids, quasiuniform spaces. The definition of Cauchy filter for these is the same, with these caveats: * for a topological monoid, there is a difference between left and right in in Definition 3; * there is no notion of quasiuniform cover to generalise Definition 5.
The most general context is that of a Cauchy space, where the notion of Cauchy filter is axiomatic.
Cauchy filters in all cases above have these properties:
These conditions form the abstract definition of a Cauchy space.
Furthermore, all of these have a notion of convergence given as follows:
In this way, every Cauchy space becomes a convergence space, which agrees with the usual convergence on metric spaces, uniform spaces, etc.
In nonstandard analysis, the hyperpoints of a (quasi)uniform space have a relation of adequality?. A proper filter is Cauchy iff its nonstandard extension contains a hyperset (a collection of hyperpoints) whose elements are all adequal. So compared to the other definitions, a single of infinitesimal diameter suffices.
A hyperpoint is finite? (or limited) if there is a proper filter (necessarily Cauchy) such that contains a hyperset whose elements are all adequal to . In the analogy between hyperpoints and ultrafilters, the finite hyperpoints correspond to the Cauchy ultrafilters.
A map between (quasi)uniform spaces is Cauchy-continuous iff its nonstandard extension has the property that and are adequal whenever and are adequal and is finite. Presumably one can define a Cauchy space in nonstandard analysis by specifying the finite hyperpoints and the relation of adequality only with these (although perhaps not every Cauchy space arises in this way).