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:
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. (Compare that is uniformly continuous iff has this property regardless of whether 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).