nLab
Cauchy space

Idea

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.

Definitions

A Cauchy space is a set $S$ together with a collection of proper filters declared to be Cauchy filters. These must satisfy axioms:

1. Centred: The principal ultrafilter $F_x=\{A\mid x\in A\}$ at $x$ is Cauchy;
2. Isotone: If $F\subseteq G$ and $F$ is a Cauchy filter (and $G$ is at least a proper filter), then $G$ is Cauchy;
3. Locally filtered: If $F$ and $G$ are Cauchy filters and $F\vee G$ (the filter generated by $F\cup G$) is proper, then $F\cap G$ is Cauchy.

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 definition can also be phrased in terms of nets; a Cauchy net is a net whose eventuality filter is Cauchy. In particular, a Cauchy sequence is a sequence whose eventuality filter is Cauchy.

The morphisms of Cauchy spaces are the Cauchy-continuous functions; a function $f$ between Cauchy spaces is Cauchy-continuous if $f(F)$ is a (base of a) Cauchy filter whenever $F$ is. In this way, Cauchy spaces form a concrete category $\mathrm{Cau}$.

Examples

Any metric space is a Cauchy space: $F$ 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 $\mathrm{Met}$ (metric spaces and short maps) to $\mathrm{Cau}$ is faithful but not full.

More generally, any uniform space is a Cauchy space: $F$ is a Cauchy filter if, given any entourage $U$, $A\times A\subseteq U$ for some $A\in F$. 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 $\mathrm{Unif}$ (uniform spaces and uniformly continuous maps) to $\mathrm{Cau}$ 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 $U$ of free ultrafilters and define a proper filter $F$ to be Cauchy if (hence iff) every free ultrafilter that refines $F$ belongs to $U$.

Properties

Every Cauchy space is a convergence space; $F\to x$ if the intersection of $F$ with the principal ultrafilter $F_x$ 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 $\mathrm{Cau}$. 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.

A Cauchy space $S$ is precompact (or totally bounded) if every proper filter is contained in a Cauchy filter. Equivalently (assuming the ultrafilter principle), $S$ 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.

References

• Eva Lowen-Colebunder, Function Classes of Cauchy Continuous Maps Dekker, New York, 1989