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 Cauchy, then $G$ is Cauchy;
3. Filtered: If $F$ and $G$ are Cauchy 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 filter of proper filters that contains all principal ultrafilters (sort of a tongue twister).

Notice that by this definition, there are two Cauchy structures on the empty set (or rather, one for each truth value), depending on whether the improper filter is Cauchy; on an inhabited set, the improper filter must be Cauchy. Often one sees a definition where Cauchy filters are required to be proper instead.

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 category $\mathrm{Cau}$.

Examples

Any metric space is a Cauchy space: $F$ is a Cauchy filter if 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 if 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.

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 filter must be Cauchy. Conversely, if every Cauchy filter is convergent, then the Cauchy space is called complete.

The set of proper Cauchy filters on a Cauchy space has a natural Cauchy structure which is complete and (as a convergence space) Hausdorff; this is the (Hausdorff) completion of the original Cauchy space. The complete 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 filter is contained in a Cauchy filter. Equivalently (assuming the ultrafilter theorem, a weak form of the axiom of choice), $S$ is precompact iff every ultrafilter is Cauchy. A Cauchy space is compact (as a convergence space) if and only if it is both complete and precompact. Conversely, it is precompact iff its completion is compact.

References

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