nLab
Cauchy space

Cauchy spaces

Idea

A Cauchy space is a generalisation of a metric space with a bare minimum of structure for the concepts of Cauchy net(A generalization 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 SS together with a collection of proper filters declared to be Cauchy filters. These must satisfy axioms:

  1. Centred: The principal ultrafilter F x={AxA}F_x = \{ A \;|\; x \in A \} at xx is Cauchy;
  2. Isotone: If FGF \subseteq G and FF is a Cauchy filter (and GG is at least a proper filter), then GG is Cauchy;
  3. Locally filtered: If FF and GG are Cauchy filters and FGF \vee G (the filter generated by FGF \cup G) is proper, then FGF \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 ff between Cauchy spaces is Cauchy-continuous if f(F)f(F) is a (base of a) Cauchy filter whenever FF is. In this way, Cauchy spaces form a concrete category CauCau.

Examples

Any metric space is a Cauchy space: FF 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 MetMet (metric spaces and short maps) to CauCau is faithful but not full.

More generally, any uniform space is a Cauchy space: FF is a Cauchy filter if, given any entourage UU, A×AUA \times A \subseteq U for some AFA \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 UnifUnif (uniform spaces and uniformly continuous maps) to CauCau 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 UU of free ultrafilters and define a proper filter FF to be Cauchy if (hence iff) every free ultrafilter that refines FF belongs to UU.

Properties

Every Cauchy space is a convergence space; FxF \to x if the intersection of FF with the principal ultrafilter F xF_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 CauCau. 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 SS is precompact (or totally bounded) if every proper filter is contained in a Cauchy filter. Equivalently (assuming the ultrafilter principle), SS 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

Revised on September 26, 2013 21:31:56 by Anonymous Coward (88.104.175.102)