| The natural duality between "regular" and "topological" as convergence space properties extends to the more general properties " p-regular" and "p-topological". This duality is explored for the case of Cauchy spaces and Cauchy completions. Of particular interest are Cauchy completions which are not Hausdorff. As the T2 requirement can not simply be dropped, a weaker requirement, "stable", is introduced for these completions. A generalization of Reed's family of completions for non-Hausdorff spaces is given.; A p-topological Cauchy space has a p-topological, stable completion if and only if it is "cushioned," meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing a p-topological completion, we show that a certain class of Reed completions preserve the p-topological property. This class includes the finest and coarsest p-topological completions, the Wyler and Kowalsky completions, respectively. However, not all p-topological completions are Reed completions. Any Cauchy-continuous map between Cauchy spaces allowing p-topological and p'-topological completions, respectively, can always be extended to a theta-continuous map between any p-topological completion of the first space and any p'-topological completion of the second.; Diagonal axioms are given for both p-regular and p-topological Cauchy spaces. Further diagonal axioms are established for a space to allow p-regular and p-topological Reed completions.; For a p-regular Cauchy space properties are found that are equivalent to the space having a stable p-regular completion and to having a stable, p-regular Reed completion. Further, a p-regular, cushioned space has a p-regular Kowalsky completion if and only if it is relatively round. For a p-regular and p-topological space, it is equivalent to say that it has a stable, p-regular and p-topological completion; that its Kowalsky completion is p-regular and p-topological; that it is cushioned and relatively round; and that it satisfies certain diagonal conditions. A comparison of results obtained for topological and qc-topological completions with results possible for regular and qc-regular completions is made. |
