p-topological Cauchy completions

• Main Authors: J. Wig, D. C. Kent
• Format: Article
• Language: English
• Published: Hindawi Limited 1999-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: p-topological completion; strict completion; Wyler completion; Kowalsky completion; Reed completion; θ-continuous map.
• Online Access: http://dx.doi.org/10.1155/S0161171299224970
• ISSN: 0161-1712; 1687-0425

The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “p-regular” and “p-topological.” Since earlier papers have investigated regular, p-regular, and topological Cauchy completions, we hereby initiate a study of p-topological Cauchy completions. A p-topological Cauchy space has a p-topological 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, it is shown that a certain class of Reed completions preserve the p-topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsest p-topological completions. However, not all p-topological completions are Reed completions. Several extension theorems for p-topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowing p-topological and p′-topological completions, respectively, can always be extended to a θ-continuous map between any p-topological completion of the first space and any p′-topological completion of the second.