On the Regularity of the Kowalsky Completion

Abstract

Cauchy spaces were introduced by Kowalsky in 1954 [9]. In that paper a first completion method for these spaces was given. In 1968 Keller [5] has shown that the Cauchy space axioms characterize the collections of Cauchy filters of uniform convergence spaces in the sense of [1]. Moreover in the completion theory of uniform convergence spaces the associated Cauchy structures play an essential role [12]. This fact explains why in the past ten years in the theory of Cauchy spaces, much attention has been given to the study of completions.

p‐regular Cauchy completions

“…It was shown in [7] that for a regular Cauchy space (X, Ꮿ) and λ ∈ Λ, X * , Ꮿ * λ is a regular completion if and only if Ᏺ ∈ Ꮿ implies s λ Ᏺ ∈ Ꮿ. Combining this result with Proposition 3.4 and Theorem 3.5, we obtain the next corollary.…”

“…In 1984, Colebunders [7], studied conditions under which Kowalsky completion and other Reed completions are regular. We shall apply her approach to the study of pregular Reed completions.…”

A note on Cauchy spaces

“…In the completion theory of uniform convergence spaces and convergence vector spaces, Cauchy spaces play an essential role ( [23], [30], [46]). This fact explains why most work on Cauchy spaces deals mainly with completions ( [21], [22], [33]). …”

T3 AND T4-objects in the topological category of cauchy spaces

“…In the completion theory of uniform convergence spaces and convergence vector spaces, Cauchy spaces play an essential role ( [19], [25], [39]). This fact explain why most work on Cauchy spaces deals mainly with completions ( [17], [18], [29]). Thus, Cauchy spaces form a useful tool for investigating completions.…”