Regular completions of Cauchy spaces via function algebras

Abstract

A regular completion with the universal property is obtained for each member of a certain class of Cauchy spaces by embedding the Cauchy space in a complete function algebra with the continuous convergence structure.convergence in C{X) ", Pacific J. Math, (to appear).

On Products of CAUCHY Completions

“…A CAUCHY completion was constructed in 1974 by R. J. GAZIK and one of the authors (see [4]) by embedding a CAUCHY space in its double dual function space, equipped with the continuous CAUCHY structure. The functional properties of this completion were further studied in [2], where it was named the natural completion.…”

“…The resulting CAUCHY space (C" ( X ) , U " ) , (called the dual space of (X, U)) is denoted more concisely by C : ( X ) . It is shown in [4] that C : ( X ) is complete and CAUCHY separated.…”

“…Questions of this type (involving commutativity of extensions relative to products) have been studied for compactifications (e.g., [5], [6] and [15]) as well as other types of topological extensions (e.g., [l], [3] and [7]). In this study, we consider three CAUCHY completions: the WYLER (or fine) completions introduced in [I61 for uniform convergence spaces and in [14] for CAUCHY spaces, the natural completion introduced in [4], and the natural ordered completion introduced in [I 11. All of these completions are defined on categories which are closed under arbitrary products.…”

Completion functors for Cauchy spaces

“…Both of these properties are known to be hereditary and productive, and, in each case, P is a modification functor. In the case where (P) is the C-embedded property, the P-spaces are the C-embedded spaces which were originally introduced and internally characterized in [4]. In the second case, the P-spaces are the sequentially regular which are defined and characterized in [3].…”

“…Background information on Cauchy spaces and Cauchy space completions is available in references [3], [4], and [8]. However a review of this material will be given in this preliminary section.…”