On certain new types of completeness properties using infinite chainability and associated metrization problems in uniform spaces

Abstract

This article is in line with earlier investigations done in [10–12,14,15,18] and several such works. Here our aim is to introduce and study two new completeness-like properties, namely, Bourbaki quasi-completeness and cofinally Bourbaki quasi-completeness (we use infinite chains instead of finite ones), which strictly lie between compactness and completeness, primarily in the setting of uniform spaces. We use the concept of finite-component covers [19] to define a new type of modification of a uniform space, which plays a crucial role throughout the paper. In Section 2, we relate cBq-completeness to the existing notion of superparacompactness [19]. Another significant and very natural problem we deal with is the topological problem of metrizability of a uniform space using a Bq-complete and a cBq-complete metric. We obtain results similar to the classical Čech theorem about the complete metrizability of a metric space X in terms of its Stone-Čech compactification βX, which are presented in sections 3 and 4 of the article.