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

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 Cech theorem about the complete metrizability of a metric space X in terms of its Stone-Cech compactification beta X , which are presented in sections 3 and 4 of the article.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This article is in line with previous research and aims to introduce and study two new properties similar to completeness in uniform spaces. It also explores the topological problem of metrizability of a uniform space using different types of completeness-like properties. The article discusses the use of finite-component covers and their relation to superparacompactness, as well as the metrizability of a uniform space using Bq-complete and cBq-complete metrics.