Uniformly Star Superparacompact Subsets and Spaces

TL;DR

This paper introduces a new class of subsets called uniformly star superparacompact in metric spaces, characterizes them using finite-component covers, and explores their relationships with other topological properties.

Contribution

It defines and investigates uniformly star superparacompact subsets, establishes their bornology structure, and provides new characterizations and relationships with other metric space properties.

Findings

Collection of such subsets forms a bornology with a closed base.

Conditions when bornologies of uniformly star superparacompact and paracompact subsets coincide.

New characterizations of uniformly star superparacompact metric spaces.

Abstract

Uniformly star superparacompactness, which is a topological property between compactness and completeness, can be characterized using finite-component covers and a measure of strong local compactness. Using these finite-component covers and the associated functional, we introduce and investigate a variational notion of uniformly star superparacompact subsets in metric spaces in the spirit of studies on uniformly paracompact subset and UC-subset. We show that the collection of all such subsets forms a bornology with a closed base, which is contained in the bornology of uniformly paracompact subsets. Conditions under which these two bornologies coincide are specified. Furthermore, we provide several new characterizations of uniformly star superparacompact metric spaces also known as cofinally Bourbaki-quasi complete spaces in terms of some geometric functionals. As a consequence, we establish new relationships among metric spaces that lie between compactness and completeness.