Uniformly Star Superparacompact Subsets and Spaces

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.