Uniformly MetriBornologies: A Unified Characterization and Applications to Fundamental Bornologies in Metric Spaces

Hasan Khaled  AL omar; Zahir Dobeas  AL-Nafie

doi:10.31642/JoKMC/2018/130107

Authors

Hasan Khaled AL omar university of babylon
Zahir Dobeas AL-Nafie Departement of Mathematics,college of Education of Pure Sciene, University of Babylon

DOI:

https://doi.org/10.31642/JoKMC/2018/130107

Keywords:

Bornology, Uniform Metrizability, Characteristic Function, Compact Bornology, Totally Bounded Bornology, Bourbaki-Bounded Set, Heine-Borel Property.

Abstract

This paper presents a study of bornological structures within metric spaces, with a central focus on the property of being a "uniform metric" bornology. The primary objectives are to characterize such bornologies and to establish necessary and sufficient conditions for this property to hold for several fundamental types. We provide a characterization theorem for uniform metric bornologies and apply this framework to analyze specific cases, including the compact, totally bounded, and Bourbaki-bornological structures. Furthermore, we demonstrate that several previously known results emerge as direct corollaries of our general theorems. The analysis employs tools from mathematical analysis and topology, utilizing the construction of characteristic functions alongside concepts of metric extensions and uniform restrictions. This work yields a deeper theoretical understanding of uniform metric bornologies and provides applicable tools for their investigation in varied contexts. The results are subsequently applied to the study of metric spaces possessing special properties, such as locally bounded and complete spaces.

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