This monograph, for the first time in book form, considers the large structure of metric spaces as captured by bornologies: families of subsets that contain the singletons, that are stable under finite unions, and that are stable under taking subsets of its members. The largest bornology is the power set of the space and the smallest is the bornology of its finite subsets. Between these lie (among others) the metrically bounded subsets, the relatively compact subsets, the totally bounded subsets, and the Bourbaki bounded subsets.
Classes of functions are intimately connected to various bornologies; e.g., (1) a function is locally Lipschitz if and only if its restriction to each relatively compact subset is Lipschitz; (2) a subset is Bourbaki bounded if and only if each uniformly continuous function on the space is bounded when restricted to the subset. A great deal of attention is given to the variational notions of strong uniform continuity and strong uniform convergence with respect to the members of a bornology, leading to the bornology of UC-subsets and UC-spaces. Spaces on which its uniformly continuous real-valued functions are stable under pointwise product are characterized in terms of the coincidence of the Bourbaki bounded subsets with a usually larger bornology.
Special attention is given to Lipschitz and locally Lipschitz functions. For example, uniformly dense subclasses of locally Lipschitz functions within the real-valued continuous functions, Cauchy continuous functions, and uniformly continuous functions are presented. It is shown very generally that a function between metric spaces has a particular metric property if and only if whenever it is followed in a composition by a real-valued Lipschitz function, the composition has the property. Bornological convergence of nets of closed subsets, having Attouch-Wets convergence as a prototype, is considered in detail. Topologies of uniform convergence for continuous linear operators between normed spaces is explained in terms of the bornological convergence of their graphs. Finally, the idea of a bornological extension of a topological space is presented, and all regular extensions can be so realized.
Author(s): Gerald Beer
Publisher: CRC Press/Science Publishers
Year: 2023
Language: English
Pages: 242
City: Boca Raton
Cover
Title Page
Copyright Page
Preface
Table of Contents
List of Symbols
Introduction
1. Background Material
2. Continuous Functions on Metric Spaces
3. Extension of Real-Valued Continuous Functions on Subsets of a Metric Space
4. The Lipschitz Norm for the Vector Space of Lipschitz Real-Valued Functions
5. Nets and Uniformities
6. Some Basic Bornologies
7. Total Boundedness Revisited and Bourbaki Boundedness
8. Locally Lipschitz Functions
9. Common Sets of Boundedness for Classes of Continuous Functions
10. Hejcman’s Theorem and its Analog for Totally Bounded Subsets
11. General Constructions
12. Properties of Bornologies
13. Approximation by Members of a Bornology
14. Selected Topological Properties of the One-Point Extension
15. Bornologies of Metrically Bounded Sets
16. Bornologies of Totally Bounded Sets
17. Strong Uniform Continuity
18. UC-Subsets
19. UC-Spaces
20. Pointwise Products of Uniformly Continuous Real-Valued Functions
21. Strong Uniform Convergence on Bornologies
22. Uniform Convergence on Totally Bounded Subsets
23. Where Must Each Member of a Class of Locally Lipschitz Functions be Lipschitz?
24. Real-Valued Lipschitz Functions and Classes of Locally Lipschitz Functions
25. Metrically Convex Spaces and Coarse Maps
26. Some Density Results
27. More on our Four Classes of Locally Lipschitz Functions
28. Real-Valued Functionals and Bornologies
29. Uniformly Paracompact Subsets
30. Uniformly Paracompact Spaces and Uniformly Locally Lipschitz Functions
31. Bornological Convergence of Nets of Closed Subsets
32. Attouch-Wets Convergence
33. Topologies of Uniform Convergence on B(X, Y) and Convergence of Graphs
34. Bornological Convergence and Uniform Convergence of Distance Functionals
35. Bornological Convergence with Respect to the Compact Bornology
36. When is Bornological Convergence Topological?
37. Uniformizability and Metrizability
38. Ideals, Bornologies and Extensions
39. When is an Extension Bornological?
References
Index
