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Bornologies are a useful tool in studying problems in mathematical analysis and general topology. For example, they can be used to characterize those metric spaces on which the real-valued uniformly continuous functions are stable under pointwise products.
List of contents
Background material. Continuous functions on metric spaces. Extension of real-valued continuous functions on subsets of a metric space. The Arens-Eells theorem and the Hausdorff extension theorem. Nets and uniformities. Some basic bornologies. Total boundedness revisited and Bourbaki boundedness. Localy Lipschitz functions. Common sets of boundedness for classes of continuous functions. Hejcman's theorem and its analog for totally bounded subsets. General constructions with bornologies. Properties of bornologies. Approximation by members of a bornology. Selected topological properties of the one-point extension. Bornologies of metrically bounded sets. Bornologies of totally bounded sets. Strong uniform continuity. UC-subsets. UC-spaces. Pointwise products of uniformly continuous real-valued functions. Strong uniform convergence on bornologies. Uniform convergence on totally bounded subsets. Where must each member of a class of locally Lipschitz functions be Lipschitz? Real-valued Lipschitz functions and classes of locally Lipschitz functions. Coarse maps on metrically convex spaces. Some density results. More on our 4 classes of locally Lipschitz functions. Real-valued functionals and bornologies. Uniformly paracompact subsets. Uniformly paracompact spaces and uniformly locally Lipschitz functions. Bornological convergence of nets of closed subsets. Attouch-Wets convergence. Topologies of uniform convergence on
B(X,Y) and convergence of graphs. Bornological convergence and uniform convergence of distance functionals. Bornological convergence with respect to the compact bornology. When is bornological convergence topological? Uniformizability and metrizability. Ideals, bornologies and extensions. When is an extension bornological?
About the author
Gerald Beer, PhD UCLA 1971 won the faculty prize for teaching assistants at UCLA. He was a full professor at California State University Los Angeles, where he won the Presidents Distinguished Professor Award. He has around 140 papers in refereed journals plus two books: (1) Applied Calculus for Business and Economics; (2) Topologies on Closed and Closed Convex Sets. He is on the editorial boards of
The Journal of Convex Analysis and
Set-Valued and Variational Analysis. In 1983-1984, he was a Fulbright Professor associated with the Mathematical Consortium of Manila, and in 1986, he was a National Academy of Sciences exchange scholar at the Bulgarian Academy of Sciences. He had visiting positions at the University of California, Davis, University of Minnesota, University of Milan, University of Salerno, University of Naples II, University of Limoges, University of Montpellier II, University of Perpignan, University of Complutense Madrid, Politecnica Valencia, UMH Elche, and Auckland Institute of Technology. A conference honoring Professor Beer was held in Varenna, Lake Como, Italy on his 65th birthday.
Summary
Bornologies are a useful tool in studying problems in mathematical analysis and general topology. For example, they can be used to characterize those metric spaces on which the real-valued uniformly continuous functions are stable under pointwise products.
