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Set-Valued and Variational Analysis

, Volume 26, Issue 1, pp 49–65 | Cite as

Uniform Continuity and a New Bornology for a Metric Space

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Abstract

In the context of functions between metric spaces, continuity is preserved by uniform convergence on the bornology of relatively compact subsets while Cauchy continuity is preserved under uniform convergence on the bornology of totally bounded subsets. We identify a new bornology for a metric space containing the bornology of Bourbaki bounded sets on which uniform convergence preserves uniform continuity. Further, for real-valued uniformly continuous functions, the function space is a ring (with respect to pointwise multiplication) if and only if the two bornologies agree. We show that Cauchy continuity is preserved by uniform convergence on compact subsets if and only if the domain space is complete, and that uniform continuity is preserved under uniform convergence on totally bounded subsets if and only if the domain space has UC completion. Finally, uniform continuity is preserved under uniform convergence on compact subsets if and only if the domain space is a UC-space. We prove a simple omnibus density result for Lipschitz functions within a larger class of continuous functions equipped with a topology of uniform convergence on a bornology and apply that to each of our three function classes.

Keywords

Continuous function Cauchy continuous function Uniformly continuous function Lipschitz function Ring of functions Bornology UC-space Relatively compact set Totally bounded set Bourbaki bounded set Infinitely nonuniformly isolated set 

Mathematics Subject Classifications (2010)

Primary 54C35, 26A16, 46A17 Secondary 54E50, 54E35 

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Notes

Acknowledgements

The authors would like to thank the referees for their close reading of our manuscript and their many helpful suggestions.

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Gerald Beer
    • 1
  • M. Isabel Garrido
    • 2
  • Ana S. Meroño
    • 3
  1. 1.Department of MathematicsCalifornia State University Los AngelesLos AngelesUSA
  2. 2.Instituto de Matemática Interdisciplinar (IMI), Departamento de Geometría y TopologíaUniversidad Complutense de MadridMadridSpain
  3. 3.Departamento de Análisis MatemáticoUniversidad Complutense de MadridMadridSpain

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