Uniform Continuity Preserved With Equivalent Metrics

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.

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Indonesia)

References

1. Atsuji, M.: Uniform continuity of continuous functions of metric spaces. Pacific J. Math. 8, 11–16 (1958)

   MathSciNet  Article  MATH  Google Scholar

2. Beer, G.: More about metric spaces on which continuous functions are uniformly continuous. Bull. Aust. Math. Soc. 33, 397–406 (1986)

   MathSciNet  Article  MATH  Google Scholar

3. Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dordrecht (1993)

   Book  MATH  Google Scholar

4. Beer, G., Garrido, M.I.: Bornologies and locally Lipschitz functions. Bull. Aust. Math. Soc. 90, 257–263 (2014)

   MathSciNet  Article  MATH  Google Scholar

5. Beer, G., Levi, S.: Strong uniform continuity. J. Math. Anal. Appl. 350, 568–589 (2009)

   MathSciNet  Article  MATH  Google Scholar

6. Beer, G., Naimpally, S.: Uniform continuity of a product of real functions. Real Anal. Exch. 37, 213–220 (2011)

   MathSciNet  Article  MATH  Google Scholar

7. Bouleau, N.: Une structure uniforme sur un espace F(E, F). Cahiers Topol. Géom. Diff. 11, 207–214 (1969)

   MathSciNet  MATH  Google Scholar

8. Bourbaki, N.: Elements of Mathematics, General Topology, Part 1. Hermann, Paris (1966)

   MATH  Google Scholar

9. Cabello-Sánchez, J.: \(\mathcal {U}(X)\) as a ring for metric spaces X. Filomat 31, 1981–1984 (2017)

   MathSciNet  Article  Google Scholar

10. Di Maio, G., Meccariello, E., Naimpally, S.: Decompositions of UC spaces. Questions and Answers in Gen. Top. 22, 13–22 (2004)

    MathSciNet  MATH  Google Scholar

11. Garrido, M.I., Meroño, A.S.: New types of completeness in metric spaces. Ann. Acad. Sci. Fenn. Math. 39, 733–758 (2014)

    MathSciNet  Article  MATH  Google Scholar

12. Goldberg, R.: Methods of Real Analysis, 2nd edn. Wiley, New York (1976)

    MATH  Google Scholar

13. Heinonen, J.: Lectures on Analysis in Metric Spaces. Springer, New York (2001)

    Book  MATH  Google Scholar

14. Hejcman, J.: Boundedness in uniform spaces and topological groups. Czech. Math. J. 9, 544–563 (1959)

    MathSciNet  MATH  Google Scholar

15. Hogbe-Nlend, H.: Bornologies and Functional Analysis. North-Holland, Amsterdam (1977)

    MATH  Google Scholar

16. Hohti, A., Junnila, H., Meroño, A.S.: On strongly Čech-complete spaces, manuscript

17. Jain, T., Kundu, S.: Atsuji spaces: equivalent conditions. Top. Proc. 30, 301–325 (2006)

    MathSciNet  MATH  Google Scholar

18. Jain, T., Kundu, S.: Atsuji completions: equivalent characterizations. Top. Appl. 154, 28–38 (2007)

    Article  MATH  Google Scholar

19. Lindenstrauss, J.: On nonlinear projections in Banach spaces. Mich. Math. J. 11, 263–287 (1964)

    MathSciNet  Article  MATH  Google Scholar

20. Lowen-Colebunders, E.: Function Classes of Cauchy Continuous Functions. Marcel Dekker, New York (1989)

    MATH  Google Scholar

21. McCoy, R., Ntantu, E.: Topological Properties of Spaces of Continuous Functions. Springer, New York (1988)

    Book  MATH  Google Scholar

22. McShane, E.: Extension of range of functions. Bull. Amer. Math. Soc. 40, 837–842 (1934)

    MathSciNet  Article  MATH  Google Scholar

23. Munkres, J.: Topology, 2nd edn. Prentice-Hall, Upper Saddle River (2000)

    Google Scholar

24. Nadler, S.: Pointwise products of uniformly continuous functions. Sarajevo J. Math. 13, 117–127 (2005)

    MathSciNet  MATH  Google Scholar

25. Rudin, W.: Principles of Mathematical Analysis, 2nd edn. McGraw-Hill, New York (1964)

    MATH  Google Scholar

26. Oxtoby, J.C.: Measure and Category. Springer, New York (1971)

    Book  MATH  Google Scholar

27. Sierpinski, W.: General Topology. University of Toronto Press, Toronto (1952)

    MATH  Google Scholar

28. Snipes, R.: Functions that preserve Cauchy sequences. Nieuw Archief Voor Wiskunde 25, 409–422 (1977)

    MathSciNet  MATH  Google Scholar

29. Toader, G.: On a problem of Nagata. Mathematica (Cluj) 20, 78–79 (1978)

    MathSciNet  MATH  Google Scholar

30. Valentine, F.: A Lipschitz condition preserving extension for a vector function. Amer. J. Math. 67, 83–93 (1945)

    MathSciNet  Article  MATH  Google Scholar

Download references

Acknowledgements

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

Author information

Authors and Affiliations

1. Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA, 90032, USA

   Gerald Beer

2. Instituto de Matemática Interdisciplinar (IMI), Departamento de Geometría y Topología, Universidad Complutense de Madrid, 28040, Madrid, Spain

   M. Isabel Garrido

3. Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040, Madrid, Spain

   Ana S. Meroño

Corresponding author

Correspondence to Gerald Beer.

Additional information

to Jonathan Borwein, in memoriam

The second and third authors were partially supported by MINECO Project-MTM2012-34341 (Spain)

Rights and permissions

About this article

Cite this article

Beer, G., Garrido, M.I. & Meroño, A.S. Uniform Continuity and a New Bornology for a Metric Space. Set-Valued Var. Anal 26, 49–65 (2018). https://doi.org/10.1007/s11228-017-0429-4

Download citation

• Received: 10 January 2017

• Accepted: 31 May 2017

• Published: 08 June 2017

• Issue Date: March 2018

• DOI : https://doi.org/10.1007/s11228-017-0429-4

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

griffintine1940.blogspot.com

Source: https://link.springer.com/article/10.1007/s11228-017-0429-4

Bagikan Artikel ini