Skip to main content
SpringerLink
Log in

The bornology of cofinally complete subsets

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Cofinal completeness of a metric space, which is a property between completeness and compactness, can be characterized in terms of a measure of local compactness functional [7]. Using this functional, we introduce and then study the variational notion of cofinally complete subset of a metric space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  MathSciNet  Google Scholar 

  2. H. Attouch, R. Lucchetti and R. Wets, The topology of the ρ-Hausdorff distance, Ann. Mat. Pura Appl., 160 (1991), 303–320.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc., 95 (1985), 653–658.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Beer, UC spaces revisited, Amer. Math. Monthly, 95 (1988), 737–739.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  7. G. Beer, Between compactness and completeness, Top. Appl., 155 (2008), 503–514.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Beer, Operator topologies and graph convergence, J. Convex Anal., 16 (2009), 687–698.

    MATH  MathSciNet  Google Scholar 

  9. G. Beer, C. Costantini and S. Levi, Bornological convergence and shields, to appear in Mediterranean J. Math.

  10. G. Beer, C. Costantini and S. Levi, Total boundedness in metrizable spaces, to appear in Houston J. Math.

  11. G. Beer and A. Di Concilio, Uniform continuity on bounded sets and the Attouch–Wets topology, Proc. Amer. Math. Soc., 112 (1991), 235–243.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Beer and G. Di Maio, Cofinal completeness of the Hausdorff metric topology, Fund. Math., 208 (2010), 75–85.

    Article  MATH  MathSciNet  Google Scholar 

  13. G. Beer and S. Levi, Pseudometrizable bornological convergence is Attouch–Wets convergence, J. Convex Anal., 15 (2008), 439–453.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Beer and S. Levi, Uniform continuity, uniform convergence, and shields, Set-Valued and Var. Anal., 18 (2010), 251–275.

    Article  MATH  MathSciNet  Google Scholar 

  16. G. Beer, S. Naimpally and J. Rodríguez-López, -topologies and bounded convergences, J. Math. Anal. Appl., 339 (2008), 542–552.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Borwein and J. Vanderwerff, Epigraphical and uniform convergence of convex functions, Trans. Amer. Math. Soc., 348 (1996), 1617–1631.

    Article  MATH  MathSciNet  Google Scholar 

  18. B. Burdick, On linear cofinal completeness, Top. Proc., 25 (2000), 435–455.

    MATH  MathSciNet  Google Scholar 

  19. H. Corson, The determination of paracompactness by uniformities, Amer. J. Math., 80 (1959), 185–190.

    Article  MathSciNet  Google Scholar 

  20. G. Di Maio, E. Meccariello and S. Naimpally, Uniformizing (proximal) △-topologies, Top. Appl., 137 (2004), 99–113.

    Article  MATH  Google Scholar 

  21. R. Engelking, General Topology, Polish Scientific Publishers (Warsaw, 1977).

    MATH  Google Scholar 

  22. A. García-Máynez and S. Romaguera, Perfect pre-images of cofinally complete metric spaces, Comment. Math. Univ. Carolinae, 40 (1999), 335–342.

    MATH  Google Scholar 

  23. A. Hohti, On uniform paracompactness, Ann. Acad. Sci. Fenn. Series A Math. Diss., 36 (1981), 1–46.

    MathSciNet  Google Scholar 

  24. N. Howes, On completeness, Pacific J. Math., 38 (1971), 431–440.

    MathSciNet  Google Scholar 

  25. N. Howes, Modern Analysis and Topology, Springer (New York, 1995).

    Book  MATH  Google Scholar 

  26. T. Jain and S. Kundu, Atsuji spaces: equivalent conditions, Topology Proc., 30 (2006), 301–325.

    MATH  MathSciNet  Google Scholar 

  27. E. Klein and A. Thompson, Theory of Correspondences, Wiley (New York, 1984).

    MATH  Google Scholar 

  28. A. Lechicki, S. Levi and A. Spakowski, Bornological convergences, J. Math. Anal. Appl., 297 (2004), 751–770.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Lucchetti, Convexity and Well-posed Problems, Springer Verlag (Berlin, 2006).

    MATH  Google Scholar 

  30. S. Nadler and T. West, A note on Lesbesgue spaces, Topology Proc., 6 (1981), 363–369.

    MathSciNet  Google Scholar 

  31. S. Naimpally, Proximity Approach to Problems in Topology and Analysis, Oldenbourg (Munich, 2009).

    Book  MATH  Google Scholar 

  32. J. Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math, 9 (1959), 567–570.

    MATH  MathSciNet  Google Scholar 

  33. M. Rice, A note on uniform paracompactness, Proc. Amer. Math. Soc., 62 (1977), 359–362.

    Article  MATH  MathSciNet  Google Scholar 

  34. S. Romageura, On cofinally complete metric spaces, Q & A in Gen. Top., 16 (1998), 165–170.

    Google Scholar 

  35. G. Toader, On a problem of Nagata, Mathematica (Cluj), 20 (1978), 77–79.

    MathSciNet  Google Scholar 

  36. W. Waterhouse, On UC spaces, Amer. Math. Monthly, 72 (1965), 634–635.

    Article  MATH  MathSciNet  Google Scholar 

  37. S. Willard, General Topology, Addison-Wesley (Reading, MA, 1970).

    MATH  Google Scholar 

Download references

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. Facoltà di Scienze, Dipartimento di Matematica, Seconda Univerità degli Studi di Napoli, Via Vivaldi 43, 81100, Caserta, Italy

    Giuseppe Di Maio

Authors
  1. Gerald Beer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Di Maio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerald Beer.

Additional information

Corresponding author.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beer, G., Di Maio, G. The bornology of cofinally complete subsets. Acta Math Hung 134, 322–343 (2012). https://doi.org/10.1007/s10474-011-0156-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-011-0156-5

Keywords

2000 Mathematics Subject Classification

Search

Navigation