Skip to main content
SpringerLink
Log in

Ultrafilter Completeness in \({{\varepsilon}}\) -approach Nearness Spaces

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

This paper presents a new approach to the proof of the Niemytzki–Tychonoff theorem for symmetric topological spaces. The proof uses the concept of completeness in \({\varepsilon}\) -approach nearness spaces which was introduced by Peters and Tiwari (Appl Math Lett 25:1544–1547, 2012), and of clusters that are a generalization of Cauchy sequences.

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. Peters J., Tiwari S.: Completing extended metric spaces: an alternative approach. Appl. Math. Lett. 25(10), 1544–1547 (2012). doi:10.1016/j.aml.2012.01.012

    Article  MathSciNet  MATH  Google Scholar 

  2. Herrlich H.: A concept of nearness. Gen. Top. Appl. 4, 191–212 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kuratowski C.: Sur l’opération \({\bar{A}}\) opération de l’analysis situs. Fundamenta Mathematica 3, 182–199 (1922)

    MATH  Google Scholar 

  4. Kuratowski, K.:Introduction to calculus, p 316. Pergamon Press, Oxford (1961)

  5. Naimpally, S.A., Warrack, B.D.: Proximity spaces. Cambridge Tract No. 59, Cambridge (1970)

  6. Ivanova V.M., Ivanov A.A.: Contiguity spaces and bicompact extensions of topological spaces. Izv. Akad. Nauk. SSSR Ser. Mat 23, 613–634 (1959)

    MathSciNet  MATH  Google Scholar 

  7. Willard, S.: General topology. Addison-Wesley Publising Company (1970)

  8. Tiwari, S.: α*-Uniformities and their order structure. Afr. Mat. (2012). doi:10.1007/s13370-012-0065-y

  9. Katětov M.: On continuity structures and spaces of mappings. Comment. Math. Univ. Carolinae 6(2), 257–278 (1965)

    MathSciNet  MATH  Google Scholar 

  10. Peters J.F.: Near sets. Special theory about nearness of objects. Fundamenta Informaticae 75(1-4), 407–433 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Peters J.F., Wasilewski P.: Foundations of near sets. Inform. Sci. 179, 3091–3109 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lowen, R.: Approach spaces: the missing link in the topology-uniformity-metric triad. Oxford mathematical monographs, p 253. Oxford University Press, Oxford (1997)

  13. Khare, M., Tiwari, S.: Completion in a common supercategory of Met, UAP, wsAP and Near. Demonstr. Math. 46(1) (2013) (to appear)

  14. Khare M., Tiwari S.: L-approach merotopies and their categorical perspective. Demonstr. Math. 45(3), 699–716 (2012)

    MathSciNet  Google Scholar 

  15. Khare, M., Tiwari, S.: Approach merotopological spaces and their completion. Int. J. Math. Math. Sci. 2010(409804), 16 (2010) doi:10.1155/2010/409804

  16. Khare M., Tiwari S.: Grill determined L-approach merotopological spaces. Fundam. Inform. 99(1), 1–12 (2010). doi:10.3233/FI-2010-234

    MathSciNet  MATH  Google Scholar 

  17. Carlson J.: B-completeness in nearness spaces. Gen. Top. Appl. 5(3), 263–278 (1975)

    Article  MATH  Google Scholar 

  18. Bentley H.L., Herrlich H.: Merotopological spaces. Appl. Categ. Struct. 12, 155–180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Peters J., Tiwari S.: Approach merotopies and near filters. Theory and application. Gen. Math. Notes 3(1), 32–45 (2011)

    Google Scholar 

  20. Čech, E.: Topological spaces. Revised edition by Z. Frolik and M. Katetov, p 893. John Wiley and Sons, London (1966)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad, 211004, UP, India

    Surabhi Tiwari

Authors
  1. Surabhi Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Surabhi Tiwari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tiwari, S. Ultrafilter Completeness in \({{\varepsilon}}\) -approach Nearness Spaces. Math.Comput.Sci. 7, 107–111 (2013). https://doi.org/10.1007/s11786-013-0148-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-013-0148-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation