Skip to main content
SpringerLink
Log in

Topological Sorting of Finitely Near Sets

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

Abstract

This paper introduces a natural approach in the evaluation of the nearness of sets in topological spaces. The objective is to classify levels of nearness of sets relative to each given set. The main result is a proximity measure of nearness for disjoint sets in an extremally disconnected topological space. Another result is that if A, B are nonempty semi-open sets such that \(A\ \delta \ B\), then \((\text{int }A)\ \delta \ (\text{int }B)\).

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. Arkhangel’skiǐ, A.V.: Extremally-disconnected space, Encyclopedia of Mathematics 2. Kluwer Academic Publishers, London, UK (1995)

    Google Scholar 

  2. Arkhangel’skiǐ, A.V.: A study of extremally-disconnected space topological spaces. Bull. Math. Sci. 1(1), 3–12 (2011)

    Article  MathSciNet  Google Scholar 

  3. Arkhangel’skiǐ, A.V.: On topological and algebraic structure of extremally disconnected semitopological groups. Comment. Math. Univ. Carolin. 41(4), 803–810 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Crossley, S.G., Hildebrand, S.K.: Semi-Topological Properties. Fundam. Math. 74, 233–254 (1972)

    MathSciNet  MATH  Google Scholar 

  5. Dochviri, I.: Computer Mathematics: Logical and Point-Set Topological Constructions. Georgian Technical University Press, Tbilisi (2008). (Georgian)

    Google Scholar 

  6. Duszinsky, Z., Noiri, T.: Semi-Open, Semi-Closed and Semi-Continuity of Functions. Math. Pannonica 23(2), 195–200 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Edelsbrunner, H., Harer, J.L.: Computational Topology. An Introduction, American Mathematical Society, Providence, RI, 2010. xii+241 pp

  8. Dey, T.K., Edelsbrunner, H., Guha, S.: Computational Topology. Advances in Discrete and Computational Geometry, 109143, Contemp. Math., 223, Amer. Math. Soc., Providence, RI, (1999)

  9. Efremovich, V.A.: The Gometry of Proximity. I., Mat. Sbornik. (N.S.) 31(73), 189–200 (1952)

    Google Scholar 

  10. Frolik, Z.: Maps of extremally disconnected spaces, theory of types, and applications, General topology and its relations to modern analysis and algebra III(Proc. Conf., Kanpur, 1968) (1971), 131–142

  11. Guadagni, C.: Bornological Convergences on Local Proximity Spaces and \(\omega _{\mu }\)-Metric Spaces, Ph.D. Thesis, Università degli Studi di Salerno, Dipartimento di Matematica, supervisor: A. Di Concilio, 2015, 79pp

  12. Hettiarachchi, R., Peters, J.F.: Multi-manifold LLE learning in pattern recognition. Pattern Recognition 48, 2947–2960 (2015)

    Article  Google Scholar 

  13. Kuratowski, C.: Topologie I. Panstwowe Wydawnictwo Naukowe, Warsaw (1958)

    MATH  Google Scholar 

  14. Leader, S.: On Clusters in Proximity Spaces. Fundam. Math. 47, 205–213 (1959)

    MathSciNet  MATH  Google Scholar 

  15. Levine, N.: A decomposition of continuity in topological spaces. Amer. Math. Monthly. 68, 44–46 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  16. Levine, N.: Semi-Open Sets and Semi-Continuity in Topological Spaces. Amer. Math. Monthly. 70, 71–80 (1963)

    MathSciNet  MATH  Google Scholar 

  17. Maki, H., Mohammed, A.M., Karim, A., Nishi, A., Ogata, H.: On Operation Approaches of \(\alpha \)-Open Sets and Semi-Open Sets. Sci. Math. Japon. 73(1), 1–10 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Naimpally, S.A., Warrack, B.D.: Proximity Spaces, Cambridge University Press, Cambridge Tract in Mathematics and Mathematical Physics 59, Cambridge, UK, 1970

  19. Naimpally, S.A., Peters, J.F.: Topology with applications. Topological spaces via near and far, With a foreword by Iskander A. Taimanov. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. xvi+277 pp

  20. Noiri, T.: On Weakly Continuous Mappings. Proc. Amer. Math. Soc. 46(1), 120–124 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  21. Noiri, T.: A Note on Extremally Disconnected Spaces. Proc. Amer. Math. Soc. 79(2), 327–330 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Peters, J.F., Naimpally, S.A.: Applications of Near Sets. Notices Am. Math. Soc. 59(4), 536–542 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Peters, J.F.: Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces, Intelligent Systems Reference Library, vol. 63, Springer, : xv + 411 pp. Zentralblatt MATH Zbl 1295, 68010 (2014)

  24. Peters, J.F.: Computational Proximity. Excursions in the Topology of Digital Images, Intelligent Systems Reference Library, vol. 102, Springer Int. Pub., Switzerland, 2016, xxviii + 433pp, doi:10.1007/978-3-319-30262-1

  25. Riesz, F.: Stetigkeitsbegriff und abstrakte Mengenlehr, Atti IV Congr. Internat. Mat. 2 (Roma 1908), pp. 18-24

  26. Smirnov, Y.M.: On Proximity Spaces. Mat. Sb. (N.S.) 31(73), 543–574 (1952)

    MathSciNet  MATH  Google Scholar 

  27. Wolski, M.: Toward foundations of near sets: (Pre-)sheaf theoretic approach. Math. Comput. Sci. 7(1), 125–136 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Caucasus International University, 73, Chargali Str, 0192, Tbilisi, Georgia

    Irakli Dochviri

  2. Department of Electrical and Computer Engineering, University of Manitoba, R3T 5V6, Winnipeg, Canada

    James F. Peters

  3. Department of Mathematics, Faculty of Arts and Sciences, Adıyaman University, 02040, Adıyaman, Turkey

    James F. Peters

Authors
  1. Irakli Dochviri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James F. Peters
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James F. Peters.

Additional information

Dedicated to the Memory of V.A. Efremovich and S.A. Naimpally.

Irakli Dochviri was supported by Shota Rustaveli Georgian NSF Grant FR/291/5-103/14.

James F. Peters was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) discovery Grant 185986.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dochviri, I., Peters, J.F. Topological Sorting of Finitely Near Sets. Math.Comput.Sci. 10, 273–277 (2016). https://doi.org/10.1007/s11786-016-0273-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-016-0273-1

Keywords

Mathematics Subject Classification

Search

Navigation