Sufficiently Near Sets of Neighbourhoods

  • James F. Peters
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6954)


The focus of this paper is on sets of neighbourhoods that are sufficiently near each other as yet another way to consider near sets. This study has important implications in M. Katětov’s approach to topologising a set. A pair of neighbourhoods of points are sufficiently near, provided that the C̆ech distance between the neighbourhoods is less than some number ε. Sets of neighbourhoods are sufficiently near, provided the C̆ech distance between the sets of neighbourhoods is less than some number ε.


Approach space C̆ech distance collection ε-near collections merotopy near sets neighbourhood topology 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Katětov, M.: On continuity structures and spaces of mappings. Comment. Math. Univ. Carolinae 6, 257–278 (1965)MathSciNetMATHGoogle Scholar
  2. 2.
    Tiwari, S.: Some Aspects of General Topology and Applications. Approach Merotopic Structures and Applications, supervisor: M. Khare. PhD thesis, Department of Mathematics, Allahabad (U.P.), India (January 2010)Google Scholar
  3. 3.
    Lowen, R.: Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs, pp. viii + 253. Oxford University Press, Oxford (1997)MATHGoogle Scholar
  4. 4.
    Lowen, R., Vaughan, D., Sioen, M.: Completing quasi metric spaces: an alternative approach. Houstan J. Math. 29(1), 113–136 (2003)MathSciNetMATHGoogle Scholar
  5. 5.
    Peters, J., Tiwari, S.: Approach merotopies and near filters. Gen. Math. Notes 2(2), 1–15 (2011)Google Scholar
  6. 6.
    Peters, J., Tiwari, S.: Completion of ε-approach nearness spaces (communicated) (2011) Google Scholar
  7. 7.
    Khare, M., Tiwari, S.: L-approach merotopies and their categorical perspective. Demonstratio Math., 1–16 (2010), doi: 10.1155/2010/409804Google Scholar
  8. 8.
    Čech, E.: Topological Spaces, revised Ed. by Z. Frolik and M. Katătov. John Wiley & Sons, NY (1966)Google Scholar
  9. 9.
    Beer, G., Lechnicki, A., Levi, S., Naimpally, S.A.: Distance functionals and suprema of hyperspace topologies. Annali di Matematica Pura ed Applicata CLXII(IV), 367–381 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hausdorff, F.: Grundzüge der Mengenlehre, pp. viii + 476. Veit and Company, Leipzig (1914)Google Scholar
  11. 11.
    Leader, S.: On clusters in proximity spaces. Fundamenta Mathematicae 47, 205–213 (1959)MathSciNetMATHGoogle Scholar
  12. 12.
    Bourbaki, N.: Topologie générale, pp. 1–4. Hermann, Paris (1971); Springer-Verlag published a new edition, Heidelberg 2007Google Scholar
  13. 13.
    Bourbaki, N.: Elements of Mathematics. General Topology, Part 1, pp. i-vii, 437. Hermann & Addison-Wesley, Paris & Reading (1966)Google Scholar
  14. 14.
    Hausdorff, F.: Set Theory, p. 352. AMS Chelsea Publishing, Providence (1914)MATHGoogle Scholar
  15. 15.
    Engelking, R.: General Topology, Revised & completed edition. Heldermann Verlag, Berlin (1989)MATHGoogle Scholar
  16. 16.
    Hocking, J., Young, G.: Topology. Dover, NY (1988)MATHGoogle Scholar
  17. 17.
    Henry, C.: Near Sets: Theory and Applications, Ph.D. dissertation, supervisor: J.F. Peters. PhD thesis, Department of Electrical & Computer Engineering (2010)Google Scholar
  18. 18.
    Peters, J.: How near are Zdzisław Pawlak’s paintings? Merotopic distance between regions of interest. In: Skowron, A., Suraj, S. (eds.) Intelligent Systems Reference Library Volume Dedicated to Prof. Zdzisław Pawlak, pp. 1–19. Springer, Berlin (2011)Google Scholar
  19. 19.
    Sutherland, W.: Introduction to Metric & Topological Spaces. Oxford University Press, Oxford (1974, 2009); 2nd edn., 2008Google Scholar
  20. 20.
    Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, The Netherlands (1993)CrossRefMATHGoogle Scholar
  21. 21.
    Henry, C., Peters, J.: Arthritic hand-finger movement similarity measurements: Tolerance near set approach. Comp. & Math. Methods in Medicine 2011, Article ID 569898, 1–14 (2011), doi:10.1155/2011/569898Google Scholar
  22. 22.
    Peters, J.: Near sets. Special theory about nearness of objects. Fund. Inf. 75(1-4), 407–433 (2007)MathSciNetMATHGoogle Scholar
  23. 23.
    Peters, J.F., Wasilewski, P.: Foundations of near sets. Info. Sci. 179, 3091–3109 (2009)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Tiwari, S., Peters, J.: Supercategories εANear and εAMer. Int. J. of Computer Math. (communicated) (2011)Google Scholar
  25. 25.
    Tiwari, S., Peters, J.: Almost near L-fuzzy sets. Fuzzy Sets and Systems (communicated) (2011) Google Scholar
  26. 26.
    Peters, J.: ε-near collections. In: Yao, J.-T., Ramanna, S., Wang, G., Suraj, Z. (eds.) RSKT 2011. LNCS, vol. 6954, pp. 533–542. Springer, Heidelberg (2011)Google Scholar
  27. 27.
    Ramanna, S., Peters, J.F.: Approach space framework for image database classification. In: Hruschka Jr., E.R., Watada, J., do Carmo Nicoletti, M. (eds.) INTECH 2011. Communications in Computer and Information Science, vol. 165, pp. 75–89. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • James F. Peters
    • 1
  1. 1.Computational Intelligence Laboratory, Department of Electrical & Computer EngineeringUniv. of ManitobaWinnipegCanada

Personalised recommendations