Arabian Journal of Mathematics

, Volume 8, Issue 2, pp 125–132 | Cite as

Upper neighbours of Čech closure operators

  • T. KavithaEmail author
  • M. Kunheenkutty
  • P. T. Ramachandran
Open Access


Here, we investigate the existence of upper neighbours in the lattice of \(T_{1}\) Čech closure operators on a fixed set. In this paper, we prove that a first countable \(T_{1}\) closure operator has no upper neighbour in the lattice of Čech closure operators.

Mathematics Subject Classification

54A05 03G10 



The authors express gratitude to the referee(s) and editor(s) for their valuable comments and suggestions which improved this paper. The first author acknowledges the financial support from U.G.C., Govt of India.


  1. 1.
    Agashe, P.; Levine, N.: Adjacent topologies. J. Math. Tokushima Univ. 7, 21–35 (1973)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloq. Pub. Vol. 25, Third Edn. (1967)Google Scholar
  3. 3.
    Čech, E.: Topological Spaces, Rev edn. Wiley, New York (1966)zbMATHGoogle Scholar
  4. 4.
    Hewitt, E.: A problem of set theoretic topology. Duke Math. J. 10, 309–333 (1943)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kunheenkutty, M.: On simple expansions of topologies. In: Proceedings, K. M. A. National Seminar on New Vistas in Topology, KKTM Govt College, Pullut, Kodungallur, Kerala, pp. 11–15 (2005)Google Scholar
  6. 6.
    Kunheenkutty, M.: Comparison of simple expansions of topologies. In: Proceedings, International Seminar on Recent Trends in Topology and its Applications, St. Joseph’s college, Irinjalakuda, Kerala, 99–105 (2009).Google Scholar
  7. 7.
    Kunheenkutty, M.: Some problems on local properties of the lattice of topologies. Thesis for Ph. D. Degree, University of Calicut (2013)Google Scholar
  8. 8.
    Kunheenkutty, M.; Kavitha, T.; Ramachandran, P.T.: Adjacency in the lattice of Čech closure operators. Int. J. Pure Appl. Math. 105(1), 73–86 (2015)CrossRefGoogle Scholar
  9. 9.
    Larson, R.E.; Andima, S.J.: The lattice of topologies: a survey. Rocky Mt. J. Math. 5, 177–198 (1975)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Levine, N.: Simple extensions of topologies. Am. Math. Mon. 71, 22–25 (1964)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ramachandran, P.T.: Some problems in set topology relating group of homeomorphisms and order. Thesis for Ph. D. Degree, CUSAT (1985)Google Scholar
  12. 12.
    Ramachandran, P.T.: Complementation in the lattice of Čech closure operators. Indian J. Pure. App. Math. 18(2), 152–158 (1987)zbMATHGoogle Scholar
  13. 13.
    Ramachandran, P. T.: Fixed Points of the Lattice of Čech Closure Operators. Proceedings, U.G.C. Sponsored National Seminar on Discrete Analysis at Sri Ramakrishna Mission Vidyalaya College of Arts and Science (Autonomous), Coimbatore 641020, (1999)Google Scholar
  14. 14.
    Ramachandran, P.T.: Complemented Elements in the Lattice of Čech Closure Operators. J. Comp. & Math. Sci. 3(5), 498–501 (2012)Google Scholar
  15. 15.
    Stoll, R.R.: Set Theory and Logic. Dover Publication, Inc., New York (1961)Google Scholar
  16. 16.
    Steiner, A.K.: The Lattice of Topologies: Structure and Complementation. Trans. Amer. Math. Soc. 122, 379–398 (1966)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • T. Kavitha
    • 1
    Email author
  • M. Kunheenkutty
    • 2
  • P. T. Ramachandran
    • 1
  1. 1.Department of MathematicsUniversity of CalicutCalicutIndia
  2. 2.CalicutIndia

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