Advertisement

On maximal, minimal and mean open sets

  • Kallol Bhandhu Bagchi
  • Ajoy Mukharjee
Article
  • 7 Downloads

Abstract

We see that the real numbers system with the usual topology contains no minimal open sets. This observation instigates us to study topological spaces having no minimal and maximal open sets. We find that such topological spaces if connected are not cut-point spaces. We also characterize mean open sets in \(T_{1}\) connected topological spaces.

Keywords

Minimal open set Maximal open set Mean open set Cut point Cut-point space Connected space 

Mathematics Subject Classification

54D20 54A05 54D30 

Notes

Acknowledgements

The authors are grateful to the reviewers for some kind comments and suggestions toward a few modifications in the contents of the paper.

References

  1. 1.
    Honari, B., Bahrampour, Y.: Cut-point spaces. Proc. Am. Math. Soc. 127(9), 2797–2803 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Mukharjee, A.: On maximal and minimal clopen sets. An. Univ. Oradea Fasc. Mat. 23(2), 127–130 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Mukharjee, A., Bagchi, K.B.: On mean open and closed sets. Kyungpook Math. J. 56(4), 1259–1265 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Mukharjee, A., Bagchi, K.B.: Some results on mean open, closed and clopen sets. Hacet. J. Math. Stat. 47(1), 101–106 (2018)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Nakaoka, F., Oda, N.: Some applications of minimal open sets. Int. J. Math. Math. Sci. 27(8), 471–476 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Nakaoka, F., Oda, N.: Some properties of maximal open sets. Int. J. Math. Math. Sci. 2003(21), 1331–1340 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Nakaoka, F., Oda, N.: Minimal closed sets and maximal closed sets. Int. J. Math. Math. Sci. (2006).  https://doi.org/10.1155/IJMMS/2006/18647 (article ID 18647, 8 pages)

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKalipada Ghosh Tarai MahavidyalayaDarjeelingIndia
  2. 2.Department of MathematicsSt. Joseph’s CollegeDarjeelingIndia

Personalised recommendations