A new approach for some applications of generalized fuzzy closed sets

Abstract

Generalized closed sets play an important role in the study of topological spaces. In the literature we have seen the relation between closed set and generalized closed set is linear in nature, both in crisp and fuzzy environment. In particular, every closed set is a generalized closed set. In this present treatise, a new type of generalized closed set via \((i,j)^{*}\)-fuzzy \(\gamma \)-open set, namely \((i,j)^{*}\)-\(\gamma gf\gamma \)-closed set have been introduced in a fuzzy bitopological space and interrelationships of it with the existing ones have been made. It has been proved that this newly defined generalized fuzzy closed set is totally independent with \((i,j)^{*}\)-fuzzy closed set. Various characterizations of this set have been established and some interesting results are cited. As applications of this set various types of continuities are studied in the same environment. More applications are proposed by introducing a new kind of closure operator namely \((i,j)^{*}\)-\(\gamma \)-\(cl_{\gamma }^{*}\)-operator and important results have been reported based on this concept.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. Balasubramanian G, Sundaram P (1997) On some generalizations of fuzzy continuous functions. Fuzzy Sets Syst 86:93–100

    MathSciNet  Article  Google Scholar 

  2. Bhattacharya S (2011) On generalized regular closed sets. Int J Contemp Math Sci 6(3):145–152

    MathSciNet  MATH  Google Scholar 

  3. Bhattacharya B (2017) Fuzzy independent topological spaces generated by fuzzy \(\gamma ^{*}\)-open sets and their applications. Afrika Mathematika 28(5–6):909–928

    MathSciNet  Article  Google Scholar 

  4. Bhattacharya B, Chakaraborty J (2015) Generalized regular fuzzy closed sets and their applications. Int J Fuzzy Math 23(1):227–239

    Google Scholar 

  5. Bhattacharya B, Paul A (2013) On bitopological \(\gamma \)-open sets. IOSR J. Math. 5(2):10–14

    Article  Google Scholar 

  6. Cao J, Ganster M, Reilly I (2002) On generalized closed sets. Topol Appl 123:37–46

    MathSciNet  Article  Google Scholar 

  7. Das B, Bhattacharya B (2019) On \((i, j )\) generalized fuzzy \(\gamma \)-closed set in fuzzy bitopological spaces, advances in intelligent systems and computing. Springer Nature Singapore Pte Ltd., Berlin, pp 661–673. https://doi.org/10.1007/978-981-13-1592-3_52

  8. Das B, Chakraborty J, Bhattacharya B (2019) A new type of generalized closed set via \(\gamma \)-open set in a fuzzy bitopological space. Proyecc J Math 38(3):511–536

    MathSciNet  MATH  Google Scholar 

  9. El-Shafei ME, Zakari AH (2006) \(\theta \)-generalized closed set in fuzzy topological spaces. Arab J Sci Eng 31(2A):197–206

    MathSciNet  MATH  Google Scholar 

  10. Fukutuke T (1986) On generalized closed sets in bitopological spaces. Bull Fukuoka Univ Educ 35(III):19–28

    MathSciNet  Google Scholar 

  11. Kandil A, Nouh AA, El-Sheikh SA (1995) On fuzzy bitopological spaces. Fuzzy Sets Syst 74:353–363

    MathSciNet  Article  Google Scholar 

  12. Kelly JC (1963) Bitopological Spaces. Proc Lond Math Soc 13:71–89

    MathSciNet  Article  Google Scholar 

  13. Levine N (1970) Generalized closed sets in topology. Rendiconti Del Circolo Mathematico Palermo 19:89–96

    MathSciNet  Article  Google Scholar 

  14. Palaniappan N, Rao KC (1993) Regular generalized closed sets. Kyungpook Math J 33(2):211–219

    MathSciNet  MATH  Google Scholar 

  15. Park JH, Park JK (2003) On regular generalized fuzzy closed sets and generalization of fuzzy continuous functions. Indian J Pure Appl Math 34(7):1013–1024

    MathSciNet  MATH  Google Scholar 

  16. Salama AS (2010) Bitopological rough approximations with medical applications. J King Saud Univer. (Science) 22:177–183

    Article  Google Scholar 

  17. Thivagar ML, Ravi O (2004) On strong forms of \((1,2)^{*}\)-quotient mappings in a bitopological space. Int J Math Game Theory Algebra 14(6):481–492

    MathSciNet  MATH  Google Scholar 

  18. Tripathy BC, Acharjee S (2014) On \((\gamma,\delta )\)-Bitopological semi-closed set via topological ideal. Proyecc J Math 33(3):245–257

    MathSciNet  MATH  Google Scholar 

  19. Tripathy BC, Debnath S (2013) \(\gamma \)-open sets and \(\gamma \)-continuous mappings in fuzzy bitopological spaces. J Intell Fuzzy Syst 24:631–635

    MathSciNet  Article  Google Scholar 

  20. Tripathy BC, Debnath S (2015) On fuzzy \(b\)-locally open sets in bitopological space. Songklanakarin J Sci Technol 37(1):93–96

    Google Scholar 

  21. Tripathy BC, Debnath S (2019) Fuzzy \(m\)-structures, \(m\)-open multifunctions and bitopological spaces. Boletin da Sociedade Paranaense de Matematica 37(4):119–128

    MathSciNet  Article  Google Scholar 

  22. Tripathy BC, Ray GC (2013) Mixed fuzzy ideal topological spaces. Appl Math Comput 220:602–607

    MathSciNet  MATH  Google Scholar 

  23. Tripathy BC, Ray GC (2014) On \(\delta \)-continuity in mixed fuzzy topological spaces. Boletim da Sociedade Paranaense de Matematica 32(2):175–187

    MathSciNet  Article  Google Scholar 

  24. Tripathy BC, Sarma DJ (2012) On pairwise \(b\)-locally open and \(b\)-locally closed functions in bitopological spaces. Tamkang J Math 43(4):533–539

    MathSciNet  Article  Google Scholar 

  25. Tripathy BC, Sarma DJ (2013) On weakly \(b\)-continuous functions in bitopological spaces. Acta Sci Technol 35(3):521–525

    Article  Google Scholar 

  26. Tripathy BC, Sarma DJ (2014) Generalized \(b\)-closed sets in Ideal bitopological spaces. Proyecc J Math 33(3):315–324

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are highly grateful to each one of the esteemed learned reviewers of this work for their valuable constructive suggestions towards the improvement of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Baby Bhattacharya.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Regivan Hugo Nunes Santiago.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Das, B., Chakraborty, J., Paul, G. et al. A new approach for some applications of generalized fuzzy closed sets. Comp. Appl. Math. 40, 44 (2021). https://doi.org/10.1007/s40314-021-01432-7

Download citation

Keywords

  • \((i</Keyword> <Keyword>j)^{*}\)-\(\gamma gf\gamma \)-closed set
  • \((i</Keyword> <Keyword>j)^{*}\)-\(\gamma \)-\(gf\gamma \)-q-neighbourhood
  • \((i</Keyword> <Keyword>j)^{*}\)-\(\gamma gf\gamma \)-continuous function
  • \((i</Keyword> <Keyword>j)^{*}\)-\(\gamma \)-\(cl_\gamma ^{*}\)-operator

Mathematics Subject Classification

  • 54E55
  • 54A40