A new type of generalized closed set via γ-open set in a fuzzy bitopological space
DOI:
https://doi.org/10.22199/issn.0717-6279-2019-03-0033Keywords:
(i, j)*-fuzzy γ-open set, (i, j)*-generalized fuzzy γ-closed set, (i, j)*-γ-generalized fuzzy closed set, (i, j)*-generalized fuzzy γ-continuous function, (i, j)*-generalized fuzzy γ-irresolute functionAbstract
This paper aims to present the notion of (i, j)*-fuzzy γ-open set in a fuzzy bitopological space as a parallel form of (i, j)-fuzzy γ-open set due to Tripathy and Debnath (2013) [17] and show that both of them are independent concepts. Then we extend our study to (i, j)*-generalized fuzzy γ-closed set and (i, j)*-γ-generalized fuzzy closed set. We show that (i, j)*-γ-generalized fuzzy closed set and (i, j)*-generalized fuzzy γ-closed set are also independent of each other in nature. Though every (i, j)*-fuzzy γ-closed set is a (i, j)*-generalized fuzzy γ-closed set but with (i, j)*-γ-generalized fuzzy closed set, the same relation is not linear. Similarly though every (i, j)*-fuzzy closed set is (i, j)*-γ-generalized fuzzy closed set but it is independent to (i, j)*-generalized fuzzy γ-closed set. Various properties related to (i, j)*-generalized fuzzy γ-closed set are also studied. Finally, (i, j)*-generalized fuzzy γ-continuous function and (i, j)*-generalized fuzzy γ-irresolute functions are introduced and interrelationships among them are established. We characterized these functions in different directions for different applications.References
G. Balasubramanian and P. Sundaram, “On some generalizations of fuzzy continuous functions”, Fuzzy Sets and Systems, vol. 86, no. 1, pp. 93–100, Feb. 1997, doi: 10.1016/0165-0114(95)00371-1.
B. Bhattacharya and J. Chakaraborty, “Generalized regular fuzzy closed sets and their applications”, International Journal of Fuzzy Mathematics, vol. 23, no. 1, pp. 227-239, 2015. [On line]. Available: http://bit.ly/33oiRLc
S. Bhattacharya, “On generalized regular closed sets”, International Journal Contemporary Mathematical Sciences, vol. 6, no. 3, pp. 145-152, 2011. [On line]. Available: http://bit.ly/2YpHdVT
A. Shahna, “On fuzzy strong semicontinuity and fuzzy precontinuity”, Fuzzy Sets and Systems, vol. 44, no. 2, pp. 303–308, Nov. 1991, doi: 10.1016/0165-0114(91)90013-G.
J. Cao, M. Ganster, and I. Reilly, “On generalized closed sets”, Topology and its Applications, vol. 123, no. 1, pp. 37–46, Aug. 2002, doi: 10.1016/S0166-8641(01)00167-5.
M. El-Shafei and A. Zakari, “θ-Generalized closed set in fuzzy topological spaces”, The Arabian Journal for Science and Engineering, vol. 31, no. 2A, pp. 197-206, Jul. 2006. [On line]. Available: http://bit.ly/2OIcY8m
T. Fukutuke, “On generalized closed sets in bitopological spaces”, Fukuoka Kyoiku Daigaku Kiyo, Dai-3-Bu, Rika-hen. vol. 35, pp. 19-28, 1986.
A. Kandil, A. Nouh, and S. El-Sheikh, “On fuzzy bitopological spaces”, Fuzzy Sets and Systems, vol. 74, no. 3, pp. 353–363, Sep. 1995, doi: 10.1016/0165-0114(94)00333-3.
S. Kumar, “On fuzzy pairwise α-continuity and fuzzy pairwise pre-continuity”, Fuzzy Sets and Systems, vol. 62, no. 2, pp. 231–238, Mar. 1994, doi: 10.1016/0165-0114(94)90063-9.
N. Levine, “Generalized closed sets in topology”, Rendiconti del Circolo Matematico di Palermo, vol. 19, no. 1, pp. 89–96, Jan. 1970, doi: 10.1007/BF02843888.
N. Palaniappan and K. Rao, “Regular generalized closed sets”, Kyungpook Mathematical Journal, vol. 33, no. 2, pp. 211-219, 1993. [On line]. Available: http://bit.ly/2KltiaO
J. H. Park, and J. K. Park, “On regular generalized fuzzy closed sets and generalization of fuzzy continuous functions”, Indian Journal of Pure and Applied Mathematics, vol. 34, no. 7, pp. 1013-1024, Jul. 2003. [On line]. Available: http://bit.ly/2ZHk8uo
A. Paul, B. Bhattacharya, and J. Chakraborty, “On Γγ- set in fuzzy bitopological spaces”, Boletim da Sociedade Paranaense de Matemática, vol. 35, no. 3, pp. 285–299, Oct. 2017, doi: 10.5269/bspm.v35i3.28701.
M. Singal and N. Prakash, “Fuzzy preopen sets and fuzzy pre separation axioms”, Fuzzy Sets and Systems, vol. 44, no. 2, pp. 273–281, Nov. 1991, doi: 10.1016/0165-0114(91)90010-n.
M. Thivagar and O. Ravi, “On strong forms of (1, 2)∗-quotient mappings in a bitopological space”, International Journal of Mathematics, Game Theory and Algebra, vol. 14, no. 6, pp. 481-492, Jan. 2004.
B. Tripathy and S. Acharjee, “On (γ, δ) -Bitopological semi-closed set via topological ideal”, Proyecciones (Antofagasta), vol. 33, no. 3, pp. 245–257, Sep. 2014, doi: 10.4067/s0716-09172014000300002.
B. Tripathy, and S. Debnath, “γ-open sets and γ-continuous mappings in fuzzy bitopological spaces”, Journal of Intelligence and Fuzzy Systems, vol. 24, no. 3, pp. 631-635, 2013, doi: 10.3233/IFS-2012-0582.
B. Tripathy and D. Sarma, “On b-locally open sets in bitopological spaces”, Kyungpook mathematical journal, vol. 51, no. 4, pp. 429–433, Dec. 2011, doi: 10.5666/kmj.2011.51.4.429.
B. Tripathy and S. Debnath, “Fuzzy m-structures m-open multifunctions and bitopological spaces”, Boletim da Sociedade Paranaense de Matemática, vol. 37, no. 4, pp. 119–128, Jan. 2018, doi: 10.5269/bspm.v37i4.35152.
D. Sarma and B. Tripathy, “On pairwise b–locally open and pairwise b–locally closed functions in bitopological spaces”, Tamkang Journal of Mathematics, vol. 43, no. 4, pp. 533–539, Dec. 2012, doi: 10.5556/j.tkjm.43.2012.748.
B. Tripathy and D. Sarma, “On weakly b-continuous functions in bitopological spaces”, Acta Scientiarum. Technology, vol. 35, no. 3, Jun. 2013, doi: 10.4025/actascitechnol.v35i3.15612.
B. Tripathy and D. Sarma, “Generalized b-closed sets in ideal bitopological spaces”, Proyecciones (Antofagasta), vol. 33, no. 3, pp. 315–324, Sep. 2014, doi: 10.4067/S0716-09172014000300006
B. Tripathy and S. Debnath, “On fuzzy b-locally open sets in bitopological spaces”, Songklanakarin Journal of Science and Technology, vol. 37, no. 1, pp. 93-96, 2015. [On line]. Available: http://bit.ly/2M8wXL7
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