218 130
Full Length Article
Volume 10 , Issue 2, PP: 96-104 , 2020

Title

A Review of Fuzzy Soft Topological Spaces, Intuitionistic Fuzzy Soft Topological Spaces and Neutrosophic Soft Topological Spaces

Authors Names :   M. Parimala   1     M.Karthika   2     F. Smarandache   3  

1  Affiliation :  Department of Mathematics, Bannari Amman Institute of Technology, TN, India-638 401.

    Email :  rishwanthpari@gmail.com


2  Affiliation :  Department of Mathematics, Bannari Amman Institute of Technology, TN, India-638 401.

    Email :  karthikamuthusamy1991@gmail.com


3  Affiliation :  University of New Mexico, Gallup, NM 87301, USA

    Email :  fsmarandache@gmail.com



Doi   :  10.5281/zenodo.4008341

Received: April 29, 2020 Accepted: August 14, 2020

Abstract :

 

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

 

Keywords :

Soft sets , fuzzy soft topological space , intuitionistic fuzzy soft topological space , neutrosophic soft topological space.

References :

 

[1]     Abd El-Latif, A. M., Characterizations of fuzzy soft pre separation axioms, Journal of New Theory, 7, pp. 47-63, 2015

[2]     Abd El-latif, A. M., Fuzzy soft α-connectedness in fuzzy soft topological spaces, Math.Sci. Lett., 5 (1), pp. 85-91,2016.

 [3]     Abd El-latif, A. M., Fuzzy soft separation axioms based on fuzzy β-open soft sets, Ann. Fuzzy Math. Inform., 11 (2), pp. 223-239,2016.

 

[4]     Abd El-latif, A. M., Hosny, R. A., Fuzzy soft pre-connected properties in fuzzy soft topological spaces, South Asian J Math, 5 (5), pp. 202-213, 2015.

 

[5]     Agarwal, M., Biswas, K. K. and Henmandlu, M., Generalised intuitionistic fuzzy soft sets with application in decision making, Applied Soft Computing, 13, pp. 3552-3566,2013.

 

[6]     Amit Kumar Singh., Rekha Srivastava, Separation axioms in intuitionistic fuzzy topological spaces, Hin- dawi Publishing Corporation Advances in Fuzzy Systems , Article ID 604396, pp. 1-7,2012.

 

[7]     Arockiarani. I., Martina Jency. J., More on fuzzy neutrosophic sets and fuzzy neutrosophic topological spaces, Inter.J. Innov. Research & Studies, 3 (5), 643-652,2014.

 

[8]     Arockiarani, I., Sumathi, I. R. and Martina Jency, J., Fuzzy neutrosophic soft topological spaces, In- ter.J.Math.Archive, 4 (10), pp. 225-238,2013.

 

[9]     Arockiarani. I., A Fuzzy neutrosophic soft set model in medical diagnosis, in:IEEE conference on Norbert Wiener in the 21st Century, (2014), Boston, MA, USA.

 

[10]  Aygunoglu, A., Cetkin, V., Aygun, H., An introduction to fuzzy soft topological spaces, Hacettepe Journal of Mathematics and Statistics, 43 (2) ,pp. 197 – 208, 2014.

 

[11]  Al-Quran, A., Hassan, N., Neutrosophic vague soft set and its application, Malaysian Journal of Mathe- matical Science, 11 (2), pp. 141-163,2017.

 

[12]     Atanassov, K., Intuitionistic fuzzy sets, Fuzzy Sets and Syst., 20, pp. 87-96,1986.

 

[13]  Bayramov, S., Gunduz, C., On intuitionistic fuzzy soft topological spaces, TWMS. J. Pure Appl. Math., 5, pp. 66-79,2014.

 

[14]  Bayramov, S., Gunduz, C., A New approach to separability and compactness in soft topological spaces. TWMS J. Pure Appl. Math., 9, pp. 82-93,2018.

 

[15]  Bayramov,S. and Gunduz, G., On intuitionistic fuzzy soft topological spaces, Inter.J.PureAppl.Math, 5(1), pp. 66-79,2014.

 

[16]  Bora,M., Neog, T. J., Sut,D.K., Some new operations of intuitionistic fuzzy soft sets, International Jour- nals of Soft Computing and Engineering, 2(4), pp.2231-2307,2012.

 

[17]  Bera, T., Mahapatra, N. K., On neutrosophic soft topological space, Neutrosophic Sets and Systems, 19, pp. 3- 15,2018.

 

[18]  Bera, T., Mahapatra, N.K., Introduction to neutrosophic soft topological space, OPSEARCH, 54(4),pp. 841-867,2017.

 

[19]  Broumi, S., and Smarandache, F., Intuitionistic neutrosophic soft set, Journal of Information and Com- puting Science, 8 (2), pp. 130-140,2013.

 

[20]     Broumi, S., Generalized neutrosophic soft set, IJCSEIT,3 (2), pp. 17-30,2013.

 

[21]     Cagman, N., Karatas, S., Enginoglu, S., Soft topology, Computer and Mathematics with Application, 62, pp. 351-358,2011.

 

[22]     Cagman, N., Citak, F. and Enginoglu, S., Fuzzy parameterized fuzzy soft set theory and its applications, Turkish Journal of Fuzzy Systems, 1 (1), pp. 21-35,2010.

 

[23]     Cagman, N., Enginoglu, S., Citak,F., Fuzzy soft set theory and its application, Iranian Journal of fuzzy systems,8 (3), pp. 137-147,2011.

 

[24]     Chang, C. L., Fuzzy topological spaces, J. Math. Anal. Appl., 24, pp.182–190,1968.

 

[25]     Cetkin, V., Aygun, H., A Note on fuzzy soft topological spaces, Conference of the European Society for Fuzzy Logic and Technology,2013.

 

[26]     Cetkin, V., Aygun, H., A Note on intuitionistic supra fuzzy soft topological spaces, Notes on Intuitionistic Fuzzy Sets, 21 (4), pp. 48-57,2015.

 

[27]     Coker, D., An Introduction to intuitionistic fuzzy topological space, Fuzzy Sets and Systems, 88, pp. 81-89,1997.

 

[28]     Deli, I., npn-Soft sets theory and applications, Annals of Fuzzy Mathematics and Informatics, 10 (6), pp. 847 -862,2015.

 

[29]     Deli, I., Broumi, S., Neutrosophic soft relations and some properties, Annals of Fuzzy Mathematics and Informatics, 9 (1), pp. 169-182,2015.

 

[30]     Dey, A., Pal, M., Generalised multi-fuzzy soft set and its application in decision making, Pacific Science Review A: Natural Science and Engineering, 17, pp. 23-28,2015.

 

[31]     Gunduz Aras, C., Bayramov, S., Some results on fuzzy soft topological spaces, Math. Prob. in Engg., 20, pp. 1-10,2013.

 

[32]     Gunduz Aras, C., Bayramov, S., Separation axioms on neutrosophic soft topological spaces, Turk J Math, 43, pp. 498 – 510,2019.

 

[33]     Jiang, J., Tang,Y., Chen,Q., Li, H., Tang,J., Interval-valued intuitionistic fuzzy soft sets and their proper- ties, Computer and Mathematics with Application, 60, pp. 906-918,2010.

 

[34]     Lu,J., Pan,L., Yang,Y., A Group medical diagnosis model based on intuitionistic fuzzy soft sets, Applied soft computing Journal, 77, pp. 453-466,2017.

 

[35]     Kalpana, G., Kalaivani, C., Fuzzy soft topology, International Journal of Engineering Studies,9 (1), pp. 45-56,2017.

 

[36]     Kandil, A., Tantawy,O. A. E., El-Sheikh, S. A., Abd El-latif, A. M. and El-Sayed. S., Fuzzy soft α- connectedness in fuzzy soft topological spaces, Journal of Mathematics and Computer Applications Re- search, 2 (1), pp. 37-46,2015.

 

[37]     Kandil, A., Tantawy,O. A. E., El-Sheikh, S. A., Abd El-latif, A. M., Fuzzy soft semi connected properties in fuzzy soft topological spaces, Math. Sci. Lett., 4, pp. 171-179,2015.

 

[38]     Kumud Borgohain., Separation axioms in intuitionistic fuzzy soft topological space, International Journal of Mathematics Trends and Technology, 5, pp. 176-180,2014.

 

[39]     Lowen, R., Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56, pp. 621–633,1976.

 

[40]     Li, Z., Cui, R., On the topological structure of intuitionistic fuzzy soft sets, Annals of Fuzzy Mathematics and Informatics, 5 (1), pp. 229-239,2013.

 

[41]     Maji, P. K., Neutrosophic soft set, Annals of Fuzzy Mathematics and Informatics, 5, pp. 157-168,2013.

 

[42]     Maji,P.K., Biswas, R., Roy, A. R., Fuzzy soft sets, Journal of Fuzzy Mathematics, 9 (3), pp. 589- 602,2001.

 

[43]     Maji,P.K., Biswas,R., Roy, A. R., Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics, 9 (3), pp. 677-691,2001. Maji,P.K., An application of weighted neutrosophic soft sets in a decision making problem, Springer proceedings in Mathematics and Statistics, 125, pp. 215-223,2015.

 

[44]     Mahanta, J., Das, P. K., Results on fuzzy soft topological spaces, arXiv:1203.0634v1,2012.

 

[45]     Majumdar, P., Samanta, S.K., Generalised fuzzy soft sets, Computers and Mathematics with Application, 59, pp. 1425-1432,2010.

 

[46]     Molodtsov, D., Soft set theory-first results, Computers and Mathematics with Application, 37, pp. 19- 31,1999.

 

[47]     Mahanta, J., Das, P. K., Fuzzy soft topological spaces, Journal of Intelligent & Fuzzy Systems, 32, pp.443–450,2017.

 

[48]     Mukherjee, A., Das,A. K., Saha, A., Sequence in intuitionistic fuzzy soft multi topological spaces, NTM- SCI, 3( 3), pp. 184-191,2015 .

 

[49]     Mukherjee, A., Das,A. K., Topological structure formed by fuzzy soft multi sets, Bulletin of Cal. Math. Soc., 21, pp. 193-212,2013.

 

[50]     Mukherjee, A., Das,A. K., Parameterized topological space induced by an intuitionistic fuzzy soft multi topological space, Annals of Pure and Applied Math., 7 pp. 7-12,2014.

 

[51]     Mukherjee, A., Das, A. K. and Saha, A., Topological structure formed by soft multi sets and soft multi compact space, Annals of Fuzzy Mathematics and Informatics, 7, pp. 919-933,2014.

 

[52]     Muthukumar,P., Sai sundara Krishnan, G., A similarity measure of intuitionistic fuzzy soft sets and its application in medical diagnosis, Applied Soft Computing, 41, pp. 148- 156,2016.

 

[53]     Neog, T., Su, D. K., Hazarika, G. C., Fuzzy soft topological spaces, Inter J. Latest Trend Math., 2(1), pp. 87-96,2012.

 

[54]     Osmanoglu, I., Tokat, D., On intuitionistic fuzzy soft topology ,Gen. Math. Notes, 19 (2), pp. 59-70,2013.

 

[55]     Parimala, M., Karthika, M., Jafari, S., Smarandache, F. and Udhayakumar, R., Decision- making via neutrosophic support soft topological spaces, Symmetry, 217(10), pp. 1-10,2018.

 

[56]     Parimala, M., Karthika, M., Dhavaseelan, R. and Jafari, S., On neutrosophic supra pre- continuous func- tions in neutrosophic topological spaces, New Trends in Neutrosophic Theory and Applications, 2, pp. 371-383,2017.

 

[57]     Parimala, M., Indirani, C., Perumal, R., On intuitionistic fuzzy contra psi-continuous mappings in topo- logical spaces, International Journal of Pure and Applied Mathematics, 113 (12), pp. 107-114,2017.

 

[58]     Parimala, M., Indirani, C., On intuitionistic fuzzy beta-supra open set and intuitionistic fuzzy beta-supra continuous functions, Notes on Intuitionistic Fuzzy Sets, 20 (3), pp. 6-12,2014.

 

[59]     Parimala, M., Karthika, M,, Smarandache, F., Broumi,S., On αω-closed sets and its connectedness in terms of neutrosophic topological spaces, International Journal of Neutrosophic Science, 2(2), PP: 82-88 , 2020.

 

[60]     Roy, S., Samanta, T. K., A Note on fuzzy soft topological spaces, Annals of fuzzy mathematics and informatics, 3(2), pp. 305-311,2012.

 

[61]     Roy, A. R., Maji, P. K., A Fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics, 203, pp. 412–418,2007.

 

[62]     Roy, S., Samanta, T. K., An introduction to open and closed sets on fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 6 (2), pp. 425- 431,2012.

 

[63]     Sabir Hussain., On weak and strong forms of fuzzy soft open sets, Fuzzy information and Engineering, 8 (4), pp. 451-463,2016.

 

[64]     Salama, A.A., Alblowi, S.A.,Neutrosophic set and neutrosophic topological spaces, IOSRJ.Mathematics, 3 (4), pp. 31-35,2012.

 

[65]     Sahin, M., Alkhazaleh,S., Ulucay,V., Neutrosophic soft expert sets, Applied Mathematics, 6, pp. 116- 127,2015.

 

[66]     Simsekler, T., Yuksel, S., Fuzzy soft topological space, Annals of fuzzy mathematics and Informatics, 5 (1), pp. 87-96,2013.

 

[67]     Smarandache. F, Neutrosophy and neutrosophic logic, First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics, University of New Mexico, Gallup, NM 87301, USA (2002).

 

[68]     Smarandache, F., Neutrosophic set, a generalization of the intuitionistic fuzzy sets, Inter.J.Pure Appl.Math., 24, pp.287 – 297,2005.

 

[69]     Shabir, M., Naz, M., On soft topological spaces, Comput. Math. Appl., 61, pp. 1786-1799,2011.

 

[70]     Shuker Mahmood Khalil., On intuitionistic fuzzy soft β-closed sets in intuitionistic fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 10 (2), pp. 221-233,2015.

 

[71]     Tanay, B., Burc Kandemir, M., Topological structure of fuzzy soft sets, Comp. Math. Appl., 61,pp. 2952- 2957,2011.

 

[72]     Varol, B. P., Aygun, H., Fuzzy soft topology, Hacettepe Journal of Mathematics and Statistics, 41 (3), pp. 407 – 419,2012.

 

[73]     Xu, Y., Sun, Y., Li. D., Intuitionistic fuzzy soft set, 2nd International workshop on intelligent systems and applications, IEEE, China,2010.

 

[74]     Yang, C., A note on soft set theory, Computers and Mathematics with Applications, 56, pp. 1899–1900,2008.

 

[75]     Yang, X., Lin, T. S., Yang, J., Li, Y. and Yu, D., Combination of interval-valued fuzzy set and soft set, Computers and Mathematics with Applications 58 , pp. 521–527,2009.

 

[76]     Yang. Y., Tan, X. and Meng, C., The multi-fuzzy soft set and its application in decision making, Applied Mathematical Modelling, 37, pp. 4915-4923,2013.

 

[77]     Yao, B. X., Liu, J. L., Yan, R. X., Fuzzy soft set and soft fuzzy set, Fourth international conference on natural computation, IEEE.2008

 

[78]     Yin, Y., Li, H. and Jun, Y. B., On algebraic structure of intuitionistic fuzzy soft sets, Computer and Mathematics with Applications, 64, pp. 2896-2911,2012.

 

[79]     Yogalakshmi, T., On intuitionistic soft fuzzy compactness, International Journal of Pure and Applied Mathematics, 115 (9), pp. 231-237,2017.

 

[80]     Zadeh, L.A.,Fuzzy sets, Information and Control, 8, pp.338–353,1965.

 

[81]     Zhi Kong., Lifu Wang., Zhaoxia Wu., Application of fuzzy soft set in decision making problems based on grey theory, journal of Computational and applied Mathematics, 236, pp.1521-1530,2011.