Fundamental Group of Rough Topological Spaces
Abstract
In this paper we study and define the concept of the fundamental group of rough topological spaces (RTSs), which deeply depends on the concepts of rough sets (RSs) and rough topology (RT). Working towards this stated objective, we define the concept of rough path (RPt) which gives room for the introduction of rough loop (RL). We also define the concepts of rough homotopy (RH) and later shows that it is indeed an equivalence relation. We introduce the fundamental group of rough topological spaces by showing that all the group axioms satisfied. Also, this paper establish the fact that most of the results in fundamental group of ordinary topological spaces are also hold for the fundamental group of rough topological spaces.
References
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
H. Poincaré, Analysis situs and its five supplements, Papers on Topology 2 (2009), 22-28.
B.P. Mathew and S.J. John, On rough topological spaces, International Journal of Mathematical Archive 3(9) (2012), 3413-3421.
B.P. Mathew and S.J. John, I-rough topological spaces, International Journal of Rough Sets and Data 3(1) (2016), 98-311. https://doi.org/10.4018/IJRSDA.2016010106
B.P. Mathew and S.J. John, Some special properties of I-rough topological spaces, Annals of Pure and Applied Mathematical 12(2) (2016), 111-122.
B.P. Mathew and S.J. John, Some studies on I-rough topological spaces, International Journal of Mathematical Archive 11(3) (2020), 20-29.
E.A. Az-zo’bi, M.F. Marashdah and R.F. Uzabashy, The fundamental group of intuitionistic fuzzy topological spaces, Applied Mathematical Sciences 8(157) (2014), 7829-7843. https://doi.org/10.12988/ams.2014.49719
E.F. Lashin, A.M. Kozae, A.A. Abo Khadra and T. Medhat, Rough set theory for topological spaces, Internat. J. Approx. Reason. 40 (2005), 35-43. https://doi.org/10.1016/j.ijar.2004.11.007
G. Varma and S.J. John, Generalized multi-fuzzy rough sets and the induced topology, IOSR Journal of Mathematics (IOSR-JM) (2017), 2278-5728.
G. Zhang, On topological structures of IVF approximation spaces, Fuzzy Inf. Eng. 8 (2016), 217-227. https://doi.org/10.1016/j.fiae.2016.06.006
H. Zhang, L. Xiong and W. Ma, Generalized intuitionistic fuzzy soft rough set and its application in decision making, J. Comput. Anal. Appl. 20(4) (2016), 750-766.
J. Brazas, The fundamental group as a topological group, Topology Appl. 160 (2013), 170-188. https://doi.org/10.1016/j.topol.2012.10.015
J.S. Calcut, R.E. Gompf and J.D. McCarthy, On fundamental groups of quotient spaces, Topology Appl. 159 (2012), 322-330. https://doi.org/10.1016/j.topol.2011.09.038
K. Saito and T. Ishibe, Monoids in the fundamental groups of the complement of logarithmic free divisors in C^3, J. Algebra 344 (2011), 137-160. https://doi.org/10.1016/j.jalgebra.2011.07.018
L. Abdullateef, On algebraic properties of fundamental group of intuitionistic fuzzy topological spaces (IFTSs), Open Journal of Mathematical Sciences 4 (2020), 34-47. https://doi.org/10.30538/oms2020.0092
N. Xie, On topological properties of IF approximation spaces, Fuzzy Inf. Eng. 8 (2015), 183-193. https://doi.org/10.1016/j.fiae.2015.05.004
R. Mareay, Generalized rough sets based on neighborhood systems and topological spaces, J. Egyptian Math. Soc. 24 (2016), 603-608. https://doi.org/10.1016/j.joems.2016.02.002
V. Madhuri and B. Amudhambigai, The fuzzy J^*-fundamental group of fuzzy J^*-structure spaces, Int. Journal of Computational and Appl. Math. 12(1) (2017), 40-53.
W. Tang, J. Wu and D. Zheng, On fuzzy rough sets and their topological structures, Math. Probl. Eng. 2014, Art. ID 546372, 17 pp. https://doi.org/10.1155/2014/546372
Y.B. Jun, K.J. Lee and C.H. Park, Soft set theory applied to ideals in d-algebras, Comput. Math. Appl. 57 (2009), 367-378.
Z. Li and R. Cui, On topological structure of intuitionistic fuzzy sets, Fuzzy Math. Inform. 5(1) (2013), 229-239.
Z. Li, T. Xie and Q. Li, Topological structure of generalized rough sets, Comput. Math. Appl. 63 (2012), 1066-1071. https://doi.org/10.1016/j.camwa.2011.12.011
Z. Pawalak, Rough sets, Internat. J. Comput. Inform. Sci. 11 (1982), 341-356. https://doi.org/10.1007/BF01001956
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