J. Bangladesh Acad. Sci., Vol. 42, No. 1, 67-71, 2018
CERTAIN PROPERTIES OF FUZZY PATH CONNECTEDNESS AND MAXIMAL FUZZY PATH CONNECTEDNESS
RUHUL AMIN* AND SOHEL RANA
Department of Mathematics, Faculty of Science, Begum Rokeya University, Rangpur-5404, Bangladesh
ABSTRACT
This paper deals with some properties of fuzzy path connectedness and maximal fuzzy path connectedness on a fuzzy topological space by means of fuzzy points.
Key words: Fuzzy point, Fuzzy path, Fuzzy path connectedness, Maximal fuzzy path connectedness.
INTRODUCTION fuzzy path connected spaces are fuzzy The concept of fuzzy sets was first connected. Further the image of a fuzzy path introduced by L. A. Zadeh (1965). Later C. connected space and of a maximal fuzzy path L. Chang (1968), R. Lowen (1976), Halakatti connected space under fuzzy continuous and Halijol (2016) and others developed the mapping is fuzzy path connected and maximal theory of fuzzy topological spaces. One fuzzy path connected, respectively. intrinsic characteristic of fuzzy topological space is path connectedness. C. L. Chang BASIC NOTIONS AND PRELIMINARY (1968) first introduced fuzzy path and fuzzy RESULTS path connectedness and D. M. Ali (1987) introduced the new concept of fuzzy path In this section, we recall some concepts which connected space and studied its various will be needed in the sequel. In the present characteristics. The purpose of this paper is paper, X and Y always denote non-empty sets to further contribute to the development of and I = [0 1] is the closed unit interval. fuzzy path connected spaces. Definition: Let X be a set and IX denote the In the present paper, we introduced a new set of fuzzy sets in X. A fuzzy topology on X definition of fuzzy path, fuzzy path is a subset IX such that connectedness and maximal fuzzy path (i) 0, 1 connectedness depending on fuzzy point. By (ii) using these definitions, we observe that fuzzy (iii) (i)iI sup i path connectedness and maximal fuzzy path connectedness forms an equivalence relation. Then is (X, t) called a fuzzy topological Also, fuzzy path connected and maximal space or short fts. (Chang 1968).
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Definition: A fuzzy point p in X is a special MAIN RESULTS type of fuzzy set with membership function Definition: Let (X, t) be a fuzzy topological r if x x space and I = [0, 1]. A continuous mapping f p(x) 0 : I X is said to be a fuzzy path in X if for 0 if x x0 any two fuzzy points p, q X such that f(0) where 0 < r < 1. (Zadeh 1965) = p, f(1) = q. Definition: Let (X, t) and (Y, s) be two fuzzy Definition: Any fuzzy topological space (X, t) topological spaces. A function is said to be fuzzy path connected iff for every f : (X, t) (Y, s) is called fuzzy continuous fuzzy point p, q X there exists a fuzzy path f iff f–1 t for every v s. (Zadeh 1965) in X such that f(0) = p and f(1) = q.
Definition: A fuzzy topological space X is Theorem: Every fuzzy path connected space said to be fuzzy connected iff = 1, is fuzzy connected. where and are non-empty open fuzzy Proof: Let X be fuzzy path connected and let sets in X such that = 0. (Sethupathy- Lakshmivarahan 1977) X = , where and are non-empty open fuzzy sets. We have to prove that Definition: Let (X, t) be an fuzzy topological . space and I = [0 1]. A continuous mapping f : I X is said to be a fuzzy path in X if for Suppose . Taking two fuzzy points p any two points x, y X such that f(0) = x, and q . Then p, q X. As X is fuzzy f(1) = y (Ali 1987). path connected, then there exists a fuzzy path f : I X such that f(0) = p, f(1) = q. Definition: Let f is a fuzzy path from p to q i.e., f : I X such that f(0) = p, f(1) = q and As p , q and f is a fuzzy path from also let g is a fuzzy path from q to r i.e., g : I fuzzy point p to the fuzzy point q which X such that g(0) = q, g(1) = r. gives a contradiction. Hence . Therefore X is fuzzy connected. Now define h : I X by Theorem: Let X and Y be fts and f : X Y 1 f (2t); 0 t t 2 be an onto fuzzy continuous function. If X is h(t) 1 fuzzy path connected then Y is fuzzy path g(2t); t 1 2 connected. Since f(1) = r = g(0), h is well defined and it Proof: Let X is fuzzy path connected and p, q is obviously a fuzzy continuous function. be the two fuzzy points in Y. Then by Then h is a composite fuzzy path. (Halakatti- definition of fuzzy continuous function, we Halijol 2016) have f–1(p) and f–1(q) is in X. Since X is fuzzy path connected, then there exists a fuzzy path Definition: Considering a fuzzy path f : I g (say) from f–1(p) to f–1(q) is in X, i.e., g : I X such that f(x) = t, x [0, 1]. Then f is X is a fuzzy path such that g(0) = f–1(p) called a constant fuzzy path. (Halakatti- –1 Halijol 2016) and f (q). CERTAIN PROPERTIES OF FUZZY PATH CONNECTEDNESS 69
Now consider the function fg : Y. Then Symmetry : Suppose f is a fuzzy path from (fg) (0) = f(g(0)) = f–1(q) = p and (fg) (1) = the fuzzy p to the fuzzy point q such that f(0) f(g(1)) = f(f–1(q)) = p. = p, f(1) = q. Thus fg : I Y is a fuzzy path in Y from p to Now define g : I X such that g(t) = f(1 – t) q. Therefore, Y is fuzzy path connected. t [0 1].
Theorem: Let {Xi} be a collection of fuzzy Now g(0) = f(1) = q and g(1) = f(1) = p and g path connected spaces and Xi = . Then X is fuzzy continuous mapping. = X . Then X = X is fuzzy path i i Hence g : I X is a fuzzy path from q to p. connected. Therefore, fuzzy path connected relation is Proof: Let p, q be the two fuzzy points in X, symmetric relation. then for some i and i, we have 1 2 Transitivity : Consider p, q and r be the fuzzy p X , q X . i1 i2 points in X. Suppose f : I X is a fuzzy path Let r be the fuzzy point in X X . Since i1 i2 such that f(0) = p, f(1) = r and g : I X is a X is fuzzy path connected, then there exists fuzzy path such that g(0) = r, g(1) = q. i1 a fuzzy path f : I X from p to r such that i1 1 f (2t); 0 t f(0) = p and f(1) = r. 2 Now define h(t) Similarly, for X , we get a fuzzy path 1 i2 g(2t); t 1 2 g : I X from r to q such that g(0) = r, i2 g(1) = q. As f(1) = r = g(0) then h is well defined and h is fuzzy continuous. Now h(0) = f(0) = p 1 f (2t); 0 t and h(1) = f(1) = q. Thus h is a fuzzy path 2 Now define h(t) 1 from p to q. Hence fuzzy path connected g(2t); t 1 2 relation is transitive relation. Therefore, fuzzy path connectedness is an Since f(1) = r = g(0), h is well defined and h equivalence relation. is fuzzy continuous mapping. Definition: Let X be a fuzzy topological Now h(0) = f(0) = p and h(1) = g(1) = q. space and p, q, r be the fuzzy points in X. If Thus h is a fuzzy path in X. Therefore X is there exists fuzzy continuous mapping f : I fuzzy path connected. X such that Theorem: Fuzzy path connectedness of 1 fuzzy topological spaces is an equivalence f (0) p, f q, f (1) r relation. 2 Proof: Let X be a fuzzy path connected then p is a maximal fuzzy path connected to space. r. If it is true for all p, q, r X, then X is a maximal fuzzy path connected space. Reflexivity : Reflexive relation is trivial by considering a constant fuzzy path f : I X Theorem: Every maximal fuzzy path such that f(t) = p t [0 1]. connected space is fuzzy connected. 70 AMIN AND RUNA
Proof: Let X be a maximal fuzzy path This implies that fog is a maximal fuzzy path connected space. Le X = , where and in Y from p to r. Therefore, Y is maximal are non-empty open fuzzy sets in X. Now fuzzy path connected space. we have to show that . Theorem: Maximal fuzzy path Suppose . Suppose that p, q be two connectedness of a fuzzy topological space is fuzzy points in and r be the fuzzy point in an equivalence relation. . This implies that p, q, r X. Since X is Proof: Let X be a fuzzy topological space. maximal fuzzy path connected, so that there Now we shall show that maximal fuzzy path exists a maximal fuzzy path f : I X such connectedness is an equivalence relation. that Reflexivity : Reflexive relation is trivial by 1 f (0) p, f q, f (1) r. considering a constant fuzzy path f : I X 2 such that f(p) = t, p [0 1]. Therefore, This is a contradiction. maximal fuzzy path connectedness is a reflexive relation. Similarly, the same contradiction arise if we let p be the fuzzy point in and q, r be the Symmetry : Suppose f is a maximal fuzzy path from the fuzzy point p to the fuzzy point fuzzy points in . Hence . Thus X r i.e., path f : I X such that is fuzzy connected space. 1 Theorem: Let X and Y be fuzzy topological f (0) p, f q, f (1) r. 2 spaces and f : X Y be an onto fuzzy continuous function. If X is maximal fuzzy Now we define g : I X such that path connected space then Y is so. g(t) = f(1 – t), t [0 1]. It is clear that g is fuzzy continuous and hence a maximal fuzzy Proof: Let X be maximal fuzzy path path in X from r to p such that connected and p, q, r be the fuzzy points in g(0) = f(1 – 0) = f(1) = r, Y. Then by definition of fuzzy continuous –1 –1 –1 1 1 1 function, we have f (p), f (q), f (r) in X. g f 1 f q, Since X is maximal fuzzy path connected 2 2 2 space, then there exists a maximal fuzzy path g(1) = f(1 – 1) = f(0) = p. g : I X such that Therefore, maximal fuzzy path 1 connectedness is a symmetric relation. g(0) f 1 ( p), g f 1 (q), g(1) f 1 (r). 2 Transitivity : Let f is a maximal fuzzy path Consider the function fog : I Y such that from the fuzzy point p to the fuzzy point r (fog) (0) = f(g(o)) = f(f–1(p)) = p i.e., f : I X such that
1 1 1 1 ( fog) f g f ( f (q)) q f (0) p, f q, f (1) r and also let g is 2 2 2 a maximal fuzzy path from the fuzzy point r (fog) (1) = f(g(1)) = f(f–1(r)) = r. to the fuzzy point t i.e., g : I X such that CERTAIN PROPERTIES OF FUZZY PATH CONNECTEDNESS 71
1 Ali, D. M. 1992. Some other types of fuzzy g(0) r, g s, g(1) t . 2 connectedness. Fuzzy Sets and Systems. 46: 56-61. Now define h : I X such that h(t) Ajmal, N. and Kohili. 1989. Connectedness in fuzzy topological spaces. Fuzzy sets and 1 f (2t); 0 t systems. 31: 369-388. 2 Chang, C. L. 1968. Fuzzy topological spaces. J. 1 Math. Annal. Appl. 24: 182-190. g(2t); t 1 2 Halakatti, S. C. P. and A. Halijol. 2016. Different types of fuzzy path connectedness on fuzzy Since f(1) = r = g(0), then h is well defined Banach Manifold. IOSR Journal of and it is obviously a fuzzy continuous Mathematics (IOSR-JM). 12(3): 74-78. function. Lowen, R. 1976. Fuzzy topological space and compactness. J. Math. Annal. Appl. 56: 621- Now h(0) = f(0) = p 633. Ming, P. P., and L. Y. Ming. 1980. Fuzzy topology I. Neighbourhood structure of a fuzzy point 1 1 and Moore-Smith convergence. J. Math. Annal. f 2 f (1) r; 0 t Appl. 76: 571-599. 1 2 2 h Ming, P. P., and L. Y. Ming. 1980. Fuzzy 2 1 1 topology II. Product and quotient spaces. J. g2 1 g(0) r; t 1 2 2 Math. Annal. Appl. 77: 20-37. Sethupathy, K. S. R., and S. Lakshmivarahan. h(1) = g(1) = t. 1977. Connectedness in fuzzy topology. KYBERNETIKA. 13(3). Therefore, h is a maximal fuzzy path from p Sohel Rana, R. Amin, S. S. Miah and R. Islam. to t. Thus, maximal fuzzy path 2017. Studies on fuzzy connectedness. J. connectedness is transitive relation. Bangladesh Academy of Sciences. 41(2): 175- 182. Therefore, maximal fuzzy path Wuyts, P. 1987. Fuzzy path and fuzzy connectedness is an equivalence relation. connectedness. Fuzzy sets and Systems. 24: 127-128. REFERENCES Zadeh, L. A. 1965. Fuzzy sets. Information and control. 8: 338-353. Ali, D. M. and Arun K. Srivastava. 1988. On fuzzy connectedness. Fuzzy Sets and Systems. Zheng, C. Y. 1984. Fuzzy path and fuzzy 28: 203-208. connectedness. Fuzzy Sets and Systems. 14: 273-280. Ali, D. M. 1987. On fuzzy path connectedness. BUSEFAL. 26-33. (Received revised manuscript on 17 April, 2018)