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Mixed fuzzy ideal topological spaces
Applied Mathematics and Computation 220 (2013) 602–607
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Mixed fuzzy ideal topological spaces Binod Chandra Tripathy a, Gautam Chandra Ray b,⇑ a b
Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Pashim Boragaon, Guwahati 781035, Assam, India Department of Mathematics, Central Institute of Technology, Kokrajhar 783370, Assam, India
a r t i c l e
i n f o
Keywords: Fuzzy I-open sets Fuzzy b-I-open sets Semi I-open sets
a b s t r a c t The aim of this paper is to introduce a new concept of mixed fuzzy ideal topological spaces and investigate some properties of this space. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The concept of ideal in topological spaces has been introduced and studied by Jankovic and Hamlett [3], Kuratouski [4], Nasef and Mahmoud [7], Vaidyanathswamy [21], Yuksel, Caylak and Acikgoz [23], Yuksel, Acikgoz and Noiri [24] and Yuksel, Kara and Acikgoz [25]. In 1997, Sarkar [9] and Mahmoud [5] introduced the notions of fuzzy ideal and fuzzy local function in fuzzy set theory. Malakar [6] introduced the concepts fuzzy semi-irresolute and strongly irresolute functions. Hatir and Jafari [2] and Nasef and Hatir [8] defined fuzzy semi-I-open set and fuzzy pre-I-open set via fuzzy ideal. The notion of Ideal has been applied in sequence spaces and different classes of ideal convergent sequences have been introduced and investigated by Tripathy and Dutta [10], Tripathy and Hazarika [11–15], Tripathy and Mahanta [16], Tripathy and Sarma [19], Tripathy, Sen and Nath [20] and many others in the recent years. In 1995, Das and Baishya [1] introduced the concept of mixed fuzzy topological spaces. Tripathy and Ray [17] introduced and studied the concept of mixed fuzzy topological spaces in slightly different ways which is not fuzzification of classical mixed topology. Tripathy and Ray [18] introduced and investigated different properties of fuzzy weakly continuous functions, fuzzy d-continuous function between two mixed fuzzy topological spaces. 2. Preliminaries and definitions Let X be a non-empty set and I, the unit interval [0,1]. A fuzzy set A in X is characterised by a function lA: X ? I where lA is called the membership function of A and lA(x) representing the membership grade of x in A. The empty fuzzy set is defined by lØ(t) = 0 for all t 2 X. Also X can be regarded as a fuzzy set in itself defined as lX(t) = 1 for all t 2 X. Further, an ordinary subset A of X can also be regarded as a fuzzy set in X if its membership function is taken as usual characteristic function of A that is lA(t) = 1 for all t 2 X and lA(t) = 0 for all t 2 X-A. Two fuzzy sets A and B are said to be equal if lA = lB. A fuzzy set A is said to be contained in a fuzzy set B, written asA # B, if lA 6 lB. Complement of a fuzzy set A in X is a fuzzy set denoted by Ac in X. Its membership function is defined by lAc ¼ 1 lA :. We can also denote the complement of A by coA that is Ac = coA. Union and intersection of a collection {Ai:i 2 I} of fuzzy sets in X, are written as [ Ai and \ Ai , respectively. The membership i¼1 i¼1 functions are defined as follows:
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (G.C. Ray). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.05.072
B.C. Tripathy, G.C. Ray / Applied Mathematics and Computation 220 (2013) 602–607
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l[Ai ðxÞ ¼ supflAi ðxÞ : i 2 Ig for all x 2 X: i2I
and
l\Ai ðxÞ ¼ infflAi ðxÞ : i 2 Ig for all x 2 X: i2I
A fuzzy topology s on X is a collection of fuzzy sets in X such that Ø, X 2 s; if Ai 2 s, i 2 I then [ Ai 2 s and if A, B 2 s then i2I A \ B 2 s. The pair (X, s) is called a fuzzy topological space (fts). Members of s are called open fuzzy sets and the complement of an open fuzzy set is called a closed fuzzy set. If (X, s) is a fts then the closure and interior of a fuzzy set A in X, denoted by cl A and int A, respectively, are defined as cl A = \ {B: B is a closed fuzzy set in X and A # B} and int A = [ {V: V is an open fuzzy set in X and V # A}. Clearly, cl A (respectively int A) is the smallest (respectively largest) closed (respectively open) fuzzy set in X containing (respectively contained in) A. If there is more than one topology on X, then the closure and interior of A with respect to a fuzzy topology s on X will be denoted by s-cl A and s-int A. Definition 2.1. A collection B of open fuzzy sets in fts X is said to be an open base for X if every open fuzzy sets in X is a union of members of B. Definition 2.2. If A is a fuzzy set in X and B is a fuzzy set in Y then, A B is a fuzzy set in X Y defined as lA B (x,y) = min{lA(x), lB(y)} for all x 2 X and for all y 2 Y. Definition 2.3. Let f be a function from X into Y. Then for each fuzzy set B in Y, the inverse image of B under f, written as f1[B], is a fuzzy set in X defined as lf 1 ½B ðxÞ ¼ lB ðf ðxÞÞ for all x 2 X. Definition 2.4. A fuzzy set A in a fuzzy topological space (X, s) is called a neighbourhood of a point x 2 X if and only if there exists B 2 s such that B # A and A(x) = B(x) > 0. Definition 2.5. A fuzzy point xa is said to be quasi-coincident with A, denoted by xa qA, if and only if a + A(x) > 1 or a > (A(x))c. Definition 2.6. A fuzzy set A is said to be quasi-coincident with B and is denoted by AqB, if and only if there exists an x 2 X such that A(x) + B(x) > 1. It is clear that if A and B are quasi-coincident at x both A(x) and B(x) are not zero at x and hence A and B intersect at x. Definition 2.7. A fuzzy set A in a fts (X, s) is called a quasi-neighbourhood of xk if and only if A1 2 s such that A1 # A and xk qA1. The family of all Q-neighbourhood of xk is called the system of Q-neighbourhood of xk. Intersection of two quasi-neighbourhood of xk is a quasi-neighbourhood. Definition 2.8. Let (X, s1) and (X, s2) be two fuzzy topological spaces. Consider the collection of fuzzy sets s1(s2) = {A 2 IX: For any fuzzy set B in X with AqB, there exists s2-open set A1 such that A1qB and s1-closure A1 # A}. Then this family of fuzzy sets will form a topology on X and this topology we call a mixed fuzzy topology on X. Definition 2.9. A non-empty collection of fuzzy sets I of a set X is called a fuzzy ideal if the following postulates are satisfies (i) if A 2 I and B 6 A then B 2 I. (heredity) and (ii) if A, B 2 I then A _ B 2 I. (finite additivity) The triplet (X, s, I) is called Ideal fuzzy topological space with the ideal I and fuzzy topology s. Definition 2.10. Let A be any subset of X in a fuzzy ideal topological space (X, s, I). The fuzzy local function of A with respect to s and I is denoted by A⁄(s, I) in short A⁄. The fuzzy local function A⁄(s, I) of A is the union of all fuzzy points xk such that U 2 Nq(xk ) and E 2 I, then there is at least one y 2 X for which U(y) + A(y) 1 > E(y). i.e. A⁄ = _{x 2 X: A_U R I for every U 2 s(x)}. The fuzzy closure operator of a fuzzy set A in (X, s, I) is defined as cl⁄(A) = A_A⁄. In (X, s, I) the collection s⁄(I) means an extension of fuzzy topological space than s via fuzzy ideal which is constructed by considering the class b = {U E: U 2 s, E 2 I} as a base. A fuzzy subset A of a fuzzy ideal topological space (X, s, I) is said to be I-open (respectively a-I-open fuzzy set, pre-I-open fuzzy set, semi-I-open fuzzy set, b-I-open fuzzy set) if A 6 int(cl⁄(intA)) (respectively A 6 int(cl⁄(int (A))), A 6 int(cl⁄(A)), A 6 cl⁄(int(A)), A 6 cl(int(cl⁄(A)))). Definition 2.11. A fuzzy subset A of a fuzzy ideal topological space (X, s, I) is said to be fuzzy ⁄-perfect if A = A⁄.
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Lemma 2.1 (Yuksel, Kara, Acikgoz [17], Lemma 3.1). Let A and B be fuzzy subsets of a fuzzy ideal topological space (X, s, I). Then we have (a) If A 6 B, then A⁄ 6 B⁄, (b) If U 2 s, then U^A 6 (U^A)⁄ (c) A⁄ is fuzzy closed set in (X, s, I). 3. Main results We introduce the mixed fuzzy ideal topological space as follows. Definition 3.1. Let X be a non-empty set and I be a fuzzy ideal in X. Consider a mixed fuzzy topological space (X, s1(s2)). Then the triplet (X, s1(s2), I) is said to be mixed fuzzy ideal topological space. Example 3.1. Let X = {x, y} and the fuzzy sets are defined by A1 = {(x,.2),(y,.8)}, A2 = {(x,.2),(y,.2)}, A3 = {(x, .8), (y, .2)}, A4 = {(x, .8), (y, .8)} Then the collection s1 = {0; 1; A1, A2, A3, A4} will form a fuzzy topology in X. Also, consider the following fuzzy sets in X B1 = {(x,.3),(y,.7)}, B2 = {(x,.7),(y,.3)}, B3 = {(x, .3), (y, .3)}, B4 = {(x, .7), (y, .7)} Then the collection of fuzzy sets s2 = {0; 1; B1, B2, B3, B4} will form a fuzzy topology on X and (X, s1(s2)) is a mixed fuzzy topological space. Let I = {0; 1}, then I is a fuzzy ideal in X. Therefore (X, s1(s2), I) is a mixed fuzzy ideal topological space. Definition 3.2. Let (X, s1(s2), I) be a mixed fuzzy ideal topological space. A fuzzy set A in X is said to be fuzzy I-open set if A 6 int (A⁄), where A⁄ is the fuzzy local function of A defined as A⁄ = _{x 2 X: A^U R I for every U 2 s(x)}. Example 3.2. Let us consider a non-empty set X = {x, y} and consider the following fuzzy sets in X. A = {(x, .7), (y, .3)} and B = {(x, .3), (y, .7)}. Then the collection of fuzzy sets s1 = {0;1;B} and s2 = {0;1;A} are two fuzzy topologies on X. Now, we construct the mixed fuzzy topology on X from these two fuzzy topologies s1 and s2. Since the mixed fuzzy topology is coarser than s2, so we need only to verify whether the fuzzy set A is in s1(s2) or not. Let us consider a fuzzy set S in X such that AqS. Now, the only s2-open sets are 1 and A such that AqS and 1 qS, Again, s1-closure of A = ^fF : F is s1-closed and A # F}
¼ 1 ^ A ¼ A # A: Hence, A 2 s1(s2) and so s1(s2) = {0; 1;A}. Now, we consider an ideal I = {0} on X. Then the triplet (X, s1(s2), I) is a mixed fuzzy topological space. Let C = {(x, .1), (y, .8)} be a fuzzy set in X. Then C⁄ = _{xk: C^U R I, for every U 2 s1(s2) and x 2 U}. Now, consider the fuzzy point x.3 in X. The open set in s1(s2) containing x.3 are A and 1. Then C^A R I and C^1 R I and so x.3 2 C⁄. Again consider the fuzzy point y.7 2 C. The open set containing y.7 is 1. Then C^1 R I and so y.7 2 C⁄. Similarly we can show that for any fuzzy point xa > .3 and yb > .7 in X is a member of C⁄. Therefore C 6 int(C⁄) and so C is an I-open set in (X, s1(s2), I). Definition 3.3. Let (X, s1(s2), I) be a mixed fuzzy ideal topological space. A fuzzy set A in X is said to be fuzzy a-I-open set (respectively fuzzy pre-I-open set, fuzzy semi-I-open set, fuzzy b-I-open set) if A 6 int(cl⁄(int(A))) (respectively A 6 cl(int(cl⁄(A))), A 6 cl⁄(int(A)), A 6 cl(int(cl⁄(A)))). Definition 3.4. A fuzzy set A in a mixed fuzzy ideal topological space (X, s1(s2), I) is said to be fuzzy regularly I-open set if A = int(cl⁄(A)) where cl⁄(A) = A_A⁄. Definition 3.5. A fuzzy point xk in a mixed fuzzy ideal topological space (X, s1(s2), I) is said to be d-I-cluster point of a fuzzy set A in X if Aq(int(cl⁄V)) where V is any fuzzy I-regularly open Q-neighbourhood of xk. The set of all d-I-cluster point of A is called d-I-closure of A and is denoted by CldI(A). A fuzzy set A is said to be d-I-closed if A = CldI(A).
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Theorem 3.1. Let (X, s1(s2), I) be a mixed fuzzy ideal topological space and A and B be two fuzzy sets in X. Then the following results hold (i) If A 6 B, then A⁄ 6 B⁄. (ii) If A 6 B, then cl⁄(A) 6 cl⁄(B). (b) If U 2 s, then U^A 6 (U^A)⁄ (c) A⁄ is fuzzy closed set in (X, s, I). Proof. (i) Let x 2 X be any point. Since A 6 B so A(x) 6 B(x) for any x 2 X. Now, A⁄ = {x 2 X: A^U R I for every U 2 s1(s2) and x 2 U} 6 {x 2 X: B^U R I for every U 2 s1(s2) and x 2 U} = B⁄. Thus A 6 B ) A⁄ 6 B⁄. (ii) Let x 2 X be any point. We have cl⁄(A) = A^A⁄. Therefore, cl⁄(A)(x) = (A^A⁄)(x) = max {A(x), A⁄(x)} 6 max {B(x), B⁄(x)} = cl⁄(B). Thus A 6 B ) cl⁄(A) 6 cl⁄(B). Proof of (iii) and (iv) are similar to the proof of Lemma 3.1. of Yuksel, Kara, Acikgoz [17]. h Theorem 3.2. In a mixed fuzzy ideal topological space (X, s1(s2), I) a fuzzy I-open set need not be a fuzzy d-I-open set. Proof: The result follows from the following example. Example 3.3. Consider the mixed fuzzy ideal topological space as defined in Example 3.2. Then the fuzzy set C = {(x, .1), (y, .8)} in X is an fuzzy I-open set in X and so Cc = {(x, .9), (y, .2)} is fuzzy I-closed. Now, we show CC is not fuzzy d–I-closed set in X. i.e. every fuzzy point in Cc is not fuzzy d-I-cluster point. Consider the fuzzy point x.9 in Cc. Let U = {(x, .3), (y, .8)} be any fuzzy set in X. Then by Theorem 3.1. U⁄ is fuzzy closed set in X. But the only fuzzy closed set in s1(s2) are 0; 1and Ac = {(x, .3), (y, .7)}. We have U⁄ is the union of all fuzzy points xk such that V 2 Nq(xk ) and E 2 I, then there is at least one y 2 X for which V(y) + U(y) 1 > E(y). Since I = {0} so E(y) = 0. Therefore in this case U⁄ is the union of all fuzzy point xa such that for any V 2 s1(s2)(x) there exists r 2 X such that VqU. Consider the fuzzy point x.4, then A 2 s1(s2) be such that x.4 2 A and y 2 X such that A(y) + U(y) = .4 + .8 > 1 So AqU and consequently x.4 2 U⁄. Therefore U⁄ – 0: Similarly we show that U⁄ – 1: Thus we must have U⁄ = {(x, .3), (y, .7)}. Hence cl⁄U = U_U⁄ = {(x, .3), (y, .8)}_{(x, .3), (y, .7)}. = {(x, .3), (y, .8)}. Therefore int(cl⁄U) = int({(x, .3), (y, .8)}) = 1. Hence U is not fuzzy I-regularly open set. Thus x.9 is not a d-I-cluster point of Cc and so Cc is not a d-I-closed. So C is not fuzzy d-I-open. Now, we give an example to show that in a mixed fuzzy ideal topological space a fuzzy d-I-open set may be a fuzzy Iclosed set. Example 3.4. Consider the non-empty set X = {x, y} and the fuzzy ideal I = {0} and fuzzy sets A = {(x, .7), (y, .7)} and B = {(x, .3), (y, .3)} in X. Then s1 = {0;1} and s2 = {0,1;A, B} are fuzzy topologies on X. Consider the mixed fuzzy topology from these two topologies s1 and s2, then s1(s2) = {0;1;A, B} and (X, s1(s2), I) is a mixed fuzzy ideal topological space. Consider the fuzzy set U = {(x, .1), (y, .8)}, then V = Uc = {(x, .9), (y, .2)}. We prove that V is fuzzy d-I-closed. Consider the fuzzy point y.2 in V and C = {(x, .7), (y, .3)} a fuzzy set in X. Then C is fuzzy I-regularly open set because C⁄ – 0 and C⁄ – 1 so we must have C⁄ = {(x, .3), (y, .7)} (Since C⁄ is fuzzy closed in s1(s2)) Hence C_C⁄ = {(x, .7), (y, .7)} ) cl⁄(C) = {(x, .7), (y, .7)} Therefore int(cl⁄(C) = C. Thus C is fuzzy I-regularly open Q-neighbourhood of the fuzzy point y.2 such that CqV. Hence y.2 is a fuzzy d-I-cluster point of V. Also consider the fuzzy point x.9 in X. Then 1 is the only fuzzy I-regularly open Q-neighbourhood of the fuzzy point x.9 such that 1qV and so x.9 is a fuzzy d-I-cluster point of V.
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Thus all point of the fuzzy set V are fuzzy d-I-cluster point and hence V is fuzzy d-I-closed and consequently U is fuzzy d-Iopen. Now, we show that U is fuzzy I-closed. We have V⁄is fuzzy closed in (X, s1(s2), I). By definition V⁄ = _{xk 2 X: V^E R I where E 2 s1(s2) and x 2 E}. Hence V⁄ – 0: We show that V⁄ – A. Consider the fuzzy point x.7, then A and 1 are the only fuzzy open set in X such that V^A R I and V^1 R I and so x.7 e V⁄. Further, if we consider the fuzzy point x.3, then B and 1 are the only fuzzy open set in X such that VB R I and V 1 R I and hence x.3 2 V⁄. Hence V⁄ – A. Similarly we can show that V⁄ – B. But V⁄ is a fuzzy closed set in s1(s2) so V⁄ = 1. Now, int(V⁄) = int(1) =1: Hence V 6 int(V⁄) and so V is fuzzy I-open and consequently U is fuzzy I-closed set. Thus in a mixed fuzzy ideal topological space a fuzzy d-I-open set may be a fuzzy I-closed. Now, we establish a relation between fuzzy I-regularly open set and fuzzy I-open set in a mixed fuzzy ideal topological space. Theorem 3.3. In a mixed fuzzy ideal topological space a fuzzy I-regularly open set is not fuzzy I-open. Proof. Let (X, s1(s2), I) be a mixed fuzzy ideal topological space. Let A be any fuzzy I-regularly open set in X. Then by definition of fuzzy I-regularly open set we get A = int(cl⁄(A)) = int(A_A⁄) P int (A⁄). Thus we get A P int (A⁄). Hence A is not fuzzy I-open set. Now, we give an example that in a mixed fuzzy ideal topological space a fuzzy I-open set may not be fuzzy I-regularly open set. h Example 3.5. Consider the non-empty set X = {x, y} and the fuzzy ideal I = {0} and fuzzy sets A = {(x, .7), (y, .7)} and B = {(x, .3), (y, .3)} in X. Then s1 = {0;1 and s2 = {0;1;A, B} are fuzzy topologies on X. Now, if we consider the mixed fuzzy topology from these two topologies s1 and s2, then s1(s2) = {0;1; A, B} and (X, s1(s2), I) is a mixed fuzzy ideal topological space. Consider the fuzzy set U = {(x, .1), (y, .8)}, then V = Uc = {(x, .9), (y, .2)}. Then V is fuzzy I-open set in the mixed fuzzy ideal topological space. But V is not fuzzy I-regularly open set because cl⁄(V) = V^V⁄ = V^1 = 1: Also int(cl⁄(V)) = int(1) = 1 Thus V – int(cl⁄(V)) and so V is not fuzzy I-regularly open set. Theorem 3.4. In a mixed fuzzy ideal topological space (X, s1(s2), I), if I = P(X) then A⁄ = 0 for any fuzzy set A in X. Proof. We have A⁄ = _{xk 2 X: A^E R I where E 2 s1(s2) and x 2 E} for any fuzzy set A in X. Since I = P(X) so for any fuzzy set A in X and for any fuzzy open set in (X, s1(s2), I), we get A^U 2 I. Hence any fuzzy point does not contain in A⁄. i.e. A⁄ = 0. h Theorem 3.5. In a mixed fuzzy ideal topological space (X, s1(s2), I), intersection of two fuzzy I-open sets is I-open. Proof: Let A and B be any two fuzzy I-open sets in X. Then be definition of fuzzy I-open set in mixed fuzzy ideal topological spaces, we have A 6 int(A⁄) and B 6 int(B⁄). Now, A^B 6 int(A⁄) int(B⁄) 6 int(A⁄^B⁄) 6 int((A^B)⁄). Hence A^B is a fuzzy I-open set in X. 4. Conclusion In this article we have introduced the notion of ideal mixed fuzzy topological spaces. We have investigated some of its properties. There are many other properties of the introduced notion, those can be investigated and applied for investigations in other branches of technology.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Mixed fuzzy ideal topological spaces Binod Chandra Tripathy a, Gautam Chandra Ray b,⇑ a b
Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Pashim Boragaon, Guwahati 781035, Assam, India Department of Mathematics, Central Institute of Technology, Kokrajhar 783370, Assam, India
a r t i c l e
i n f o
Keywords: Fuzzy I-open sets Fuzzy b-I-open sets Semi I-open sets
a b s t r a c t The aim of this paper is to introduce a new concept of mixed fuzzy ideal topological spaces and investigate some properties of this space. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The concept of ideal in topological spaces has been introduced and studied by Jankovic and Hamlett [3], Kuratouski [4], Nasef and Mahmoud [7], Vaidyanathswamy [21], Yuksel, Caylak and Acikgoz [23], Yuksel, Acikgoz and Noiri [24] and Yuksel, Kara and Acikgoz [25]. In 1997, Sarkar [9] and Mahmoud [5] introduced the notions of fuzzy ideal and fuzzy local function in fuzzy set theory. Malakar [6] introduced the concepts fuzzy semi-irresolute and strongly irresolute functions. Hatir and Jafari [2] and Nasef and Hatir [8] defined fuzzy semi-I-open set and fuzzy pre-I-open set via fuzzy ideal. The notion of Ideal has been applied in sequence spaces and different classes of ideal convergent sequences have been introduced and investigated by Tripathy and Dutta [10], Tripathy and Hazarika [11–15], Tripathy and Mahanta [16], Tripathy and Sarma [19], Tripathy, Sen and Nath [20] and many others in the recent years. In 1995, Das and Baishya [1] introduced the concept of mixed fuzzy topological spaces. Tripathy and Ray [17] introduced and studied the concept of mixed fuzzy topological spaces in slightly different ways which is not fuzzification of classical mixed topology. Tripathy and Ray [18] introduced and investigated different properties of fuzzy weakly continuous functions, fuzzy d-continuous function between two mixed fuzzy topological spaces. 2. Preliminaries and definitions Let X be a non-empty set and I, the unit interval [0,1]. A fuzzy set A in X is characterised by a function lA: X ? I where lA is called the membership function of A and lA(x) representing the membership grade of x in A. The empty fuzzy set is defined by lØ(t) = 0 for all t 2 X. Also X can be regarded as a fuzzy set in itself defined as lX(t) = 1 for all t 2 X. Further, an ordinary subset A of X can also be regarded as a fuzzy set in X if its membership function is taken as usual characteristic function of A that is lA(t) = 1 for all t 2 X and lA(t) = 0 for all t 2 X-A. Two fuzzy sets A and B are said to be equal if lA = lB. A fuzzy set A is said to be contained in a fuzzy set B, written asA # B, if lA 6 lB. Complement of a fuzzy set A in X is a fuzzy set denoted by Ac in X. Its membership function is defined by lAc ¼ 1 lA :. We can also denote the complement of A by coA that is Ac = coA. Union and intersection of a collection {Ai:i 2 I} of fuzzy sets in X, are written as [ Ai and \ Ai , respectively. The membership i¼1 i¼1 functions are defined as follows:
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (G.C. Ray). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.05.072
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l[Ai ðxÞ ¼ supflAi ðxÞ : i 2 Ig for all x 2 X: i2I
and
l\Ai ðxÞ ¼ infflAi ðxÞ : i 2 Ig for all x 2 X: i2I
A fuzzy topology s on X is a collection of fuzzy sets in X such that Ø, X 2 s; if Ai 2 s, i 2 I then [ Ai 2 s and if A, B 2 s then i2I A \ B 2 s. The pair (X, s) is called a fuzzy topological space (fts). Members of s are called open fuzzy sets and the complement of an open fuzzy set is called a closed fuzzy set. If (X, s) is a fts then the closure and interior of a fuzzy set A in X, denoted by cl A and int A, respectively, are defined as cl A = \ {B: B is a closed fuzzy set in X and A # B} and int A = [ {V: V is an open fuzzy set in X and V # A}. Clearly, cl A (respectively int A) is the smallest (respectively largest) closed (respectively open) fuzzy set in X containing (respectively contained in) A. If there is more than one topology on X, then the closure and interior of A with respect to a fuzzy topology s on X will be denoted by s-cl A and s-int A. Definition 2.1. A collection B of open fuzzy sets in fts X is said to be an open base for X if every open fuzzy sets in X is a union of members of B. Definition 2.2. If A is a fuzzy set in X and B is a fuzzy set in Y then, A B is a fuzzy set in X Y defined as lA B (x,y) = min{lA(x), lB(y)} for all x 2 X and for all y 2 Y. Definition 2.3. Let f be a function from X into Y. Then for each fuzzy set B in Y, the inverse image of B under f, written as f1[B], is a fuzzy set in X defined as lf 1 ½B ðxÞ ¼ lB ðf ðxÞÞ for all x 2 X. Definition 2.4. A fuzzy set A in a fuzzy topological space (X, s) is called a neighbourhood of a point x 2 X if and only if there exists B 2 s such that B # A and A(x) = B(x) > 0. Definition 2.5. A fuzzy point xa is said to be quasi-coincident with A, denoted by xa qA, if and only if a + A(x) > 1 or a > (A(x))c. Definition 2.6. A fuzzy set A is said to be quasi-coincident with B and is denoted by AqB, if and only if there exists an x 2 X such that A(x) + B(x) > 1. It is clear that if A and B are quasi-coincident at x both A(x) and B(x) are not zero at x and hence A and B intersect at x. Definition 2.7. A fuzzy set A in a fts (X, s) is called a quasi-neighbourhood of xk if and only if A1 2 s such that A1 # A and xk qA1. The family of all Q-neighbourhood of xk is called the system of Q-neighbourhood of xk. Intersection of two quasi-neighbourhood of xk is a quasi-neighbourhood. Definition 2.8. Let (X, s1) and (X, s2) be two fuzzy topological spaces. Consider the collection of fuzzy sets s1(s2) = {A 2 IX: For any fuzzy set B in X with AqB, there exists s2-open set A1 such that A1qB and s1-closure A1 # A}. Then this family of fuzzy sets will form a topology on X and this topology we call a mixed fuzzy topology on X. Definition 2.9. A non-empty collection of fuzzy sets I of a set X is called a fuzzy ideal if the following postulates are satisfies (i) if A 2 I and B 6 A then B 2 I. (heredity) and (ii) if A, B 2 I then A _ B 2 I. (finite additivity) The triplet (X, s, I) is called Ideal fuzzy topological space with the ideal I and fuzzy topology s. Definition 2.10. Let A be any subset of X in a fuzzy ideal topological space (X, s, I). The fuzzy local function of A with respect to s and I is denoted by A⁄(s, I) in short A⁄. The fuzzy local function A⁄(s, I) of A is the union of all fuzzy points xk such that U 2 Nq(xk ) and E 2 I, then there is at least one y 2 X for which U(y) + A(y) 1 > E(y). i.e. A⁄ = _{x 2 X: A_U R I for every U 2 s(x)}. The fuzzy closure operator of a fuzzy set A in (X, s, I) is defined as cl⁄(A) = A_A⁄. In (X, s, I) the collection s⁄(I) means an extension of fuzzy topological space than s via fuzzy ideal which is constructed by considering the class b = {U E: U 2 s, E 2 I} as a base. A fuzzy subset A of a fuzzy ideal topological space (X, s, I) is said to be I-open (respectively a-I-open fuzzy set, pre-I-open fuzzy set, semi-I-open fuzzy set, b-I-open fuzzy set) if A 6 int(cl⁄(intA)) (respectively A 6 int(cl⁄(int (A))), A 6 int(cl⁄(A)), A 6 cl⁄(int(A)), A 6 cl(int(cl⁄(A)))). Definition 2.11. A fuzzy subset A of a fuzzy ideal topological space (X, s, I) is said to be fuzzy ⁄-perfect if A = A⁄.
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Lemma 2.1 (Yuksel, Kara, Acikgoz [17], Lemma 3.1). Let A and B be fuzzy subsets of a fuzzy ideal topological space (X, s, I). Then we have (a) If A 6 B, then A⁄ 6 B⁄, (b) If U 2 s, then U^A 6 (U^A)⁄ (c) A⁄ is fuzzy closed set in (X, s, I). 3. Main results We introduce the mixed fuzzy ideal topological space as follows. Definition 3.1. Let X be a non-empty set and I be a fuzzy ideal in X. Consider a mixed fuzzy topological space (X, s1(s2)). Then the triplet (X, s1(s2), I) is said to be mixed fuzzy ideal topological space. Example 3.1. Let X = {x, y} and the fuzzy sets are defined by A1 = {(x,.2),(y,.8)}, A2 = {(x,.2),(y,.2)}, A3 = {(x, .8), (y, .2)}, A4 = {(x, .8), (y, .8)} Then the collection s1 = {0; 1; A1, A2, A3, A4} will form a fuzzy topology in X. Also, consider the following fuzzy sets in X B1 = {(x,.3),(y,.7)}, B2 = {(x,.7),(y,.3)}, B3 = {(x, .3), (y, .3)}, B4 = {(x, .7), (y, .7)} Then the collection of fuzzy sets s2 = {0; 1; B1, B2, B3, B4} will form a fuzzy topology on X and (X, s1(s2)) is a mixed fuzzy topological space. Let I = {0; 1}, then I is a fuzzy ideal in X. Therefore (X, s1(s2), I) is a mixed fuzzy ideal topological space. Definition 3.2. Let (X, s1(s2), I) be a mixed fuzzy ideal topological space. A fuzzy set A in X is said to be fuzzy I-open set if A 6 int (A⁄), where A⁄ is the fuzzy local function of A defined as A⁄ = _{x 2 X: A^U R I for every U 2 s(x)}. Example 3.2. Let us consider a non-empty set X = {x, y} and consider the following fuzzy sets in X. A = {(x, .7), (y, .3)} and B = {(x, .3), (y, .7)}. Then the collection of fuzzy sets s1 = {0;1;B} and s2 = {0;1;A} are two fuzzy topologies on X. Now, we construct the mixed fuzzy topology on X from these two fuzzy topologies s1 and s2. Since the mixed fuzzy topology is coarser than s2, so we need only to verify whether the fuzzy set A is in s1(s2) or not. Let us consider a fuzzy set S in X such that AqS. Now, the only s2-open sets are 1 and A such that AqS and 1 qS, Again, s1-closure of A = ^fF : F is s1-closed and A # F}
¼ 1 ^ A ¼ A # A: Hence, A 2 s1(s2) and so s1(s2) = {0; 1;A}. Now, we consider an ideal I = {0} on X. Then the triplet (X, s1(s2), I) is a mixed fuzzy topological space. Let C = {(x, .1), (y, .8)} be a fuzzy set in X. Then C⁄ = _{xk: C^U R I, for every U 2 s1(s2) and x 2 U}. Now, consider the fuzzy point x.3 in X. The open set in s1(s2) containing x.3 are A and 1. Then C^A R I and C^1 R I and so x.3 2 C⁄. Again consider the fuzzy point y.7 2 C. The open set containing y.7 is 1. Then C^1 R I and so y.7 2 C⁄. Similarly we can show that for any fuzzy point xa > .3 and yb > .7 in X is a member of C⁄. Therefore C 6 int(C⁄) and so C is an I-open set in (X, s1(s2), I). Definition 3.3. Let (X, s1(s2), I) be a mixed fuzzy ideal topological space. A fuzzy set A in X is said to be fuzzy a-I-open set (respectively fuzzy pre-I-open set, fuzzy semi-I-open set, fuzzy b-I-open set) if A 6 int(cl⁄(int(A))) (respectively A 6 cl(int(cl⁄(A))), A 6 cl⁄(int(A)), A 6 cl(int(cl⁄(A)))). Definition 3.4. A fuzzy set A in a mixed fuzzy ideal topological space (X, s1(s2), I) is said to be fuzzy regularly I-open set if A = int(cl⁄(A)) where cl⁄(A) = A_A⁄. Definition 3.5. A fuzzy point xk in a mixed fuzzy ideal topological space (X, s1(s2), I) is said to be d-I-cluster point of a fuzzy set A in X if Aq(int(cl⁄V)) where V is any fuzzy I-regularly open Q-neighbourhood of xk. The set of all d-I-cluster point of A is called d-I-closure of A and is denoted by CldI(A). A fuzzy set A is said to be d-I-closed if A = CldI(A).
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Theorem 3.1. Let (X, s1(s2), I) be a mixed fuzzy ideal topological space and A and B be two fuzzy sets in X. Then the following results hold (i) If A 6 B, then A⁄ 6 B⁄. (ii) If A 6 B, then cl⁄(A) 6 cl⁄(B). (b) If U 2 s, then U^A 6 (U^A)⁄ (c) A⁄ is fuzzy closed set in (X, s, I). Proof. (i) Let x 2 X be any point. Since A 6 B so A(x) 6 B(x) for any x 2 X. Now, A⁄ = {x 2 X: A^U R I for every U 2 s1(s2) and x 2 U} 6 {x 2 X: B^U R I for every U 2 s1(s2) and x 2 U} = B⁄. Thus A 6 B ) A⁄ 6 B⁄. (ii) Let x 2 X be any point. We have cl⁄(A) = A^A⁄. Therefore, cl⁄(A)(x) = (A^A⁄)(x) = max {A(x), A⁄(x)} 6 max {B(x), B⁄(x)} = cl⁄(B). Thus A 6 B ) cl⁄(A) 6 cl⁄(B). Proof of (iii) and (iv) are similar to the proof of Lemma 3.1. of Yuksel, Kara, Acikgoz [17]. h Theorem 3.2. In a mixed fuzzy ideal topological space (X, s1(s2), I) a fuzzy I-open set need not be a fuzzy d-I-open set. Proof: The result follows from the following example. Example 3.3. Consider the mixed fuzzy ideal topological space as defined in Example 3.2. Then the fuzzy set C = {(x, .1), (y, .8)} in X is an fuzzy I-open set in X and so Cc = {(x, .9), (y, .2)} is fuzzy I-closed. Now, we show CC is not fuzzy d–I-closed set in X. i.e. every fuzzy point in Cc is not fuzzy d-I-cluster point. Consider the fuzzy point x.9 in Cc. Let U = {(x, .3), (y, .8)} be any fuzzy set in X. Then by Theorem 3.1. U⁄ is fuzzy closed set in X. But the only fuzzy closed set in s1(s2) are 0; 1and Ac = {(x, .3), (y, .7)}. We have U⁄ is the union of all fuzzy points xk such that V 2 Nq(xk ) and E 2 I, then there is at least one y 2 X for which V(y) + U(y) 1 > E(y). Since I = {0} so E(y) = 0. Therefore in this case U⁄ is the union of all fuzzy point xa such that for any V 2 s1(s2)(x) there exists r 2 X such that VqU. Consider the fuzzy point x.4, then A 2 s1(s2) be such that x.4 2 A and y 2 X such that A(y) + U(y) = .4 + .8 > 1 So AqU and consequently x.4 2 U⁄. Therefore U⁄ – 0: Similarly we show that U⁄ – 1: Thus we must have U⁄ = {(x, .3), (y, .7)}. Hence cl⁄U = U_U⁄ = {(x, .3), (y, .8)}_{(x, .3), (y, .7)}. = {(x, .3), (y, .8)}. Therefore int(cl⁄U) = int({(x, .3), (y, .8)}) = 1. Hence U is not fuzzy I-regularly open set. Thus x.9 is not a d-I-cluster point of Cc and so Cc is not a d-I-closed. So C is not fuzzy d-I-open. Now, we give an example to show that in a mixed fuzzy ideal topological space a fuzzy d-I-open set may be a fuzzy Iclosed set. Example 3.4. Consider the non-empty set X = {x, y} and the fuzzy ideal I = {0} and fuzzy sets A = {(x, .7), (y, .7)} and B = {(x, .3), (y, .3)} in X. Then s1 = {0;1} and s2 = {0,1;A, B} are fuzzy topologies on X. Consider the mixed fuzzy topology from these two topologies s1 and s2, then s1(s2) = {0;1;A, B} and (X, s1(s2), I) is a mixed fuzzy ideal topological space. Consider the fuzzy set U = {(x, .1), (y, .8)}, then V = Uc = {(x, .9), (y, .2)}. We prove that V is fuzzy d-I-closed. Consider the fuzzy point y.2 in V and C = {(x, .7), (y, .3)} a fuzzy set in X. Then C is fuzzy I-regularly open set because C⁄ – 0 and C⁄ – 1 so we must have C⁄ = {(x, .3), (y, .7)} (Since C⁄ is fuzzy closed in s1(s2)) Hence C_C⁄ = {(x, .7), (y, .7)} ) cl⁄(C) = {(x, .7), (y, .7)} Therefore int(cl⁄(C) = C. Thus C is fuzzy I-regularly open Q-neighbourhood of the fuzzy point y.2 such that CqV. Hence y.2 is a fuzzy d-I-cluster point of V. Also consider the fuzzy point x.9 in X. Then 1 is the only fuzzy I-regularly open Q-neighbourhood of the fuzzy point x.9 such that 1qV and so x.9 is a fuzzy d-I-cluster point of V.
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Thus all point of the fuzzy set V are fuzzy d-I-cluster point and hence V is fuzzy d-I-closed and consequently U is fuzzy d-Iopen. Now, we show that U is fuzzy I-closed. We have V⁄is fuzzy closed in (X, s1(s2), I). By definition V⁄ = _{xk 2 X: V^E R I where E 2 s1(s2) and x 2 E}. Hence V⁄ – 0: We show that V⁄ – A. Consider the fuzzy point x.7, then A and 1 are the only fuzzy open set in X such that V^A R I and V^1 R I and so x.7 e V⁄. Further, if we consider the fuzzy point x.3, then B and 1 are the only fuzzy open set in X such that VB R I and V 1 R I and hence x.3 2 V⁄. Hence V⁄ – A. Similarly we can show that V⁄ – B. But V⁄ is a fuzzy closed set in s1(s2) so V⁄ = 1. Now, int(V⁄) = int(1) =1: Hence V 6 int(V⁄) and so V is fuzzy I-open and consequently U is fuzzy I-closed set. Thus in a mixed fuzzy ideal topological space a fuzzy d-I-open set may be a fuzzy I-closed. Now, we establish a relation between fuzzy I-regularly open set and fuzzy I-open set in a mixed fuzzy ideal topological space. Theorem 3.3. In a mixed fuzzy ideal topological space a fuzzy I-regularly open set is not fuzzy I-open. Proof. Let (X, s1(s2), I) be a mixed fuzzy ideal topological space. Let A be any fuzzy I-regularly open set in X. Then by definition of fuzzy I-regularly open set we get A = int(cl⁄(A)) = int(A_A⁄) P int (A⁄). Thus we get A P int (A⁄). Hence A is not fuzzy I-open set. Now, we give an example that in a mixed fuzzy ideal topological space a fuzzy I-open set may not be fuzzy I-regularly open set. h Example 3.5. Consider the non-empty set X = {x, y} and the fuzzy ideal I = {0} and fuzzy sets A = {(x, .7), (y, .7)} and B = {(x, .3), (y, .3)} in X. Then s1 = {0;1 and s2 = {0;1;A, B} are fuzzy topologies on X. Now, if we consider the mixed fuzzy topology from these two topologies s1 and s2, then s1(s2) = {0;1; A, B} and (X, s1(s2), I) is a mixed fuzzy ideal topological space. Consider the fuzzy set U = {(x, .1), (y, .8)}, then V = Uc = {(x, .9), (y, .2)}. Then V is fuzzy I-open set in the mixed fuzzy ideal topological space. But V is not fuzzy I-regularly open set because cl⁄(V) = V^V⁄ = V^1 = 1: Also int(cl⁄(V)) = int(1) = 1 Thus V – int(cl⁄(V)) and so V is not fuzzy I-regularly open set. Theorem 3.4. In a mixed fuzzy ideal topological space (X, s1(s2), I), if I = P(X) then A⁄ = 0 for any fuzzy set A in X. Proof. We have A⁄ = _{xk 2 X: A^E R I where E 2 s1(s2) and x 2 E} for any fuzzy set A in X. Since I = P(X) so for any fuzzy set A in X and for any fuzzy open set in (X, s1(s2), I), we get A^U 2 I. Hence any fuzzy point does not contain in A⁄. i.e. A⁄ = 0. h Theorem 3.5. In a mixed fuzzy ideal topological space (X, s1(s2), I), intersection of two fuzzy I-open sets is I-open. Proof: Let A and B be any two fuzzy I-open sets in X. Then be definition of fuzzy I-open set in mixed fuzzy ideal topological spaces, we have A 6 int(A⁄) and B 6 int(B⁄). Now, A^B 6 int(A⁄) int(B⁄) 6 int(A⁄^B⁄) 6 int((A^B)⁄). Hence A^B is a fuzzy I-open set in X. 4. Conclusion In this article we have introduced the notion of ideal mixed fuzzy topological spaces. We have investigated some of its properties. There are many other properties of the introduced notion, those can be investigated and applied for investigations in other branches of technology.
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Further reading [22] A. Wiweger, Linear spaces with mixed topology, Stud. Math. 20 (1961) 47–68.