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Int. J. of Mathematical Sciences and Applications, Vol. 1, No. 2, May 2011 Copyright Mind Reader Publications www.ijmsa.yolasite.com ON FUZZY MINIMAL OPEN AND FUZZY MAXIMAL OPEN SETS IN FUZZY TOPOLOGICAL SPACES 1 Basavaraj M. Ittanagi and R. S. Wali Department of Mathematics, Siddaganga Institute of Technology, Tumkur572 103, Karnataka State, India E-Mail: [email protected] 1 Department of Mathematics, Bhandari and Rathi College GULEDAGUDD-587 203, Karnataka, India. E-mail: [email protected] Abstract: In this paper a new class of sets called fuzzy minimal open sets and fuzzy maximal open sets in fuzzy topological spaces are introduced and studied. A nonzero fuzzy open set A (1) of a fuzzy topological space X is said to be fuzzy minimal open (resp. fuzzy maximal open) set if any fuzzy open set which is contained (resp. contains) in A is either 0 or A itself (resp. either 1 or A itself). Some properties of the new concepts have been studied. 2000 Mathematics Subject Classification: 54A40. Key words and phrases: Fuzzy minimal open sets and fuzzy maximal open sets. 1. Introduction. The concept of a fuzzy subset was introduced and studied by L.A.Zadeh [4] in the year 1965. In the year 1968, C.L.Chang [1] introduced the concept of fuzzy topological space as an application of fuzzy sets to general topological spaces. 1.1 Definition: [1] A fuzzy subset A in set X is defined to be a function A: X[0, 1]. A fuzzy subset A in set X is empty iff its membership function is identically zero on X and is denoted by 0 or . The set X can be considered -1- 1024 as a fuzzy subset of X whose membership function is 1 on X and is denoted by 1 or 1X or X. In fact, every subset of X is fuzzy subset of X but not conversely. Hence the concept of a fuzzy subset is a generalization of the concept of a subset. 1.2 Definition: [4] If A and B are any two fuzzy subsets of a set X, then “A is said to be included in B” or “A is contained in B” or “A is less then or equal to B” iff A(x) B(x) for all x in X and is denoted by A B. Equivalently, A B iff A(x) B(x) for all x in X. Note that every fuzzy subset is included itself and empty fuzzy subset is included in every fuzzy subset. 1.3 Definition: [4] Two fuzzy subsets A and B of a set X are said to be equal, written A=B, if A(x)=B(x) for every x in X. 1.4 Definition: [4] The complement of a fuzzy subset A in a set X, denoted by 1A, is the fuzzy subset of X defined by 1A(x) for all x in X. Note that [1(1A)]=A. 1.5 Definition: [4] The union of two fuzzy subsets A and B in a set X, denoted by AB, is fuzzy subset in X defined by (AB)(x)=Max {A(x), B(x)}, for all x in X. In general, the union of a family of fuzzy subsets {A: } is a fuzzy subset denoted by A and defined by ( A )(x)=Sup{A (x): }, for all x in X. -2- 1025 1.6 Definition: [4] The intersection of two fuzzy subsets A and B in a set X, denoted by AB, is fuzzy subset in X defined by (AB)(x)=Min{A(x), B(x)}, for all x in X. In general, the intersection of a family of fuzzy subsets {A: } is a fuzzy subset denoted by ( A and defined by A )(x)=Inf{A (x): }, for all x in X. 1.7 Theorem: ([2], [3] and [4]) Let X be any set and A, B, C be fuzzy subsets of X. The following results hold good. (1) A(BC)=(AB)C (2) A(BC)=(AB)C (3) A(BC)=(AB)(AC) (4) A(BC)=(AB)(AC) (5) AX=A (6) AX=X (7) 1 (AB)=(1A)(1B) (8) 1 (AB)=(1A)(1B) (9) AB=A(1B) (10) A0=0, where 0 is the empty fuzzy set (11) A0=A, where 0 is the empty fuzzy set. 1.8 Definition: [1] Let X be a set and T be a family of fuzzy subsets of X. The family T is called a fuzzy topology on X iff T satisfies the following axioms (i) 0, 1T (ii) If {A: } T then (iii) If G, HT then GHT. -3- A T and 1026 The pair (X, T) is called a fuzzy topological space (abbreviated as fts). The members of T are called fuzzy open sets in X. A fuzzy set A in X is said to be fuzzy closed set in X. iff 1A is a fuzzy open set in X. 2. Fuzzy minimal open sets and fuzzy maximal open sets. 2.1 Definition: A nonzero fuzzy open set A (1) of a fuzzy topological space (X, T) is said to be a fuzzy minimal open (briefly f-minimal open) set if any fuzzy open set which is contained in A is either 0 or A. Similarly a nonzero fuzzy closed set B (1) of a fuzzy topological space (X, T) is said to be fuzzy minimal closed (briefly f-minimal closed) set if any fuzzy closed set which is contained in B is either 0 or B. 2.2 Lemma: Let (X, T) be a fuzzy topological space. i) If A is a fuzzy minimal open and B is a fuzzy open sets in X, then A B = 0 or A < B. ii) If A and C are fuzzy minimal open sets then A C = 0 or A = C. Proof: i) Let A be any fuzzy minimal open and B be any fuzzy open sets in X. If A B = 0, then there is nothing to prove. If A B 0, then we have to prove that A < B. Suppose AB 0. Then A B < A, A B is a fuzzy open and A is a fuzzy minimal open sets in X. Therefore A B = 0 or A B = A. But A B 0 then A B = A A < B. ii) Since every fuzzy minimal open set is a fuzzy open set it follows from (i) that A < C and C < A. Therefore A = C. 2.3 Theorem: If A and Ai are fuzzy minimal open sets for any i . If A i Ai then there exists an element j of such that A = Aj. Proof: Let A i Ai , then A = A [ i Ai ] A = [A Ai]. Since A and Ai i are fuzzy minimal open sets, by Lemma 2.2(ii), A Ai = 0 or A = Ai. Now if -4- 1027 A Ai = 0 then A = 0, which contradicts the fact that A is fuzzy minimal open set. Therefore if A Ai 0, then there exists an element j of such that A = Aj. 2.4 Theorem: If A and Ai are fuzzy minimal open sets for any i . If A Ai for any element i then [ i Ai ] A = 0. Proof: Suppose [ i Ai ] A 0, then there exists an element i such that Ai A 0. By Lemma 2.2(ii), Ai = A, which contradicts the fact that A Ai. Therefore [ i Ai ] A = 0. 2.5 Theorem: If Ai is a fuzzy minimal open set for any element i of and Ai Aj for any elements i and j of with i j. Then for any element j of , [ i \ { j } A ] Aj = 0. Assume that 2. i Proof: Suppose that [ i \ { j } A ] Aj 0, then i [Ai Aj] 0. Therefore i \ { j } Ai Aj 0. By Lemma 2.2(ii), Ai = Aj. Contradiction to the hypothesis. Therefore for any element j of , [ i \ { j } A ] Aj = 0. i 2.6 Theorem: If Ai is a fuzzy minimal open set for any element i of and Ai Aj for any elements i and j of with i j. If is a proper nonempty subset of , then [ i \ Ai ] [ m Am ] = 0. Proof: Suppose [ i \ Ai ] [ m Am ] 0, then [Ai Am] 0 for i\ and m Ai Am 0 for some i and m . Ai =Am, by Lemma 2.2(ii). Hence contradiction to the fact that Ai Am. Therefore [ i \ Ai ][ m Am ]=0. -5- 1028 2.7 Theorem: If Ai and Am are fuzzy minimal open sets for any elements i and m. If there exists an element n such that Ai An for any element i, then [ n An ]≮ [ i Ai ]. Proof: Suppose that there exists an element n satisfying the condition Ai ≠ An for any element i such that [ n An ] < [ i Ai ]. An [ i Ai ] for some n. An = Aj for some j, by Theorem 2.3. Contradiction to the fact that An A j for any j. Therefore [ n An ] ≮ [ i Ai ]. 2.8 Theorem: If Ai is a fuzzy minimal open set for any element i of and Ai Aj for any elements i and j of with i j. If is a proper nonempty subset of , then m Am ≨ i Ai . Proof: Let k be any element of \ , then Ak is a fuzzy minimal open set of the family {Ak: k \ } of fuzzy minimal open sets. Then, A k If [ m m A A]= m m i [ Ak m A ] = 0 and A m k A , then 0 = A i k [ i A] i = i[ Ak Ai ] = . Contradiction to the fact that A k A k . is a fuzzy minimal open set. Therefore m Am i Ai . Hence m Am ≨ ii Ai . 2.9 Theorem: Assume that 2. If Ai is a fuzzy minimal open set for any element i of and Ai Aj for any elements i and j of with i j. Then, i) A j 1 ii) i \{ j } i \ { j } A , for some element j of . i A 1 , for any element j of . i -6- 1029 Proof: i) Let j be any element of . By hypothesis Ai Aj for any elements i and j of with i j. Then by Theorem 2.4, [i Ai ] A j 0 , which is true for any j. i( Ai A j ) 0 , for some elements i and j of . Ai A j 0 , by Lemma 2.2(ii). Ai 1 A j i \{ j } Therefore A j A 1 A i 1 i \ { j } j A , for some element j of . i ii) Let j be any element of such that, i \{ j } A 0 . Contradiction to the fact that A i i \{ j } i is fuzzy minimal open set. i Therefore A 1. A 1 , for any element j of . i 2.10 Corollary: If Ai is a fuzzy minimal open set for any element i of and Ai Aj for any elements i and j of with i j. If 3, then AA i j 1 for any elements i and j of with i j. Proof: The proof follows from Theorem 2.9(ii). 2.11 Theorem: Assume that 2. If Ai is a fuzzy minimal open set for any element i of and Ai Aj for any elements i and j of with i j. Then, A j ( i A ) (1 i i \ { j } A ) for any element j of . i Proof: Let j be any element of , then ( i A ) (1 i i \ { j } A ) = [( i A ) A ] [1 ( A ) ] i i \{ j } = [( i \ { j } = 0 j i i A ) [1 A ) ] [ A A i i j -7- i j (1 i A )] i 1030 = A j for any element j of . 2.12 Theorem: If Ai is a fuzzy minimal open set for any element i of and Ai Aj for any elements i and j of with i j. If i Ai is a fuzzy closed set then Ai is a fuzzy closed set for any element i of . Proof: Let j be any element of , then by the Theorem 2.11, A j ( i A ) (1 i i \ { j } A ) = ( A ) [ i i i (1 i \{ j} A )] i = (fuzzy closed set) (fuzzy closed set) = fuzzy closed set. 2.13 Theorem: Assume that 2. If Ai is a fuzzy minimal open set for any element i of and Ai Aj for any elements i and j of with i j. If i A 1 , then { A / i } is the set of all fuzzy minimal open sets of a fuzzy i i topological space X. Proof: Suppose that there exists another fuzzy minimal open set Am of a fuzzy topological space X which is not equal to Ai for any element i of . Then, 1 i A i i( m ) \ {m} A 1 , i by the Theorem 2.9(ii). This contradicts our assumption. Therefore { Ai / i } is the set of all fuzzy minimal open sets of a fuzzy topological space X. 2.14 Definition: A nonzero fuzzy open set A (1) of a fuzzy topological space (X, T) is said to be fuzzy maximal open (briefly f-maximal open) set if any fuzzy open set which contains A is either 1 or A. Similarly a nonzero fuzzy closed set B (1)of a fuzzy topological space (X, T) is said to be fuzzy maximal closed (briefly f-maximal closed) set if any fuzzy closed set which contains B is either 1 or B. 2.15 Theorem: i) A nonzero subset of a fuzzy topological space X is fuzzy minimal open set if and only if 1 is fuzzy maximal closed set -8- 1031 ii) A nonzero subset of a fuzzy topological space X is fuzzy maximal open set if and only if 1 is fuzzy minimal closed set Proof: i) Let be a fuzzy minimal open set in X. Suppose 1 is not a fuzzy maximal closed set in X. Then there exists a fuzzy closed set in X such that 0 1 < . That is 0 1 < and 1 is a fuzzy open set in X. This is contradiction to is a fuzzy minimal open set in X. Therefore our assumption is 1 is a fuzzy maximal closed set in X. Conversely, let 1 be a fuzzy maximal closed set in X. Suppose is not a fuzzy minimal open set in X. Then there exists a fuzzy open set such that 0 < . That is 1 < 1 and 1 is a fuzzy closed set in X. This is contradiction to 1 is a fuzzy maximal closed set in X. Therefore our assumption is a fuzzy minimal open set in X. ii) Let be a fuzzy maximal open set in X. Suppose 1 is not a fuzzy minimal closed set in X. Then there exists a fuzzy closed set in X such that 0 < 1. That is < 1 and 1 is a fuzzy open set in X. This is contradiction to is a fuzzy maximal open set in X. Therefore our assumption is 1 is a fuzzy minimal closed set in X. Conversely, let 1 be a fuzzy minimal closed set in X. Suppose is not a fuzzy maximal open set in X. Then there exists a fuzzy open set such that < 1. That is 1 < 1 and 1 is a fuzzy closed set in X. This is contradiction to 1 is a fuzzy minimal closed set in X. Therefore our assumption is a fuzzy maximal open set in X. 2.16 Lemma: Let (X, T) be a fuzzy topological space. i) If A is a fuzzy maximal open and B is a fuzzy open sets in X, then A B = 1 or B < A. -9- 1032 ii) If A and C are fuzzy maximal open sets then A C = 1 or A = C. Proof: i) Let A be any fuzzy maximal open and B be any fuzzy open sets in X. If A B = 1, then there is nothing to prove. If A B 1, then we have to prove that B < A. Suppose A B 1. Then A < A B, A B is a fuzzy open and A is a fuzzy maximal open sets in X. Therefore A B = 1 or A B = A. But AB 1 then A B = A B < A. ii) Since every fuzzy maximal open set is a fuzzy open set, it follows from (i) that A < C and C < A. Therefore A = C. 2.17 Theorem: If A, B and C are fuzzy maximal open sets such that A B and if A B C, then either A = C or B = C. Proof: Let A, B and C are fuzzy maximal open sets such that A B and A B C. If A = C then there is nothing to prove. But if A C, then we have to prove that B = C. Now B C = B (C 1) = B [C (A B)] by Lemma 2.16(ii), A B = 1. = B [(C A) (C B)] = (B C A) (B C B) = (B A) (B C) by hypothesis, A B C. = B (A C) = B 1 = B. B C = B B < C. From the definition of fuzzy maximal open sets, it follows that B = C. 2.18 Theorem: If A, B and C are fuzzy maximal open sets which are different from each other, then A B ≮ A C. Proof: Let A, B and C are fuzzy maximal open sets which are different from each other such that A B A C, then we see that - 10 - 1033 (A B) (B C) (A C) (B C) = (A C) B (A B) C =1B 1C by Lemma 2.16(ii). =BC It follows that B = C from the definition of fuzzy maximal open sets. This contradicts the fact A B C. Therefore A B ≮ A C. 2.19 Theorem: Assume that 2. If Ai is a fuzzy maximal open set for any element i of and Ai Aj for any elements i and j of with i j. Then, i) 1 i \ [ j ] A i < A j , for any element j of . A 0 , for any element j of . ii) i i \ [ j ] Proof: i) Let j be any element of . Since Ai and Aj are fuzzy maximal open sets with i j, by Lemma 2.16(ii) we have, 1 A j < Ai 1 A j < i \ [ j ] A A 1 i j A . Therefore 1 i i \ [ j ] A <A i j , for any element j of . ii) Let j be any element of such that i \ [ j ] A 0 i then from (i) Aj = 1, this contradicts the fact that Aj is fuzzy maximal open set. Therefore i \ [ j ] A 0. i 2.20 Corollary: If Ai is a fuzzy maximal open set for any element i of and Ai Aj for any elements i and j of with i j. If 3 then Ai Aj 0, for any elements i and j of with i j. Proof: The proof follows from the theorem 2.19(ii). 2.21 Theorem: If Ai is a fuzzy maximal open set for any element i of and Ai Aj for any elements i and j of with i j. Assume that 2, then i \{ j } A i ≮ A j ≮ i \{ j } A i for any element j of . - 11 - 1034 Proof: Let j be any element of such that Then 1 = (1 i \{ j } 1< A j Again let A = j A)( i i{ j } A)< A i i A i A . j by Theorem 2.19(i). j , contradiction to our assumption. Therefore i \{ j } A < j , then A i i \{ j } A < j A i ≮ A j (i) for some element i of . A i A , by the definition of fuzzy maximal open set. Which contradicts i our assumption. Therefore i \{ j } < A i{ j } ≮ A j ≮ A i i \{ j } A ≮ j i \{ j } A i (ii). From (i) and (ii) . 2.22 Corollary: If Ai is a fuzzy maximal open set for any element i of and Ai Aj for any elements i and j of with i j. If is a proper nonempty subset of , then i \ A≮ i A k k ≮ i \ A. i Proof: Let k be any element of , then by Theorem 2.21, i \ A i ≮ A k i \ A i ≮ k From (i) and (ii) we have i \ A k A i (i). Similarly ≮ k A k ≮ i \ ≮ i \ A k k A i (ii). A. i 2.23 Theorem: Assume that 2. If Ai is a fuzzy maximal open set for any element i of and Ai Aj for any elements i and j of with i j, then for any element j of , A j = ( i A ) ( 1 i i \{ j } A ). i Proof: Let j be any element of , then by Theorem 2.19(i), we have ( i A ) (1 i i \{ j } A)A = [( A ) (1 A ) = (( i i \{ j } i \{ j } i i =1 A j - 12 - j ) ( 1 i \{ j } i \{ j } A) i A )] [ A i j ( 1 i \{ j } A )] i 1035 = Aj 2.24 Theorem: If Ai is a fuzzy maximal open set for any element i of and Ai Aj for any elements i and j of with i j. If is a proper nonempty subset of , then i A≨ i k A k . Proof: Since ≩ 0, there exists an elements of such that s and an element j. If contains only one element, then we have i A i i = A then j A < j A i for any element i of . Since maximal open set for any element i of , we have contradicts our assumption. Therefore i Theorem 2.23, A j k =( k k i \{ s} A i A =A s k i ( i A) i k \{ j } = i A k k < A k < A i A k i \{ s } ). If A < i i \{ s} k \{ j } (1 A i A i i i j i < A j . If A is a fuzzy = A j i i which . If 2 then by ) and = A k , then A k . k k . Thus we see that Therefore A s A j we have . It follows that with s j. This contradicts our assumption. Therefore j A≨ A i s ) (1 A = A A= A≨A A A k k . 2.25 Theorem: If Ai is a fuzzy maximal open set for any element i of of a finite set and Ai Aj for any elements i and j of with i j and if i fuzzy closed set, then A i is a fuzzy closed set for any element i of . Proof: By Theorem 2.23 we have, A j = ( i A ) (1 i i \{ j } A ), for any element j of . i - 13 - A i is a 1036 = ( i A ) ( i i \ { j } (1 Ai ) ). Since is a finite set, we see that ( (1 Ai ) ) is a fuzzy closed set. i \ { j } Hence A j is a fuzzy closed set for any element j of . 2.26 Theorem: Assume that 2. If Ai is a fuzzy maximal open set for any element i of and Ai Aj for any elements i and j of with i j. If i A 0, i then { Ai / i } is the set of all fuzzy maximal open sets of a fuzzy topological space X. Proof: Suppose that there exists another fuzzy maximal open set Am of a fuzzy topological space X which is not equal to Ai for any element i of . Then, 0 i A i i( m ) \ {m} A 0 i by Theorem 2.19(ii). This contradicts our assumption. Therefore { Ai / i } is the set of all fuzzy maximal open sets of a fuzzy topological space X. 2.27 Proposition: Let A and B be any fuzzy subsets of X. If A B = 1, A B is a fuzzy closed set and A is a fuzzy open then B is a fuzzy closed set. Proof: If A B = 1 1 – A < B, then (A B) (1 – A) = [A (1 – A)] [B (1 – A)] = 1 [B (1 – A)] = [B (1 – A)] = B Since A is a fuzzy open set, 1–A is a fuzzy closed set, (AB) (1–A) is a fuzzy closed set. Therefore B is a fuzzy closed set. 2.28 Proposition: If Ai is a fuzzy open set for any element i of and Ai Aj = 1 for any elements i and j of with i j. If i Ai is a fuzzy closed set then i \{ j } A is a fuzzy closed set for any element i of . i - 14 - 1037 Proof: Let j be any element of . Since Ai Aj = 1 for any elements i and j of with i j we have Since Therefore A i \{ j } j ( A i \ { j } ( j A ) i i \ { j } A ) i i A i ( Aj i \{ j } A ) 1. i is a fuzzy closed set by our assumption. A is a fuzzy closed set for any element i of . i 2.29 Theorem: If Ai is a fuzzy maximal open set for any element i of and Ai Aj for any elements i and j of with i j. If i Ai is a fuzzy closed set then i \{ j } A i is a fuzzy closed set for any element i of . Proof: By hypothesis Ai Aj, then by Lemma 2.16(ii) Ai Aj = 1 for any elements i and j of with i j. By Proposition 2.28, it follows that i \{ j } A is a fuzzy closed set for any element i of . i References: [1] C.L.Chang, Fuzzy topological spaces, JI. Math. Anal. Appl., 24(1968), 182-190. [2] A.Kaufmann, Introduction to the theory of fuzzy subsets, Vol.1 Acad. Press N.Y. (1975). [3] G.J.Klir and B.Yuan, Fuzzy sets and fuzzy logic, Theory and applications, PHI (1997). [4] L.A.Zadeh, Fuzzy sets, Information and control, 8 (1965) 338-353. - 15 -
