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Intuitionistic supra fuzzy topological spaces
Chaos, Solitons and Fractals 21 (2004) 1205–1214 www.elsevier.com/locate/chaos
Intuitionistic supra fuzzy topological spaces S.E. Abbas Department of Mathematics, Faculty of Science, South Valley University, Sohag 82524, Egypt Accepted 8 December 2003
Abstract In this paper, We introduce an intuitionistic supra fuzzy closure space and investigate the relationship between intuitionistic supra fuzzy topological spaces and intuitionistic supra fuzzy closure spaces. Moreover, we can obtain intuitionistic supra fuzzy topological space induced by an intuitionistic fuzzy bitopological space. We study the relationship between intuitionistic supra fuzzy closure space and the intuitionistic supra fuzzy topological space induced by an intuitionistic fuzzy bitopological space. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries Chattopadhyay et al. [7] introduced the concept of gradation of openness (closedness) of fuzzy subsets of X and gave definitions of a fuzzy topology on X as an extension of Chang’s fuzzy topology [6]. Historically, the fundamental idea of a topology being a fuzzy subset of a power set, first appeared in 1980 [13]. This was followed in 1985 by the independent and parallel generalizations of Kubiak [17] and Sostak [21], papers in which a topology was a fuzzy subset of a powerset of fuzzy subsets. It has been independently considered by authors not aware of the historical development of [2,14,15,17,18]. Moreover, Ghanim et al., [12] introduced the gradation of supra openness as an extension of supra fuzzy topology in a sense of Abd-Elmonsef and Ramadan [1]. The concept of fuzzy topology may have very important applications in quantum particles physics particularly in connection with string theory and Eð1Þ theory [10,11]. On the other hand, Atanassov [3] introduced the idea of intuitionistic fuzzy set. Recently, much work has been done with these concepts [3–5]. C ß oker and coworker [8,9] introduced the idea of the topology of intuitionistic fuzzy sets. Samanta and Mondal [19,20] introduced the definition of the intuitionistic gradation of openness. In this paper, we define an intuitionistic supra fuzzy closure space and an intuitionistic fuzzy bitopological space. We investigate the relationship between intuitionistic supra fuzzy topological spaces and intuitionistic supra fuzzy closure spaces. Moreover, we can obtain the intuitionistic supra fuzzy topological space induced by an intuitionistic fuzzy bitopological space. We study the relationship between intuitionistic supra fuzzy closure space and the intuitionistic supra fuzzy topological space induced by an intuitionistic fuzzy bitopological space. Throughout this paper, let X be a nonempty set, I ¼ ½0; 1 and I0 ¼ ð0; 1. For a 2 I, aðxÞ ¼ a for each x 2 X . Notions and notations not described in this paper are standard and usual.
2. Intuitionistic supra fuzzy topological spaces and intuitionistic supra fuzzy closure spaces Definition 2.1. An intuitionistic supra fuzzy topology on X is an ordered pair ðT; T Þ of functions from I X to I such that
E-mail address: [email protected] (S.E. Abbas). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.070
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(IS1) TðkÞ þ T ðkÞ 6 1, 8k 2 I X , ¼ 1, T ð0Þ ¼ T ð1Þ ¼ 0, (IS2) Tð0Þ W ¼ Tð1Þ V W W (IS3) Tð i2D ki Þ P i2D Tðki Þ and T ð i2D ki Þ 6 i2D T ðki Þ, ki 2 I X , i 2 D. The triplet ðX ; T; T Þ is called an intuitionistic supra fuzzy topological space. An intuitionistic supra fuzzy topology ðT; T Þ is called an intuitionistic fuzzy topology on X iff (IT) Tðk1 ^ k2 Þ P Tðk1 Þ ^ Tðk2 Þ and T ðk1 ^ k2 Þ 6 T ðk1 Þ _ T ðk2 Þ, ki 2 I X , i ¼ 1; 2. The triplet ðX ; T; T Þ is called an intuitionistic fuzzy topological space. T and T may be interpreted as gradation of openness and gradation of nonopenness, respectively. The ðX ; ðT; T Þ; ðU; U ÞÞ is called an intuitionistic fuzzy bitopological space where ðT; T Þ and ðU; U Þ are intuitionistic fuzzy topologies on X . Let ðT; T Þ and ðU; U Þ be intuitionistic supra fuzzy topologies on X . We say that ðT; T Þ is finer than ðU; U Þ (ðU; U Þ is coarser than ðT; T Þ) iff UðkÞ 6 TðkÞ and U ðkÞ P T ðkÞ for each k 2 I X . Definition 2.2. A function C : I X I0 ! I X is called an intuitionistic supra fuzzy closure operator on X if for k; l 2 I X and r; s 2 I0 , it satisfies the following conditions: (C1) (C2) (C3) (C4) (C5)
Cð0; rÞ ¼ 0, k 6 Cðk; rÞ, Cðk; rÞ _ Cðl; rÞ 6 Cðk _ l; rÞ, Cðk; rÞ 6 Cðk; sÞ, if r 6 s, CðCðk; rÞ; rÞ ¼ Cðk; rÞ.
The pair ðX ; CÞ is called an intuitionistic supra fuzzy closure space. The intuitionistic supra fuzzy closure space ðX ; CÞ is called an intuitionistic fuzzy closure space iff (C) Cðk; rÞ _ Cðl; rÞ ¼ Cðk _ l; rÞ, for each k; l 2 I X . Let C1 and C2 be intuitionistic supra fuzzy closure operators on X . We say that C1 is finer than C2 (C2 is coarser than C1 ) iff C1 ðk; rÞ 6 C2 ðk; rÞ, for each k 2 I X and r 2 I0 . Theorem 2.3. Let ðX ; T; T Þ be an intuitionistic (respectively, intuitionistic supra) fuzzy topological space. Then for each r 2 I0 , k 2 I X we define an operator CT;T : I X I0 ! I X as follows: ^ 1 lÞ 6 1 rg: CT;T ðk; rÞ ¼ fl 2 I X : k 6 l; Tð1 lÞ P r; T ð Then ðX ; CT;T Þ is an intuitionistic (respectively, intuitionistic supra) fuzzy closure space. Proof. Let ðX ; T; T Þ be an intuitionistic supra fuzzy topological space. Then, (C1), (C2) and (C4) are trivial from the definition of CT;T . (C3) Since k; l 6 k _ l, we have CT;T ðk; rÞ 6 CT;T ðk _ l; rÞ CT;T ðk; rÞ _ CT;T ðl; rÞ 6 CT;T ðk _ l; rÞ. (C5) Suppose that there exist k 2 I X , r 2 I0 and x 2 X such that
and
CT;T ðl; rÞ 6 CT;T ðk _ l; rÞ.
Hence
CT;T ðCT;T ðk; rÞ; rÞðxÞ > CT;T ðk; rÞðxÞ: 1 qÞ P r, T ð 1 qÞ 6 1 r such that By the definition of CT;T ðk; rÞ, there exists q 2 I X with q P k and Tð CT;T ðCT;T ðk; rÞ; rÞðxÞ > qðxÞ P CT;T ðk; rÞðxÞ: On the other hand, since CT;T ðk; rÞ 6 q and Tð1 qÞ P r, T ð 1 qÞ 6 1 r, by the definition of CT;T ðCT;T ðk; rÞ; rÞ, we have CT;T ðCT;T ðk; rÞ; rÞ 6 q:
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It is a contradiction. Thus, CT;T ðCT;T ðk; rÞ; rÞ ¼ CT;T ðk; rÞ. Hence CT;T is an intuitionistic supra fuzzy closure operator on X . Let ðX ; T; T Þ be an intuitionistic fuzzy topological space. From (C2), we have k 6 CT;T ðk; rÞ;
Tð1 CT;T ðk; rÞÞ P r;
T ð 1 CT;T ðk; rÞÞ 6 1 r;
l 6 CT;T ðl; rÞ;
Tð1 CT;T ðl; rÞÞ P r;
T ð 1 CT;T ðl; rÞÞ 6 1 r:
It implies k _ l 6 CT;T ðk; rÞ _ CT;T ðl; rÞ such that Tð1 ðCT;T ðk; rÞ _ CT;T ðl; rÞÞÞ ¼ T½ð1 CT;T ðk; rÞÞ ^ ð 1 CT;T ðl; rÞÞ P Tð1 CT;T ðk; rÞÞ ^ Tð 1 CT;T ðl; rÞÞ P r; 1 CT;T ðl; rÞÞ T ð1 ðCT;T ðk; rÞ _ CT;T ðl; rÞÞÞ ¼ T ½ð1 CT;T ðk; rÞÞ ^ ð 6 T ð1 CT;T ðk; rÞÞ _ T ð 1 CT;T ðl; rÞÞ 6 1 r: Hence, CT;T ðk; rÞ _ CT;T ðl; rÞ P CT;T ðk _ l; rÞ. Therefore CT;T is an intuitionistic fuzzy closure operator on X .
h
Theorem 2.4. Let ðX ; CÞ be an intuitionistic (respectively, intuitionistic supra) fuzzy closure space. Define the functions TC ; TC : I X ! I on X by _ TC ðkÞ ¼ fr 2 I0 jCð1 k; rÞ ¼ 1 kg; ^ TC ðkÞ ¼ f1 r 2 I0 jCð1 k; rÞ ¼ 1 kg: Then: (1) ðTC ; TC Þ is an intuitionistic (respectively, intuitionistic supra) fuzzy topology on X . (2) CTC ;TC is finer than C. Proof (IS1) It is obvious from the definition. 0; rÞ ¼ 0 and Cð 1; rÞ ¼ 1, we have (IS2) Let ðX ; CÞ be an intuitionistic supra fuzzy closure space. Since for all r 2 I0 , Cð (IS2). W (IS3) Suppose that there exists k ¼ i2C ki 2 I X such that ^ _ TC ðki Þ and TC ðkÞ > TC ðki Þ: TC ðkÞ < i2C
i2C
There exists r0 2 I0 such that ^ TC ðkÞ < r0 < TC ðki Þ and
TC ðkÞ > 1 r0 >
i2C
_
TC ðki Þ:
i2C
For all i 2 C, there exist ri 2 I0 with Cð1 ki ; ri Þ ¼ 1 ki such that r0 < ri 6 TC ðki Þ and
1 r0 > 1 ri P TC ðki Þ:
On the other hand, since Cð1 ki ; r0 Þ 6 Cð1 ki ; ri Þ ¼ 1 ki , by (C2) of Definition 2.2, we have Cð1 ki ; r0 Þ ¼ 1 ki : It implies for all i 2 C Cð1 k; r0 Þ 6 Cð1 ki ; r0 Þ ¼ 1 ki : It follows: Cð1 k; r0 Þ 6
^ ð1 ki Þ ¼ 1 k: i2C
Thus, Cð1 k; r0 Þ ¼ 1 k, that is, TC ðkÞ P r0 and TC ðkÞ 6 1 r0 . It is a contradiction. Hence ðTC ; TC Þ is an intuitionistic supra fuzzy topology on X .
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Let ðX ; CÞ be an intuitionistic fuzzy closure space. Suppose that there exist k1 ; k2 2 I X such that TC ðk1 ^ k2 Þ < TC ðk1 Þ ^ TC ðk2 Þ and
TC ðk1 ^ k2 Þ > TC ðk1 Þ _ TC ðk2 Þ:
There exists r 2 I0 such that TC ðk1 ^ k2 Þ < r < TC ðk1 Þ ^ TC ðk2 Þ and TC ðk1 ^ k2 Þ > 1 r > TC ðk1 Þ _ TC ðk2 Þ: For each i 2 f1; 2g, there exist ri 2 I0 with Cð1 ki ; ri Þ ¼ 1 ki such that r < ri 6 TC ðki Þ and
TC ðki Þ 6 1 ri < 1 r:
On the other hand, since Cð1 ki ; rÞ ¼ 1 ki from (C2) and (C4) of Definition 2.2, we have for each i 2 f1; 2g Cðð1 k1 Þ _ ð1 k2 Þ; rÞ ¼ ð1 k1 Þ _ ð1 k2 Þ: It follows TC ðk1 ^ k2 Þ P r and TC ðk1 ^ k2 Þ 6 1 r. It is a contradiction. Hence, for all k; l 2 I X , TC ðk ^ lÞ P TC ðkÞ ^ TC ðlÞ and TC ðk ^ lÞ 6 TC ðkÞ _ TC ðlÞ. Therefore, ðTC ; TC Þ is an intuitionistic fuzzy topology on X . (2) Since k 6 Cðk; rÞ, TC ð1 Cðk; rÞÞ P r and TC ð 1 Cðk; rÞÞ 6 1 r from (C5) of Definition 2.2, we have CTC ;TC ðk; rÞ 6 Cðk; rÞ. Thus, CTC ;TC is finer than C. h Example 2.5. Let X ¼ fa; bg be a set. Let l; q 2 I X as follows: lðaÞ ¼ 0:2;
lðbÞ ¼ 0:3;
Define the function C : I X 8 0; > > > > > > l ^ q; > > > < l; Cðk; rÞ ¼ > > > > > > q; > > > : 1;
qðaÞ ¼ 0:5;
qðbÞ ¼ 0:1:
I0 ! I X as follows: k ¼ 0 8r 2 I0 ; 0 6¼ k 6 l ^ q; 0 < r < 1 ; 2 k 6 l; kiq; 0 < r < 12 ; or 0 6¼ k 6 l; 12 6 r < 23 ; if k 6 q; kil; 0 < r < 12 ; otherwise:
if if if
Then C is an intuitionistic supra fuzzy closure operator but not an intuitionistic fuzzy closure operator because 1 ¼ C l _ q; 1 6¼ C l; 1 _ C q; 1 ¼ l _ q: 3 3 3 From Theorem 2.4, we can obtain an intuitionistic supra fuzzy topology ðTC ; TC Þ on X as follows: 8 > 1; if k ¼ 0 or 1; > > > > 2 > > < 3 ; if k ¼ 1 l; 1 TC ðkÞ ¼ 2 ; if k ¼ 1 q; > > > 1 > ; if k ¼ ð1 lÞ _ ð1 qÞ; > 2 > > : 0; otherwise; 8 > 0; > > > > 1 > > <3; TC ðkÞ ¼ 12 ; > > > > 12 ; > > > : 1;
if k ¼ 0 or 1; if k ¼ 1 l; if k ¼ 1 q; if k ¼ ð1 lÞ _ ð1 qÞ; otherwise:
Since 0 ¼ TC ðð1 lÞ ^ ð1 qÞÞ < TC ð1 lÞ ^ TC ð 1 qÞ ¼ 12 and 1 ¼ TC ðð 1 lÞ ^ ð 1 qÞÞ > TC ð 1 lÞ_ TC ð1 qÞ ¼ 12. Then the ðTC ; TC Þ is not an intuitionistic fuzzy topology on X .
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On the other hand, since 8 0; > > > > > > l ^ q; > > > < l; CTC ;TC ðk; rÞ ¼ > > > > > > q; > > > : 1;
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k ¼ 0 8r 2 I0 ; 0 6¼ k 6 l ^ q; 0 < r 6 1 ; 2 k 6 l; kiq; 0 < r 6 12 ; or 0 6¼ k 6 l; 12 < r 6 23 ; if k 6 q; kil; 0 < r 6 12 ; otherwise;
if if if
we have CTC ;TC ðk; rÞ 6 Cðk; rÞ, but CTC ;TC 6¼ C. 3. Intuitionistic fuzzy bitopological spaces and intuitionistic supra fuzzy topological spaces Theorem 3.1. Let ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ be an intuitionistic fuzzy bitopological space. For each r 2 I0 , k 2 I X , we define the function C12 : I X I0 ! I X as follows: C12 ðk; rÞ ¼ CT1 ;T1 ðk; rÞ ^ CT2 ;T2 ðk; rÞ: Then, ðX ; C12 Þ is an intuitionistic supra fuzzy closure space. Proof. (C1), (C2) and (C4) are easily proved. (C3) We prove it from the followings: for all k; l 2 I X , r 2 I0 . C12 ðk; rÞ _ C12 ðl; rÞ ¼ ðCT1 ;T1 ðk; rÞ ^ CT2 ;T2 ðk; rÞÞ _ ðCT1 ;T1 ðl; rÞ ^ CT2 ;T2 ðl; rÞÞ 6 ðCT1 ;T1 ðk; rÞ _ CT1 ;T1 ðl; rÞÞ ^ ðCT2 ;T2 ðk; rÞ _ CT2 ;T2 ðl; rÞÞ ¼ CT1 ;T1 ðk _ l; rÞ ^ CT2 ;T2 ðk _ l; rÞ ¼ C12 ðk _ l; rÞ: (C5) We prove it from the followings: for all k 2 I X , r 2 I0 C12 ðC12 ðk; rÞ; rÞ ¼ CT1 ;T1 ðC12 ðk; rÞ; rÞ ^ CT2 ;T2 ðC12 ðk; rÞ; rÞ 6 CT1 ;T1 ðCT1 ;T1 ðk; rÞ; rÞ ^ CT2 ;T2 ðCT2 ;T2 ðk; rÞ; rÞ ¼ CT1 ;T1 ðk; rÞ ^ CT2 ;T2 ðk; rÞ ¼ C12 ðk; rÞ:
Lemma 3.2. Let ðX ; T; T Þ be an intuitionistic (respectively, intuitionistic supra) fuzzy topological space. For each r 2 I0 , k 2 I X , we define the function IT;T : I X I0 ! I X as follows: _ IT;T ðk; rÞ ¼ fl j l 6 k; TðlÞ P r; T ðlÞ 6 1 rg: Then we have IT;T ð1 k; rÞ ¼ 1 CT;T ðk; rÞ: Proof. For all k 2 I X , r 2 I0 , we have the following: ^ 1 CT;T ðk; rÞ ¼ 1 fljl P k; Tð1 lÞ P r; T ð 1 lÞ 6 1 rg _ 1 lÞ 6 1 rg ¼ f1 ljl P k; Tð1 lÞ P r; T ð _ ¼ f1 lj1 l 6 1 k; Tð1 lÞ P r; T ð 1 lÞ 6 1 rg ¼ IT;T ð 1 k; rÞ:
Theorem 3.3. Let ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ be an intuitionistic fuzzy bitopological space. For each r 2 I0 , k 2 I X , we define the function I12 : I X I0 ! I X as follows: I12 ðk; rÞ ¼ IT1 ;T1 ðk; rÞ _ IT2 ;T2 ðk; rÞ:
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Then I12 ð1 k; rÞ ¼ 1 C12 ðk; rÞ: Proof. For all k 2 I X , r 2 I0 , we have the following: 1 C12 ðk; rÞ ¼ 1 ðCT ;T ðk; rÞ ^ CT ;T ðk; rÞÞ ¼ ð 1 CT 1
2
1
2
1 ;T1
ðk; rÞÞ _ ð 1 CT2 ;T2 ðk; rÞÞ
1 k; rÞ: ¼ IT1 ;T1 ð1 k; rÞ _ IT2 ;T2 ð1 k; rÞ ¼ I12 ð
From Theorems 2.4 and 3.3, we obtain the following corollary. Corollary 3.4. Let ðX ; C12 Þ be an intuitionistic supra fuzzy closure space. Define the functions TC12 ; TC12 : I X ! I on X by _ _ TC12 ðkÞ ¼ fr 2 I0 j C12 ð1 k; rÞ ¼ 1 kg ¼ fr 2 I0 jI12 ðk; rÞ ¼ kg; ^ TC12 ðkÞ ¼ f1 r 2 I0 j I12 ðk; rÞ ¼ kg: Then ðTC12 ; TC12 Þ is an intuitionistic supra fuzzy topology on X . Lemma 3.5. Let ðX ; T; T Þ be an intuitionistic fuzzy topological space. For each r; s 2 I0 , k 2 I X , we have _ CT;T ðk; sÞ ¼ CT;T ðk; rÞ: s
Proof. It is obvious.
h
Lemma 3.6. Let ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ be an intuitionistic fuzzy bitopological space. For each r; s 2 I0 , k 2 I X , we have _ C12 ðk; sÞ ¼ C12 ðk; rÞ: s
Proof. Since CT1 ;T1 ðk; sÞ 6 CT1 ;T1 ðk; rÞ and CT2 ;T2 ðk; sÞ 6 CT2 ;T2 ðk; rÞ for all s 2 ð0; rÞ _ ðCT1 ;T1 ðk; sÞ ^ CT2 ;T2 ðk; sÞÞ 6 CT1 ;T1 ðk; rÞ ^ CT2 ;T2 ðk; rÞ: s
It follows: _ C12 ðk; sÞ 6 C12 ðk; rÞ: s
Suppose there exists x 2 X such that _ ðCT1 ;T1 ðk; sÞ ^ CT2 ;T2 ðk; sÞÞðxÞ < CT1 ;T1 ðk; rÞðxÞ ^ CT2 ;T2 ðk; rÞðxÞ: s
W Since si
Put r ¼ r1 _ r2 . Then r < r and CT1 ;T1 ðk; r1 ÞðxÞ ^ CT2 ;T2 ðk; r2 ÞðxÞ 6 CT1 ;T1 ðk; r ÞðxÞ ^ CT2 ;T2 ðk; r ÞðxÞ 6
_
C12 ðk; sÞðxÞ:
s
It is contradiction. Hence _ C12 ðk; sÞ P C12 ðk; rÞ:
s
Theorem 3.7. Let ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ be an intuitionistic fuzzy bitopological space. Let ðX ; C12 Þ be an intuitionistic supra fuzzy closure space. Define the functions Ts ; Ts : I X ! I on X by _ Ts ðkÞ ¼ fT1 ðk1 Þ ^ T2 ðk2 Þjk ¼ k1 _ k2 g;
S.E. Abbas / Chaos, Solitons and Fractals 21 (2004) 1205–1214
where
W
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is taken over all families fk1 ; k2 jk ¼ k1 _ k2 g and ^ ¼ fT1 ðk1 Þ _ T2 ðk2 Þjk ¼ k1 _ k2 g;
Ts ðkÞ where
V
is taken over all families fk1 ; k2 jk ¼ k1 _ k2 g. Then:
(1) ðTs ; Ts Þ ¼ ðTC12 ; TC12 Þ is the coarsest intuitionistic supra fuzzy topology on X which is finer than ðT1 ; T1 Þ and ðT2 ; T2 Þ. (2) C12 ¼ CTs ;Ts ¼ CTC12 ;TC . 12
Proof. (1) Suppose that there exists k 2 I X such that TC12 ðkÞ > Ts ðkÞ and
TC12 ðkÞ < Ts ðkÞ:
By the definition of ðTC12 ; TC12 Þ from Corollary 3.4, there exists r0 2 I0 with C12 ð 1 k; r0 Þ ¼ 1 k such that TC12 ðkÞ P r0 > Ts ðkÞ and
TC12 ðkÞ 6 1 r0 < Ts ðkÞ:
On the other hand, since C12 ð1 k; r0 Þ ¼ 1 k, we have 1 k; r0 ÞÞ k ¼ 1 C12 ð1 k; r0 Þ ¼ 1 ðCT1 ;T1 ð1 k; r0 Þ ^ CT2 ;T2 ð ¼ ð1 CT1 ;T1 ð1 k; r0 ÞÞ _ ð1 CT2 ;T2 ð1 k; r0 ÞÞ ¼ IT1 ;T1 ðk; r0 Þ _ IT2 ;T2 ðk; r0 Þ: Since _ T1 ðIT1 ;T1 ðk; r0 ÞÞ ¼ T1 fl j l 6 k; T1 ðlÞ P r0 ; T1 ðlÞ 6 1 r0 g ^ P fT1 ðlÞ j l 6 k; T1 ðlÞ P r0 ; T1 ðlÞ 6 1 r0 g P r0 ; _ fl j l 6 k; T1 ðlÞ P r0 ; T1 ðlÞ 6 1 r0 g T1 ðIT1 ;T1 ðk; r0 ÞÞ ¼ T1 _ 6 fT1 ðlÞ j l 6 k; T1 ðlÞ P r0 ; T1 ðlÞ 6 1 r0 g 6 1 r0 and, similarly, T2 ðIT2 ;T2 ðk; r0 ÞÞ P r0 and T2 ðIT2 ;T2 ðk; r0 ÞÞ 6 1 r0 we have Ts ðkÞ P r0 and Ts ðkÞ 6 1 r0 . It is a contradiction. Hence TC12 6 Ts and TC12 P Ts . Suppose that there exists q 2 I X such that TC12 ðqÞ < Ts ðqÞ and
TC12 ðqÞ > Ts ðqÞ:
There exists r1 2 I0 such that TC12 ðqÞ < r1 < Ts ðqÞ and
TC12 ðqÞ > 1 r1 > Ts ðqÞ:
By the definition of ðTs ; Ts Þ, there exist q1 ; q2 2 I X with q ¼ q1 _ q2 such that TC12 ðqÞ < r1 6 T1 ðq1 Þ ^ T2 ðq2 Þ 6 Ts ðqÞ; TC12 ðqÞ > 1 r1 P T1 ðq1 Þ _ T2 ðq2 Þ P Ts ðqÞ: On the other hand, since r1 6 Ti ðqi Þ and 1 r1 P Ti ðqi Þ for all i ¼ 1; 2, we have CTi ;Ti ð1 qi ; r1 Þ ¼ 1 qi : It follows that 1 q1 ; r1 Þ ^ CT2 ;T2 ð 1 q2 ; r1 Þ C12 ð1 q; r1 Þ ¼ CT1 ;T1 ð1 q; r1 Þ ^ CT2 ;T2 ð1 q; r1 Þ 6 CT1 ;T1 ð ¼ ð1 q1 Þ ^ ð1 q2 Þ ¼ 1 q: Hence TC12 ðqÞ P r1 and TC12 ðqÞ 6 1 r1 . It is a contradiction. Therefore TC12 P Ts and TC12 6 Ts . Thus ðTC12 ; TC12 Þ ¼ ðTs ; Ts Þ is an intuitionistic supra fuzzy topology on X from Corollary 3.4.
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Finally, we will show that ðTs ; Ts Þ is the coarsest intuitionistic supra fuzzy topology on X which is finer than ðT1 ; T1 Þ and ðT2 ; T2 Þ. For k ¼ k _ 0 and i 6¼ j 2 f1; 2g, we have Ts ðkÞ P Ti ðkÞ ^ Tj ð0Þ ¼ Ti ðkÞ and
Ts ðkÞ 6 Ti ðkÞ _ Tj ð 0Þ ¼ Ti ðkÞ:
If ðU; U Þ is the intuitionistic supra fuzzy topology on X which finer than ðT1 ; T1 Þ and ðT2 ; T2 Þ, for every family fk1 ; k2 g such that k ¼ k1 _ k2 we have T1 ðk1 Þ ^ T2 ðk2 Þ 6 Uðk1 Þ ^ Uðk2 Þ 6 Uðk1 _ k2 Þ; T1 ðk1 Þ _ T2 ðk2 Þ P U ðk1 Þ _ U ðk2 Þ P U ðk1 _ k2 Þ: By the definition of ðTs ; Ts Þ, we have Ts ðkÞ 6 UðkÞ and Ts ðkÞ P U ðkÞ for all k 2 I X . (2) Suppose there exist x 2 X , r 2 I0 , k 2 I X such that C12 ðk; rÞðxÞ < CTs ;Ts ðk; rÞðxÞ: 1 qi Þ P r and Ti ð 1 qi Þ 6 1 r such that By the definition of C12 , for i 2 f1; 2g there exist qi 2 I X with qi P k, Ti ð C12 ðk; rÞðxÞ 6 ðq1 ^ q2 ÞðxÞ < CTs ;Ts ðk; rÞðxÞ: Since k 6 q1 ^ q2 and 1 q1 Þ ^ T2 ð 1 q2 Þ P r; Ts ð1 ðq1 ^ q2 ÞÞ ¼ Ts ðð1 q1 Þ _ ð1 q2 ÞÞ P T1 ð 1 q1 Þ _ T2 ð 1 q2 Þ 6 1 r; Ts ð1 ðq1 ^ q2 ÞÞ ¼ Ts ðð1 q1 Þ _ ð1 q2 ÞÞ 6 T1 ð by the definition of CTs ;Ts , we have CTs ;Ts ðk; rÞ 6 q1 ^ q2 : It is a contradiction. Hence ðCTs ;Ts Þ is finer than C12 . Suppose there exist x1 2 X , r1 2 I0 , x 2 I X such that C12 ðx; r1 Þðx1 Þ > CTs ;Ts ðx; r1 Þðx1 Þ: 1 qÞ P r1 and Ts ð 1 qÞ 6 1 r1 such that By the definition of CTs ;Ts , there exists q 2 I X with q P x, Ts ð C12 ðx; r1 Þðx1 Þ > qðx1 Þ P CTs ;Ts ðx; r1 Þðx1 Þ: On the other hand, since Ts ð1 qÞ P r1 and Ts ð1 qÞ 6 1 r1 , by the definition of ðTs ; Ts Þ, for all s < r1 , there exist l1 ; l2 2 I X with 1 q ¼ l1 _ l2 such that Ts ð1 qÞ P T1 ðl1 Þ ^ T2 ðl2 Þ > s and
Ts ð1 qÞ 6 T1 ðl1 Þ _ T2 ðl2 Þ < 1 s:
It follows that, for i 2 f1; 2g, CTi ;Ti ð1 li ; sÞ ¼ 1 li : Since q 6 1 li , for i 2 f1; 2g, CTi ;Ti ðq; sÞ 6 CTi ;Ti ð1 li ; sÞ: So, for all s < r1 , 1 l1 ; sÞ ^ CT2 ;T2 ð 1 l2 ; sÞ ¼ ð 1 l1 Þ ^ ð 1 l2 Þ ¼ q: C12 ðq; sÞ ¼ CT1 ;T1 ðq; sÞ ^ CT2 ;T2 ðq; sÞ 6 CT1 ;T1 ð Thus, C12 ðq; sÞ ¼ q from (C2) of Definition 2.2. Hence, by Lemma 3.6, _ C12 ðq; sÞ ¼ C12 ðq; r1 Þ: q¼ s
Since x 6 q, we have C12 ðx; r1 Þ 6 C12 ðq; r1 Þ ¼ q: It is a contradiction. Hence C12 is finer that CTs ;Ts .
h
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Example 3.8. Let X ¼ fa; bg. Define l; q 2 I X as follows: lðaÞ ¼ 0:2;
lðbÞ ¼ 0:4;
qðaÞ ¼ 0:5;
qðbÞ ¼ 0:1:
ðT1 ; T1 Þ; ðT2 ; T2 Þ
Define intuitionistic fuzzy topologies 8 8 > > < 1; if k ¼ 0; 1; < 0; 1 T1 ðkÞ ¼ 12 ; T1 ðkÞ ¼ 2 ; if k ¼ l; > > : : 0; otherwise; 1; 8 8 < 1; if k ¼ 0; 1; < 0; T2 ðkÞ ¼ 23 ; if k ¼ q; T2 ðkÞ ¼ 13 ; : : 0; otherwise; 1;
: I X ! I as follows:
if k ¼ 0; 1; if k ¼ l; otherwise; if k ¼ 0; 1; if k ¼ q; otherwise:
Then the intuitionistic fuzzy bitopological space ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ induces ðTs ; Ts Þ as follows: 8 8 1; if k ¼ 0; 1; 0; if k ¼ 0; 1; > > > > > > >1 > 1 > > ; if k ¼ l; ; if k ¼ l; > > <2 <2 Ts ðkÞ ¼ 23 ; if k ¼ q; Ts ðkÞ ¼ 13 ; if k ¼ q; > > > > > > > 12 ; if k ¼ l _ q; > 12 ; if k ¼ l _ q; > > > > : : 0; otherwise; 1; otherwise: Since 1 0 ¼ Ts ðl ^ qÞ < Ts ðlÞ ^ Ts ðqÞ ¼ ; 2 1 1 ¼ Ts ðl ^ qÞ > Ts ðlÞ _ Ts ðqÞ ¼ ; 2 ðTs ; Ts Þ is not an intuitionistic fuzzy topology on X . On the other hand, since 8 if k ¼ 0; 8r 2 I0 ; < 0; CT1 ;T1 ðk; rÞ ¼ 1 l; if 0 6¼ k 6 1 l; 0 < r 6 12 ; : 1; otherwise and 8 < 0; CT2 ;T2 ðk; rÞ ¼ 1 q; : 1;
if k ¼ 0; 8r 2 I0 ; if 0 6¼ k 6 1 q; 0 < r 6 23 ; otherwise;
we have 8 0 > > > > > ð1 lÞ ^ ð1 qÞ; > > < 1 l; C12 ðk; rÞ ¼ > > 1 q; > > > > > : 1; Since
k ¼ 0; 8r 2 I0 ; 1 lÞ ^ ð 1 qÞ; k 6¼ 0; 0 < r 6 12 ; k 6 ð 1 q; k 6 1 l; 0 < r 6 12 ; ki 1 l; k 6 1 q; 0 < r 6 12 ; ki or 0 6¼ k 6 1 q; 12 < r 6 23 ; otherwise:
if if if if
1 1 ¼ C12 ð1 lÞ _ ð1 qÞ; 1 6¼ C12 1 l; 1 _ C12 ¼ ð 1 lÞ _ ð 1 qÞ 1 q; 2 2 2
an intuitionistic supra fuzzy closure space ðX ; C12 Þ is not an intuitionistic fuzzy closure space. We easily show that C12 ¼ CTs ;Ts . h In conclusion, we may stress once more the importance of fuzzy topology as a nontrivial extension of fuzzy sets, intuitionistic fuzzy sets and fuzzy logic [16] and the possible application in quantum physics [10,11].
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References [1] Abd-Elmonsef ME, Ramadan AE. On fuzzy supra topological spaces. Indian J Pure Appl Math 1987;18:322–9. [2] Abd El-Monsef ME, Nasef AA, Salama AA. Some fuzzy topological operators via fuzzy ideals. Chaos, Solitons & Fractals 2001;12:2509–15. [3] Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20(1):87–96. [4] Atanassov K. New operators defined over the intuitionistic fuzzy sets. Fuzzy Sets Syst 1993;61(2):131–42. [5] Atanassov K, Stoeva S. Intuitionistic fuzzy sets. In: Proceedings of the Polish Symposium on Interval and Fuzzy Mathematics, Poznan, August 1983. p. 23–6. [6] Chang CL. Fuzzy topological spaces. J Math Anal Appl 1968;24:182–90. [7] Chattopadhyay KC, Hazra RN, Samanta SK. Gradation of openess: fuzzy topology. Fuzzy Sets Syst 1992;49:237–42. [8] C ß oker D. An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets Syst 1997;88:81–9. [9] C sense. Busefal 1996;67:67–76. ß oker D, Demirsi M. An introduction to intuitionistic fuzzy topological spaces in Sostak’s [10] El Naschie MS. On the uncertainty of cantorian geometry and the two-slit experiment. Chaos, Solitons & Fractals 1998;9(3):517– 29. [11] El Naschie MS. M theory and Eð1Þ theory. Chaos, Solitons & Fractals 2000;2:2397–408. [12] Ghanim MH, Tantawy OA, Selim FM. Gradation of supra-openness. Fuzzy Sets Syst 2000;109:245–50. [13] H€ ohle U. Upper semicontinuous fuzzy sets and applications. J Math Anal Appl 1980;78:659–73. [14] H€ ohle U, Sostak AP. A general theory of fuzzy topological spaces. Fuzzy Sets Syst 1995;73:131–49. [15] H€ ohle U, Sostak AP. Axiomatic Foundations of Fixed-Basis fuzzy topology, The Handbooks of Fuzzy sets series, Volume 3, Kluwer Academic Publishers, Dordrecht (Chapter 3) 1999. [16] Kosko B. Fuzzy thinking. Glasgow: Flamingo; 1994. [17] Kubiak T. On fuzzy topologies, Phd. Thesis, A. Mickiewicz, Poznan; 1985. [18] Kubiak T, Sostak AP. Lower set-valued fuzzy topologies. Quest Math 1997;20(3):423–9. [19] Samanta SK, Mondal TK. Intuitionistic gradation of openness: intuitionistic fuzzy topology. Busefal 1997;73:8–17. [20] Samanta SK, Mondal TK. On intuitionistic gradation of openness. Fuzzy Sets Syst 2002;131:323–36. [21] Sostak AP. On a fuzzy topological structure. Suppl Rend Circ Matem Palerms ser II 1985;11:89–103.
Intuitionistic supra fuzzy topological spaces S.E. Abbas Department of Mathematics, Faculty of Science, South Valley University, Sohag 82524, Egypt Accepted 8 December 2003
Abstract In this paper, We introduce an intuitionistic supra fuzzy closure space and investigate the relationship between intuitionistic supra fuzzy topological spaces and intuitionistic supra fuzzy closure spaces. Moreover, we can obtain intuitionistic supra fuzzy topological space induced by an intuitionistic fuzzy bitopological space. We study the relationship between intuitionistic supra fuzzy closure space and the intuitionistic supra fuzzy topological space induced by an intuitionistic fuzzy bitopological space. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries Chattopadhyay et al. [7] introduced the concept of gradation of openness (closedness) of fuzzy subsets of X and gave definitions of a fuzzy topology on X as an extension of Chang’s fuzzy topology [6]. Historically, the fundamental idea of a topology being a fuzzy subset of a power set, first appeared in 1980 [13]. This was followed in 1985 by the independent and parallel generalizations of Kubiak [17] and Sostak [21], papers in which a topology was a fuzzy subset of a powerset of fuzzy subsets. It has been independently considered by authors not aware of the historical development of [2,14,15,17,18]. Moreover, Ghanim et al., [12] introduced the gradation of supra openness as an extension of supra fuzzy topology in a sense of Abd-Elmonsef and Ramadan [1]. The concept of fuzzy topology may have very important applications in quantum particles physics particularly in connection with string theory and Eð1Þ theory [10,11]. On the other hand, Atanassov [3] introduced the idea of intuitionistic fuzzy set. Recently, much work has been done with these concepts [3–5]. C ß oker and coworker [8,9] introduced the idea of the topology of intuitionistic fuzzy sets. Samanta and Mondal [19,20] introduced the definition of the intuitionistic gradation of openness. In this paper, we define an intuitionistic supra fuzzy closure space and an intuitionistic fuzzy bitopological space. We investigate the relationship between intuitionistic supra fuzzy topological spaces and intuitionistic supra fuzzy closure spaces. Moreover, we can obtain the intuitionistic supra fuzzy topological space induced by an intuitionistic fuzzy bitopological space. We study the relationship between intuitionistic supra fuzzy closure space and the intuitionistic supra fuzzy topological space induced by an intuitionistic fuzzy bitopological space. Throughout this paper, let X be a nonempty set, I ¼ ½0; 1 and I0 ¼ ð0; 1. For a 2 I, aðxÞ ¼ a for each x 2 X . Notions and notations not described in this paper are standard and usual.
2. Intuitionistic supra fuzzy topological spaces and intuitionistic supra fuzzy closure spaces Definition 2.1. An intuitionistic supra fuzzy topology on X is an ordered pair ðT; T Þ of functions from I X to I such that
E-mail address: [email protected] (S.E. Abbas). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.070
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(IS1) TðkÞ þ T ðkÞ 6 1, 8k 2 I X , ¼ 1, T ð0Þ ¼ T ð1Þ ¼ 0, (IS2) Tð0Þ W ¼ Tð1Þ V W W (IS3) Tð i2D ki Þ P i2D Tðki Þ and T ð i2D ki Þ 6 i2D T ðki Þ, ki 2 I X , i 2 D. The triplet ðX ; T; T Þ is called an intuitionistic supra fuzzy topological space. An intuitionistic supra fuzzy topology ðT; T Þ is called an intuitionistic fuzzy topology on X iff (IT) Tðk1 ^ k2 Þ P Tðk1 Þ ^ Tðk2 Þ and T ðk1 ^ k2 Þ 6 T ðk1 Þ _ T ðk2 Þ, ki 2 I X , i ¼ 1; 2. The triplet ðX ; T; T Þ is called an intuitionistic fuzzy topological space. T and T may be interpreted as gradation of openness and gradation of nonopenness, respectively. The ðX ; ðT; T Þ; ðU; U ÞÞ is called an intuitionistic fuzzy bitopological space where ðT; T Þ and ðU; U Þ are intuitionistic fuzzy topologies on X . Let ðT; T Þ and ðU; U Þ be intuitionistic supra fuzzy topologies on X . We say that ðT; T Þ is finer than ðU; U Þ (ðU; U Þ is coarser than ðT; T Þ) iff UðkÞ 6 TðkÞ and U ðkÞ P T ðkÞ for each k 2 I X . Definition 2.2. A function C : I X I0 ! I X is called an intuitionistic supra fuzzy closure operator on X if for k; l 2 I X and r; s 2 I0 , it satisfies the following conditions: (C1) (C2) (C3) (C4) (C5)
Cð0; rÞ ¼ 0, k 6 Cðk; rÞ, Cðk; rÞ _ Cðl; rÞ 6 Cðk _ l; rÞ, Cðk; rÞ 6 Cðk; sÞ, if r 6 s, CðCðk; rÞ; rÞ ¼ Cðk; rÞ.
The pair ðX ; CÞ is called an intuitionistic supra fuzzy closure space. The intuitionistic supra fuzzy closure space ðX ; CÞ is called an intuitionistic fuzzy closure space iff (C) Cðk; rÞ _ Cðl; rÞ ¼ Cðk _ l; rÞ, for each k; l 2 I X . Let C1 and C2 be intuitionistic supra fuzzy closure operators on X . We say that C1 is finer than C2 (C2 is coarser than C1 ) iff C1 ðk; rÞ 6 C2 ðk; rÞ, for each k 2 I X and r 2 I0 . Theorem 2.3. Let ðX ; T; T Þ be an intuitionistic (respectively, intuitionistic supra) fuzzy topological space. Then for each r 2 I0 , k 2 I X we define an operator CT;T : I X I0 ! I X as follows: ^ 1 lÞ 6 1 rg: CT;T ðk; rÞ ¼ fl 2 I X : k 6 l; Tð1 lÞ P r; T ð Then ðX ; CT;T Þ is an intuitionistic (respectively, intuitionistic supra) fuzzy closure space. Proof. Let ðX ; T; T Þ be an intuitionistic supra fuzzy topological space. Then, (C1), (C2) and (C4) are trivial from the definition of CT;T . (C3) Since k; l 6 k _ l, we have CT;T ðk; rÞ 6 CT;T ðk _ l; rÞ CT;T ðk; rÞ _ CT;T ðl; rÞ 6 CT;T ðk _ l; rÞ. (C5) Suppose that there exist k 2 I X , r 2 I0 and x 2 X such that
and
CT;T ðl; rÞ 6 CT;T ðk _ l; rÞ.
Hence
CT;T ðCT;T ðk; rÞ; rÞðxÞ > CT;T ðk; rÞðxÞ: 1 qÞ P r, T ð 1 qÞ 6 1 r such that By the definition of CT;T ðk; rÞ, there exists q 2 I X with q P k and Tð CT;T ðCT;T ðk; rÞ; rÞðxÞ > qðxÞ P CT;T ðk; rÞðxÞ: On the other hand, since CT;T ðk; rÞ 6 q and Tð1 qÞ P r, T ð 1 qÞ 6 1 r, by the definition of CT;T ðCT;T ðk; rÞ; rÞ, we have CT;T ðCT;T ðk; rÞ; rÞ 6 q:
S.E. Abbas / Chaos, Solitons and Fractals 21 (2004) 1205–1214
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It is a contradiction. Thus, CT;T ðCT;T ðk; rÞ; rÞ ¼ CT;T ðk; rÞ. Hence CT;T is an intuitionistic supra fuzzy closure operator on X . Let ðX ; T; T Þ be an intuitionistic fuzzy topological space. From (C2), we have k 6 CT;T ðk; rÞ;
Tð1 CT;T ðk; rÞÞ P r;
T ð 1 CT;T ðk; rÞÞ 6 1 r;
l 6 CT;T ðl; rÞ;
Tð1 CT;T ðl; rÞÞ P r;
T ð 1 CT;T ðl; rÞÞ 6 1 r:
It implies k _ l 6 CT;T ðk; rÞ _ CT;T ðl; rÞ such that Tð1 ðCT;T ðk; rÞ _ CT;T ðl; rÞÞÞ ¼ T½ð1 CT;T ðk; rÞÞ ^ ð 1 CT;T ðl; rÞÞ P Tð1 CT;T ðk; rÞÞ ^ Tð 1 CT;T ðl; rÞÞ P r; 1 CT;T ðl; rÞÞ T ð1 ðCT;T ðk; rÞ _ CT;T ðl; rÞÞÞ ¼ T ½ð1 CT;T ðk; rÞÞ ^ ð 6 T ð1 CT;T ðk; rÞÞ _ T ð 1 CT;T ðl; rÞÞ 6 1 r: Hence, CT;T ðk; rÞ _ CT;T ðl; rÞ P CT;T ðk _ l; rÞ. Therefore CT;T is an intuitionistic fuzzy closure operator on X .
h
Theorem 2.4. Let ðX ; CÞ be an intuitionistic (respectively, intuitionistic supra) fuzzy closure space. Define the functions TC ; TC : I X ! I on X by _ TC ðkÞ ¼ fr 2 I0 jCð1 k; rÞ ¼ 1 kg; ^ TC ðkÞ ¼ f1 r 2 I0 jCð1 k; rÞ ¼ 1 kg: Then: (1) ðTC ; TC Þ is an intuitionistic (respectively, intuitionistic supra) fuzzy topology on X . (2) CTC ;TC is finer than C. Proof (IS1) It is obvious from the definition. 0; rÞ ¼ 0 and Cð 1; rÞ ¼ 1, we have (IS2) Let ðX ; CÞ be an intuitionistic supra fuzzy closure space. Since for all r 2 I0 , Cð (IS2). W (IS3) Suppose that there exists k ¼ i2C ki 2 I X such that ^ _ TC ðki Þ and TC ðkÞ > TC ðki Þ: TC ðkÞ < i2C
i2C
There exists r0 2 I0 such that ^ TC ðkÞ < r0 < TC ðki Þ and
TC ðkÞ > 1 r0 >
i2C
_
TC ðki Þ:
i2C
For all i 2 C, there exist ri 2 I0 with Cð1 ki ; ri Þ ¼ 1 ki such that r0 < ri 6 TC ðki Þ and
1 r0 > 1 ri P TC ðki Þ:
On the other hand, since Cð1 ki ; r0 Þ 6 Cð1 ki ; ri Þ ¼ 1 ki , by (C2) of Definition 2.2, we have Cð1 ki ; r0 Þ ¼ 1 ki : It implies for all i 2 C Cð1 k; r0 Þ 6 Cð1 ki ; r0 Þ ¼ 1 ki : It follows: Cð1 k; r0 Þ 6
^ ð1 ki Þ ¼ 1 k: i2C
Thus, Cð1 k; r0 Þ ¼ 1 k, that is, TC ðkÞ P r0 and TC ðkÞ 6 1 r0 . It is a contradiction. Hence ðTC ; TC Þ is an intuitionistic supra fuzzy topology on X .
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Let ðX ; CÞ be an intuitionistic fuzzy closure space. Suppose that there exist k1 ; k2 2 I X such that TC ðk1 ^ k2 Þ < TC ðk1 Þ ^ TC ðk2 Þ and
TC ðk1 ^ k2 Þ > TC ðk1 Þ _ TC ðk2 Þ:
There exists r 2 I0 such that TC ðk1 ^ k2 Þ < r < TC ðk1 Þ ^ TC ðk2 Þ and TC ðk1 ^ k2 Þ > 1 r > TC ðk1 Þ _ TC ðk2 Þ: For each i 2 f1; 2g, there exist ri 2 I0 with Cð1 ki ; ri Þ ¼ 1 ki such that r < ri 6 TC ðki Þ and
TC ðki Þ 6 1 ri < 1 r:
On the other hand, since Cð1 ki ; rÞ ¼ 1 ki from (C2) and (C4) of Definition 2.2, we have for each i 2 f1; 2g Cðð1 k1 Þ _ ð1 k2 Þ; rÞ ¼ ð1 k1 Þ _ ð1 k2 Þ: It follows TC ðk1 ^ k2 Þ P r and TC ðk1 ^ k2 Þ 6 1 r. It is a contradiction. Hence, for all k; l 2 I X , TC ðk ^ lÞ P TC ðkÞ ^ TC ðlÞ and TC ðk ^ lÞ 6 TC ðkÞ _ TC ðlÞ. Therefore, ðTC ; TC Þ is an intuitionistic fuzzy topology on X . (2) Since k 6 Cðk; rÞ, TC ð1 Cðk; rÞÞ P r and TC ð 1 Cðk; rÞÞ 6 1 r from (C5) of Definition 2.2, we have CTC ;TC ðk; rÞ 6 Cðk; rÞ. Thus, CTC ;TC is finer than C. h Example 2.5. Let X ¼ fa; bg be a set. Let l; q 2 I X as follows: lðaÞ ¼ 0:2;
lðbÞ ¼ 0:3;
Define the function C : I X 8 0; > > > > > > l ^ q; > > > < l; Cðk; rÞ ¼ > > > > > > q; > > > : 1;
qðaÞ ¼ 0:5;
qðbÞ ¼ 0:1:
I0 ! I X as follows: k ¼ 0 8r 2 I0 ; 0 6¼ k 6 l ^ q; 0 < r < 1 ; 2 k 6 l; kiq; 0 < r < 12 ; or 0 6¼ k 6 l; 12 6 r < 23 ; if k 6 q; kil; 0 < r < 12 ; otherwise:
if if if
Then C is an intuitionistic supra fuzzy closure operator but not an intuitionistic fuzzy closure operator because 1 ¼ C l _ q; 1 6¼ C l; 1 _ C q; 1 ¼ l _ q: 3 3 3 From Theorem 2.4, we can obtain an intuitionistic supra fuzzy topology ðTC ; TC Þ on X as follows: 8 > 1; if k ¼ 0 or 1; > > > > 2 > > < 3 ; if k ¼ 1 l; 1 TC ðkÞ ¼ 2 ; if k ¼ 1 q; > > > 1 > ; if k ¼ ð1 lÞ _ ð1 qÞ; > 2 > > : 0; otherwise; 8 > 0; > > > > 1 > > <3; TC ðkÞ ¼ 12 ; > > > > 12 ; > > > : 1;
if k ¼ 0 or 1; if k ¼ 1 l; if k ¼ 1 q; if k ¼ ð1 lÞ _ ð1 qÞ; otherwise:
Since 0 ¼ TC ðð1 lÞ ^ ð1 qÞÞ < TC ð1 lÞ ^ TC ð 1 qÞ ¼ 12 and 1 ¼ TC ðð 1 lÞ ^ ð 1 qÞÞ > TC ð 1 lÞ_ TC ð1 qÞ ¼ 12. Then the ðTC ; TC Þ is not an intuitionistic fuzzy topology on X .
S.E. Abbas / Chaos, Solitons and Fractals 21 (2004) 1205–1214
On the other hand, since 8 0; > > > > > > l ^ q; > > > < l; CTC ;TC ðk; rÞ ¼ > > > > > > q; > > > : 1;
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k ¼ 0 8r 2 I0 ; 0 6¼ k 6 l ^ q; 0 < r 6 1 ; 2 k 6 l; kiq; 0 < r 6 12 ; or 0 6¼ k 6 l; 12 < r 6 23 ; if k 6 q; kil; 0 < r 6 12 ; otherwise;
if if if
we have CTC ;TC ðk; rÞ 6 Cðk; rÞ, but CTC ;TC 6¼ C. 3. Intuitionistic fuzzy bitopological spaces and intuitionistic supra fuzzy topological spaces Theorem 3.1. Let ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ be an intuitionistic fuzzy bitopological space. For each r 2 I0 , k 2 I X , we define the function C12 : I X I0 ! I X as follows: C12 ðk; rÞ ¼ CT1 ;T1 ðk; rÞ ^ CT2 ;T2 ðk; rÞ: Then, ðX ; C12 Þ is an intuitionistic supra fuzzy closure space. Proof. (C1), (C2) and (C4) are easily proved. (C3) We prove it from the followings: for all k; l 2 I X , r 2 I0 . C12 ðk; rÞ _ C12 ðl; rÞ ¼ ðCT1 ;T1 ðk; rÞ ^ CT2 ;T2 ðk; rÞÞ _ ðCT1 ;T1 ðl; rÞ ^ CT2 ;T2 ðl; rÞÞ 6 ðCT1 ;T1 ðk; rÞ _ CT1 ;T1 ðl; rÞÞ ^ ðCT2 ;T2 ðk; rÞ _ CT2 ;T2 ðl; rÞÞ ¼ CT1 ;T1 ðk _ l; rÞ ^ CT2 ;T2 ðk _ l; rÞ ¼ C12 ðk _ l; rÞ: (C5) We prove it from the followings: for all k 2 I X , r 2 I0 C12 ðC12 ðk; rÞ; rÞ ¼ CT1 ;T1 ðC12 ðk; rÞ; rÞ ^ CT2 ;T2 ðC12 ðk; rÞ; rÞ 6 CT1 ;T1 ðCT1 ;T1 ðk; rÞ; rÞ ^ CT2 ;T2 ðCT2 ;T2 ðk; rÞ; rÞ ¼ CT1 ;T1 ðk; rÞ ^ CT2 ;T2 ðk; rÞ ¼ C12 ðk; rÞ:
Lemma 3.2. Let ðX ; T; T Þ be an intuitionistic (respectively, intuitionistic supra) fuzzy topological space. For each r 2 I0 , k 2 I X , we define the function IT;T : I X I0 ! I X as follows: _ IT;T ðk; rÞ ¼ fl j l 6 k; TðlÞ P r; T ðlÞ 6 1 rg: Then we have IT;T ð1 k; rÞ ¼ 1 CT;T ðk; rÞ: Proof. For all k 2 I X , r 2 I0 , we have the following: ^ 1 CT;T ðk; rÞ ¼ 1 fljl P k; Tð1 lÞ P r; T ð 1 lÞ 6 1 rg _ 1 lÞ 6 1 rg ¼ f1 ljl P k; Tð1 lÞ P r; T ð _ ¼ f1 lj1 l 6 1 k; Tð1 lÞ P r; T ð 1 lÞ 6 1 rg ¼ IT;T ð 1 k; rÞ:
Theorem 3.3. Let ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ be an intuitionistic fuzzy bitopological space. For each r 2 I0 , k 2 I X , we define the function I12 : I X I0 ! I X as follows: I12 ðk; rÞ ¼ IT1 ;T1 ðk; rÞ _ IT2 ;T2 ðk; rÞ:
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Then I12 ð1 k; rÞ ¼ 1 C12 ðk; rÞ: Proof. For all k 2 I X , r 2 I0 , we have the following: 1 C12 ðk; rÞ ¼ 1 ðCT ;T ðk; rÞ ^ CT ;T ðk; rÞÞ ¼ ð 1 CT 1
2
1
2
1 ;T1
ðk; rÞÞ _ ð 1 CT2 ;T2 ðk; rÞÞ
1 k; rÞ: ¼ IT1 ;T1 ð1 k; rÞ _ IT2 ;T2 ð1 k; rÞ ¼ I12 ð
From Theorems 2.4 and 3.3, we obtain the following corollary. Corollary 3.4. Let ðX ; C12 Þ be an intuitionistic supra fuzzy closure space. Define the functions TC12 ; TC12 : I X ! I on X by _ _ TC12 ðkÞ ¼ fr 2 I0 j C12 ð1 k; rÞ ¼ 1 kg ¼ fr 2 I0 jI12 ðk; rÞ ¼ kg; ^ TC12 ðkÞ ¼ f1 r 2 I0 j I12 ðk; rÞ ¼ kg: Then ðTC12 ; TC12 Þ is an intuitionistic supra fuzzy topology on X . Lemma 3.5. Let ðX ; T; T Þ be an intuitionistic fuzzy topological space. For each r; s 2 I0 , k 2 I X , we have _ CT;T ðk; sÞ ¼ CT;T ðk; rÞ: s
Proof. It is obvious.
h
Lemma 3.6. Let ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ be an intuitionistic fuzzy bitopological space. For each r; s 2 I0 , k 2 I X , we have _ C12 ðk; sÞ ¼ C12 ðk; rÞ: s
Proof. Since CT1 ;T1 ðk; sÞ 6 CT1 ;T1 ðk; rÞ and CT2 ;T2 ðk; sÞ 6 CT2 ;T2 ðk; rÞ for all s 2 ð0; rÞ _ ðCT1 ;T1 ðk; sÞ ^ CT2 ;T2 ðk; sÞÞ 6 CT1 ;T1 ðk; rÞ ^ CT2 ;T2 ðk; rÞ: s
It follows: _ C12 ðk; sÞ 6 C12 ðk; rÞ: s
Suppose there exists x 2 X such that _ ðCT1 ;T1 ðk; sÞ ^ CT2 ;T2 ðk; sÞÞðxÞ < CT1 ;T1 ðk; rÞðxÞ ^ CT2 ;T2 ðk; rÞðxÞ: s
W Since si
Put r ¼ r1 _ r2 . Then r < r and CT1 ;T1 ðk; r1 ÞðxÞ ^ CT2 ;T2 ðk; r2 ÞðxÞ 6 CT1 ;T1 ðk; r ÞðxÞ ^ CT2 ;T2 ðk; r ÞðxÞ 6
_
C12 ðk; sÞðxÞ:
s
It is contradiction. Hence _ C12 ðk; sÞ P C12 ðk; rÞ:
s
Theorem 3.7. Let ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ be an intuitionistic fuzzy bitopological space. Let ðX ; C12 Þ be an intuitionistic supra fuzzy closure space. Define the functions Ts ; Ts : I X ! I on X by _ Ts ðkÞ ¼ fT1 ðk1 Þ ^ T2 ðk2 Þjk ¼ k1 _ k2 g;
S.E. Abbas / Chaos, Solitons and Fractals 21 (2004) 1205–1214
where
W
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is taken over all families fk1 ; k2 jk ¼ k1 _ k2 g and ^ ¼ fT1 ðk1 Þ _ T2 ðk2 Þjk ¼ k1 _ k2 g;
Ts ðkÞ where
V
is taken over all families fk1 ; k2 jk ¼ k1 _ k2 g. Then:
(1) ðTs ; Ts Þ ¼ ðTC12 ; TC12 Þ is the coarsest intuitionistic supra fuzzy topology on X which is finer than ðT1 ; T1 Þ and ðT2 ; T2 Þ. (2) C12 ¼ CTs ;Ts ¼ CTC12 ;TC . 12
Proof. (1) Suppose that there exists k 2 I X such that TC12 ðkÞ > Ts ðkÞ and
TC12 ðkÞ < Ts ðkÞ:
By the definition of ðTC12 ; TC12 Þ from Corollary 3.4, there exists r0 2 I0 with C12 ð 1 k; r0 Þ ¼ 1 k such that TC12 ðkÞ P r0 > Ts ðkÞ and
TC12 ðkÞ 6 1 r0 < Ts ðkÞ:
On the other hand, since C12 ð1 k; r0 Þ ¼ 1 k, we have 1 k; r0 ÞÞ k ¼ 1 C12 ð1 k; r0 Þ ¼ 1 ðCT1 ;T1 ð1 k; r0 Þ ^ CT2 ;T2 ð ¼ ð1 CT1 ;T1 ð1 k; r0 ÞÞ _ ð1 CT2 ;T2 ð1 k; r0 ÞÞ ¼ IT1 ;T1 ðk; r0 Þ _ IT2 ;T2 ðk; r0 Þ: Since _ T1 ðIT1 ;T1 ðk; r0 ÞÞ ¼ T1 fl j l 6 k; T1 ðlÞ P r0 ; T1 ðlÞ 6 1 r0 g ^ P fT1 ðlÞ j l 6 k; T1 ðlÞ P r0 ; T1 ðlÞ 6 1 r0 g P r0 ; _ fl j l 6 k; T1 ðlÞ P r0 ; T1 ðlÞ 6 1 r0 g T1 ðIT1 ;T1 ðk; r0 ÞÞ ¼ T1 _ 6 fT1 ðlÞ j l 6 k; T1 ðlÞ P r0 ; T1 ðlÞ 6 1 r0 g 6 1 r0 and, similarly, T2 ðIT2 ;T2 ðk; r0 ÞÞ P r0 and T2 ðIT2 ;T2 ðk; r0 ÞÞ 6 1 r0 we have Ts ðkÞ P r0 and Ts ðkÞ 6 1 r0 . It is a contradiction. Hence TC12 6 Ts and TC12 P Ts . Suppose that there exists q 2 I X such that TC12 ðqÞ < Ts ðqÞ and
TC12 ðqÞ > Ts ðqÞ:
There exists r1 2 I0 such that TC12 ðqÞ < r1 < Ts ðqÞ and
TC12 ðqÞ > 1 r1 > Ts ðqÞ:
By the definition of ðTs ; Ts Þ, there exist q1 ; q2 2 I X with q ¼ q1 _ q2 such that TC12 ðqÞ < r1 6 T1 ðq1 Þ ^ T2 ðq2 Þ 6 Ts ðqÞ; TC12 ðqÞ > 1 r1 P T1 ðq1 Þ _ T2 ðq2 Þ P Ts ðqÞ: On the other hand, since r1 6 Ti ðqi Þ and 1 r1 P Ti ðqi Þ for all i ¼ 1; 2, we have CTi ;Ti ð1 qi ; r1 Þ ¼ 1 qi : It follows that 1 q1 ; r1 Þ ^ CT2 ;T2 ð 1 q2 ; r1 Þ C12 ð1 q; r1 Þ ¼ CT1 ;T1 ð1 q; r1 Þ ^ CT2 ;T2 ð1 q; r1 Þ 6 CT1 ;T1 ð ¼ ð1 q1 Þ ^ ð1 q2 Þ ¼ 1 q: Hence TC12 ðqÞ P r1 and TC12 ðqÞ 6 1 r1 . It is a contradiction. Therefore TC12 P Ts and TC12 6 Ts . Thus ðTC12 ; TC12 Þ ¼ ðTs ; Ts Þ is an intuitionistic supra fuzzy topology on X from Corollary 3.4.
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Finally, we will show that ðTs ; Ts Þ is the coarsest intuitionistic supra fuzzy topology on X which is finer than ðT1 ; T1 Þ and ðT2 ; T2 Þ. For k ¼ k _ 0 and i 6¼ j 2 f1; 2g, we have Ts ðkÞ P Ti ðkÞ ^ Tj ð0Þ ¼ Ti ðkÞ and
Ts ðkÞ 6 Ti ðkÞ _ Tj ð 0Þ ¼ Ti ðkÞ:
If ðU; U Þ is the intuitionistic supra fuzzy topology on X which finer than ðT1 ; T1 Þ and ðT2 ; T2 Þ, for every family fk1 ; k2 g such that k ¼ k1 _ k2 we have T1 ðk1 Þ ^ T2 ðk2 Þ 6 Uðk1 Þ ^ Uðk2 Þ 6 Uðk1 _ k2 Þ; T1 ðk1 Þ _ T2 ðk2 Þ P U ðk1 Þ _ U ðk2 Þ P U ðk1 _ k2 Þ: By the definition of ðTs ; Ts Þ, we have Ts ðkÞ 6 UðkÞ and Ts ðkÞ P U ðkÞ for all k 2 I X . (2) Suppose there exist x 2 X , r 2 I0 , k 2 I X such that C12 ðk; rÞðxÞ < CTs ;Ts ðk; rÞðxÞ: 1 qi Þ P r and Ti ð 1 qi Þ 6 1 r such that By the definition of C12 , for i 2 f1; 2g there exist qi 2 I X with qi P k, Ti ð C12 ðk; rÞðxÞ 6 ðq1 ^ q2 ÞðxÞ < CTs ;Ts ðk; rÞðxÞ: Since k 6 q1 ^ q2 and 1 q1 Þ ^ T2 ð 1 q2 Þ P r; Ts ð1 ðq1 ^ q2 ÞÞ ¼ Ts ðð1 q1 Þ _ ð1 q2 ÞÞ P T1 ð 1 q1 Þ _ T2 ð 1 q2 Þ 6 1 r; Ts ð1 ðq1 ^ q2 ÞÞ ¼ Ts ðð1 q1 Þ _ ð1 q2 ÞÞ 6 T1 ð by the definition of CTs ;Ts , we have CTs ;Ts ðk; rÞ 6 q1 ^ q2 : It is a contradiction. Hence ðCTs ;Ts Þ is finer than C12 . Suppose there exist x1 2 X , r1 2 I0 , x 2 I X such that C12 ðx; r1 Þðx1 Þ > CTs ;Ts ðx; r1 Þðx1 Þ: 1 qÞ P r1 and Ts ð 1 qÞ 6 1 r1 such that By the definition of CTs ;Ts , there exists q 2 I X with q P x, Ts ð C12 ðx; r1 Þðx1 Þ > qðx1 Þ P CTs ;Ts ðx; r1 Þðx1 Þ: On the other hand, since Ts ð1 qÞ P r1 and Ts ð1 qÞ 6 1 r1 , by the definition of ðTs ; Ts Þ, for all s < r1 , there exist l1 ; l2 2 I X with 1 q ¼ l1 _ l2 such that Ts ð1 qÞ P T1 ðl1 Þ ^ T2 ðl2 Þ > s and
Ts ð1 qÞ 6 T1 ðl1 Þ _ T2 ðl2 Þ < 1 s:
It follows that, for i 2 f1; 2g, CTi ;Ti ð1 li ; sÞ ¼ 1 li : Since q 6 1 li , for i 2 f1; 2g, CTi ;Ti ðq; sÞ 6 CTi ;Ti ð1 li ; sÞ: So, for all s < r1 , 1 l1 ; sÞ ^ CT2 ;T2 ð 1 l2 ; sÞ ¼ ð 1 l1 Þ ^ ð 1 l2 Þ ¼ q: C12 ðq; sÞ ¼ CT1 ;T1 ðq; sÞ ^ CT2 ;T2 ðq; sÞ 6 CT1 ;T1 ð Thus, C12 ðq; sÞ ¼ q from (C2) of Definition 2.2. Hence, by Lemma 3.6, _ C12 ðq; sÞ ¼ C12 ðq; r1 Þ: q¼ s
Since x 6 q, we have C12 ðx; r1 Þ 6 C12 ðq; r1 Þ ¼ q: It is a contradiction. Hence C12 is finer that CTs ;Ts .
h
S.E. Abbas / Chaos, Solitons and Fractals 21 (2004) 1205–1214
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Example 3.8. Let X ¼ fa; bg. Define l; q 2 I X as follows: lðaÞ ¼ 0:2;
lðbÞ ¼ 0:4;
qðaÞ ¼ 0:5;
qðbÞ ¼ 0:1:
ðT1 ; T1 Þ; ðT2 ; T2 Þ
Define intuitionistic fuzzy topologies 8 8 > > < 1; if k ¼ 0; 1; < 0; 1 T1 ðkÞ ¼ 12 ; T1 ðkÞ ¼ 2 ; if k ¼ l; > > : : 0; otherwise; 1; 8 8 < 1; if k ¼ 0; 1; < 0; T2 ðkÞ ¼ 23 ; if k ¼ q; T2 ðkÞ ¼ 13 ; : : 0; otherwise; 1;
: I X ! I as follows:
if k ¼ 0; 1; if k ¼ l; otherwise; if k ¼ 0; 1; if k ¼ q; otherwise:
Then the intuitionistic fuzzy bitopological space ðX ; ðT1 ; T1 Þ; ðT2 ; T2 ÞÞ induces ðTs ; Ts Þ as follows: 8 8 1; if k ¼ 0; 1; 0; if k ¼ 0; 1; > > > > > > >1 > 1 > > ; if k ¼ l; ; if k ¼ l; > > <2 <2 Ts ðkÞ ¼ 23 ; if k ¼ q; Ts ðkÞ ¼ 13 ; if k ¼ q; > > > > > > > 12 ; if k ¼ l _ q; > 12 ; if k ¼ l _ q; > > > > : : 0; otherwise; 1; otherwise: Since 1 0 ¼ Ts ðl ^ qÞ < Ts ðlÞ ^ Ts ðqÞ ¼ ; 2 1 1 ¼ Ts ðl ^ qÞ > Ts ðlÞ _ Ts ðqÞ ¼ ; 2 ðTs ; Ts Þ is not an intuitionistic fuzzy topology on X . On the other hand, since 8 if k ¼ 0; 8r 2 I0 ; < 0; CT1 ;T1 ðk; rÞ ¼ 1 l; if 0 6¼ k 6 1 l; 0 < r 6 12 ; : 1; otherwise and 8 < 0; CT2 ;T2 ðk; rÞ ¼ 1 q; : 1;
if k ¼ 0; 8r 2 I0 ; if 0 6¼ k 6 1 q; 0 < r 6 23 ; otherwise;
we have 8 0 > > > > > ð1 lÞ ^ ð1 qÞ; > > < 1 l; C12 ðk; rÞ ¼ > > 1 q; > > > > > : 1; Since
k ¼ 0; 8r 2 I0 ; 1 lÞ ^ ð 1 qÞ; k 6¼ 0; 0 < r 6 12 ; k 6 ð 1 q; k 6 1 l; 0 < r 6 12 ; ki 1 l; k 6 1 q; 0 < r 6 12 ; ki or 0 6¼ k 6 1 q; 12 < r 6 23 ; otherwise:
if if if if
1 1 ¼ C12 ð1 lÞ _ ð1 qÞ; 1 6¼ C12 1 l; 1 _ C12 ¼ ð 1 lÞ _ ð 1 qÞ 1 q; 2 2 2
an intuitionistic supra fuzzy closure space ðX ; C12 Þ is not an intuitionistic fuzzy closure space. We easily show that C12 ¼ CTs ;Ts . h In conclusion, we may stress once more the importance of fuzzy topology as a nontrivial extension of fuzzy sets, intuitionistic fuzzy sets and fuzzy logic [16] and the possible application in quantum physics [10,11].
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