هدفنا من هذا البحث هو تعميم تعريف بديهية فصل تيكنوف وبديهية الفصل هاوسدورف&nbsp; في الفضاءات التبولوجية&nbsp; الحدسية الخاصة ودراسة العلاقات بين هذه الفضاءات مع الفضاءات التبولوجية الحدسية الخاصة&nbsp; من نوع &nbsp;(X, τ2) و (X, τ1) من جهة والفضاءات التبولوجية الحدسية الخاصة &nbsp;من نوع (X, τ 0,2)و &nbsp;(X, τ0,1) &nbsp;من جهة اخرىOur goal in this paper is to give definitions of generalized of Tichonov and Hausdorff separation axioms in intuitionistic fuzzy special topological spaces, and study relationships between these spaces with the intuitionistic special topological spaces (X, τ0,1) and (X, τ 0,2) on one hand and intuitionistic fuzzy special topological spaces (X, τ1) and (X, τ2) on the other han

Mohammed, Fatimah  ‎ M.

Yaseen, Younis J.

Journal of University of Babylon for Pure and Applied Sciences

English

Journals of University of Babylon

Journal of University of Babylon, Pure and Applied Sciences, Vol.(26), No.(4): 2018 75  Generalization of Tichonov and Hausdorff Separation Axiomes in Intuitionistic Fuzzy Special Topological Spaces   Fatimah M. Mohammed    Younis J. Yaseen  Department of Mathematics, College of Education for Pure Sciences,  Tikrit University, Tikrit, IRAQ. nafea_y2011@yahoo.com   Abstract Our goal in this paper is to give definitions of generalized of Tichonov and Hausdorff separation axioms in intuitionistic fuzzy special topological spaces, and study relationships between these spaces with the intuitionistic special topological spaces (X, τ0,1) and (X, τ 0,2) on one hand and intuitionistic fuzzy special topological spaces (X, τ1) and (X, τ2) on the other hand.  Keywords: intuitionistic fuzzy special topological spaces, (X, τ0,1)space,  (X, τ 0,2) space, (X, τ1) space, (X, τ2) space.  ةصلاخلا انفده ثحبلا اذه نم تاءاضفلا يف  فرودسواه لصفلا ةيهيدبو فونكيت لصف ةيهيدب فيرعت ميمعت وه  ةصاخلا ةيسدحلا  ةيجولوبتلا ةساردوةصاخلا ةيسدحلا ةيجولوبتلا تاءاضفلا عم تاءاضفلا هذه نيب تاقلاعلا  م ن عون (X, τ2)  و(X, τ1)  تاءاضفلاو ةهج نمةيجولوبتلا  ةصاخلا ةيسدحلا  عون نم(X, τ 0,2)و  (X, τ0,1)   نم ىرخا ةهج.  ةيحاتفملا تاممكلاءاضف ،ةصاخلا ةيسدحلا ةيجولوبتلا تاءاضفلا : (X, τ0,1) ءاضف (X, τ0,2)  ءاضف(X, τ1)   ,ءاضف (X, τ2)   .      1.Introduction The term of fuzzy sets studied by many authors. Firstly, [Zadeh, 1965] had    introduced it, then [Chang, 1968], introduced the concept of  fuzzy topological space.  Later, as an extension of Zadeh’s study of fuzzy sets, [Atanassov, 1986] the concept of intuitionistic fuzzy sets was introduced. Later,  [Coker,  1997] defined the topology of intuitionistic fuzzy sets.   In this paper we study the new concepts of Tichonov and Hausdorff separation axioms in intuitionistic fuzzy special topological spaces and generalized it with the spaces (X, τ0,1), (X, τ 0,2), (X, τ1) and (X, τ2) on the intuitionistic fuzzy special topological spaces.  2. Preliminaries  Firstly, we shall present the fundamental definitions. Let X be a non-empty fixed set. An intuitionistic fuzzy special sets (IFSS, for short) and A is an object having the form < x, A1, A2 > , where A1 and A2 are sub set of X and satisfying the following condition A1   A2 =  . Whenever, the set A1 is called the set of member of A, while A2 is called the set of non-member of A. Every subset A of a non-empty set Journal of University of Babylon, Pure and Applied Sciences, Vol.(26), No.(4): 2018 75  X is an IFSS  having the form < x, A, Ac > where Ac refers to the complement of the set A.  [Coker, 1987] defined some Boolean algebra operations in an IFSS as followes: Let X be any non-empty set and an IFS’ A, B where < x, A1, A2 >, B = < x, B1, B2 > respectively. Furthermore, let {Ai: i   J} be an arbitrary family of IFSS in X. Then:  (i).  A  B   A1   B1   and   A2  B2 ; (ίί)    A = B      A  B   and   A   B   ; (ίίί)   Ac = < x, A2, A1 >  ; (iv)   FA= < x,A1,Ac1 > , SA = < x, Ac2, A2 > ; (v)   UAί = < x, UA ί(1)  , ∩A ί (2)  > , ∩A ί = < x, ∩A ί(1) , UAί(2)  > ; (vί)   ̃ = < x, Ø, X> ,  ̃  = < x, X, Ø >.   Recall that an intuitionistic fuzzy special topology (IFTS, for short) on a non-empty set X is a family   of IFSS in X containing  ̃,  ̃ and closed under finite intersection and arbitrary union. So, in this case the pair ( X,  ) is called an intuitionistic fuzzy special topological space (IFSTS, for short).   Let ( X,  ) be an IFSTS, we can construction two IFSTS  ’ s as follow as: τ0,1 = { FG: G    τ } and τ0,2 = { SG: G    τ }. Also, the two topological spaces τ1 = {G1: G = < x,G1, G2 >   τ } and τ2 = {G2c: G = < x,G1, G2 >   τ }. The element x in an IFSS is belong to A is define by:  x   A = < x, A1, A2 >  x   A1 and x   A2.  3. Tichonov Separation Axioms on Intuitionistic Fuzzy Special                                             Topological  Spaces. In this section, we introduced Tichonov separation axiom on IFSTS.  Definition 3.1:  Let ( X,  ) be an IFSTS, then  ( X,  ) is said to be satisfy Tichonove separation axioms if (Tich., for short) if for each x, y   X and x   y, there exist A and B   τ such  That x   A, y   B and  x    B, y    A. Theorem 3.2  Let ( X,  ) be an IFSTS, then the following are equivalent: (i) ( X,  )  is a Tich. Space; (ii) ( X, τ0,1 )  is a Tich. Space; (iii) ( X, τ1 )  is a Tich. Space.  Proof : (i)  (ii)           Journal of University of Babylon, Pure and Applied Sciences, Vol.(26), No.(4): 2018 75  Let x, y   X such that x   y, then there exist Ux = < x, A1, A2 >, Vy = < y, B1, B2 > such that x   Ux , y   Vy , x   Vy and y   Ux.  Thus,  x   A1,  x   A2 and y   B1,  y   B2, also, x   B1, y   A1.  Since, FUX = < x,A1,Ac1 > and FVy = < x, B1, Bc1 >, then x   A1 and x   Ac1. So, x   FU X  ,  x    V y   x    B 1  o r  x    B 2 .  N o w ,  i f   x    B 1  t h en  x    B 1c. Therefore, x   FVy. If x   B2, then x   B2 and since B1  B2 = Ø. So, x   B1c. Thus, x   FVy. Similarly, y   FVy and x   FVy. Therefore, ( X, τ0,1 )  is a Tich. Space.  (ii)  (iii)  Suppose that  x, y   X such that x   y, then there exist FUx = < x, A1, A1c > and FVy = < y, B1, B1c >  in τ0,1 where, exist Ux = < x, A1, A2 > and Vy = < y, B1, B2 > in    such that x   FUx , y   FVy , x   FVy and y   FUx. Thus, x   A1 and not in B1 and y in B1 not in A1. Therefore, ( X, τ1 )  is a Tich. Space.  (iii)  (i) Let x, y   X such that x   y, then there exist x   A1 and x   B1 and y   B1, y  A1 where A1 and B1 are in τ1. Put, Ux = < x, A1, A2 >, Vy = < y, B1, B2 >. So, Ux and Vy are in   and satisfy Tich. Axiom, therefore ( X,  )  is a Tich. Space.  Proposition 3.3  Let ( X,  ) be a Tich. IFSTS, then, (i). ( X, τ0,2 )  is a Tich. space; (ii). ( X, τ2)  is a Tich. space. Proof: (i)              Let ( X,  ) be a Tich. IFSTS, and let x and y be any two elements in X such that x  y, then there exists exist Ux = < x, A1, A2 >, Vy = < y, B1, B2 > such that x   Ux , y   Vy , x   Vy and y   Ux. Thus,  x   A1,  x   A2 and y   B1,  y   B2, also, x   B1, y   A1.   Since, SUx = < x, A2c, A2 > and SVy = < x, B2c, B2 >, so x   A2c and y   B2c, thus x   SUx and  y   SVy . Similarly, we can show that, x   SVy and y   SUx. Therefore, ( X, τ0,2 )  is a  Tich. space; in similar way we can prove, (ii).  Remark 3.4  The converse of proposition 3.3 is not true in general. The following example  show the case.  Example 3.5          Let X = {a, b, c} and A = < x,  , {c} >, B= < x, {b}, {a}> and C = < x, {c}, {b} >. Put τ = { ̃,  ̃ , A, B, C, A   C, A   B, B  C, A  B, A   C, B   C}. So, we can see easily that ( X, τ0,2 ) and ( X, τ2 ) is Tich. spaces, but not ( X, τ).  Remark 3.6  (i). If ( X, τ2 ) is Tich. space, then ( X, τ1 ) is not necessary to be Tich. space. Journal of University of Babylon, Pure and Applied Sciences, Vol.(26), No.(4): 2018 06   (ii). If ( X, τ1 ) is Tich. space, then ( X, τ1 ) is not. Example 3.5 shows (i) and the example 3.7 below shows(ii).  Example 3.7          Let X = {a, b, c} and A = < x, {a}, {b} >, B= < x, {b}, {a}> and C = < x, {c}, {a, b} >. Define,  τ = { ̃,  ̃ , A, B, C, A   C, A   B, B  C, A  B, A   C, B   C}. So, we can see very easily that ( X, τ1) a is Tich. space, but ( X, τ2 ) is not.  4. Hausdorff Space on Intuitionistic Fuzzy Special Topological spaces. In this section, we introduced Hausdorff separation axiom on IFSTS.  Definition 4.1  Let ( X, τ ) be an IFSTS, then ( X, τ ) is said to be satisfy Hausdorff separation axioms ( Haus.,  for short) if for each x and y in the set X such that x   y, there exist A, B   τ such that x   A and y   B, A   B =  ̃.  Proposition 4.2  Let ( X,  ) be a Haus. IFSTS, then: (i). ( X, τ0,1 )  is a Haus. space; (ii). ( X, τ1)  is a Haus. space; (iii). ( X, τ0,2 )  is a Haus. space; (ii). ( X, τ2)  is a Haus. space.  Proof: direct.  Remark 4.3  The converse of proposition 4.2 is not true in general. The following example show the cases.  Example 4. Let X = {a, b, c} and A = < x, {a}, {c} >, B= < x, {c}, {b}> and C = < x, {b}, {a } >. Define,  τ = { ̃,  ̃ , A, B, C, A   C, A   B, B  C, A  B, A   C, B   C}. So, we can see very easily that τ is not Haus.. space, but ( X, τ0,1 )  and ( X, τ2 ) are Haus. space.  Example 4.5 Let X = {a, b, c, d} and A = < x, {d}, {b, c} >, B = < x, {c}, {a, b}>, C = < x, {b}, {a, d } > and D = < x, {a}, {c, d} >. Define,  τ = { ̃,  ̃ , A, B, C}   {all pair wise  intersection and unioin}. So, we can see that ( X, τ )  is not Haus.. space, but ( X, τ0,2 ) is  Haus. space.  Journal of University of Babylon, Pure and Applied Sciences, Vol.(26), No.(4): 2018 06  Example 4.6 Let X = {a, b, c } and A = < x, {a}, {b } >, B = < x, {b}, {a }> and C = < x, {c}, {a, c } > . Define,  τ = { ̃,  ̃ , A, B, C A   C, B  C, A  B, A   C}. So, we can see that τ  is not Haus. space, but ( X, τ1 ) is Haus. space.  Proposition 4.7  Let ( X,  ) be an IFSTS such that ( X, τ0,1 ), (( X, τ0,2 )) is Haus. space, then ( X, τ) is also Haus. space.  Proof:  Let ( X, τ0,1 ) be a Haus. space, then for each x, y in X such that x  y, then there exist A = < x, A1, A2 >, B = < y, B1, B2 >   τ  and x   FA , y   FB, FA   FB =  ̃. So, x in A and y in B such that A and B are disjoint i.e ( X, τ) is Haus. space. Similarly, we can prove ( X, τ) is Haus. space when ( X, τ0,2 ) is Haus. space.  Proposition 4.8  Let ( X,  1) be a Haus. space, then for each x, y in X  such that ( X, τ1 ), (( X, τ2 )) is Haus. space, then ( X, τ) is also Haus. space.  Proof: Let ( X, τ0,1 ) be a Haus. space, then for each x, y in X such that   x  y, then there exist A = < x, A1, A2 >, B = < y, B1, B2 >   τ such that x   A1 , y   B1 and A1   B1  =  ̃. So, x in A and y in B whenever, A and B are disjoints i.e ( X, τ ) is a Haus. space.   Now, to prove ( X, τ ) is a Haus. space when  ( X, τ2) is a Haus. space, let x, y in X such that x  y, so there exist A = < x, A1, A2 >, B = < y, B1, B2 >   τ  such that x    A2c , y   B2c and  A2c   B2c =  ̃. Then A2   B2 =  ̃ and we have A1   A2 =  ̃, B1   B2 =   ̃ then, from this we can use the set theory operation to get, [ (A1   A2)   B1 ]   [ (A1   A2)   B2 ] =  ̃. And  A1   B1 =  ̃. So, A   B =  ̃. i.e. ( X, τ) is also Haus. space.  Remark 4.9 Hasdroff  property is independent between ( X, τ1) and ( X, τ2) see Example 3.5 and Example 4.6.  Remark 4.10 Every Huas. space is Tich. space, but the converse is not true in general.   Journal of University of Babylon, Pure and Applied Sciences, Vol.(26), No.(4): 2018 06  Example 4.11 Let X = {a, b } and A = < x, {a}, {b } > and B = < x, {b},   >. Define,  τ = { ̃,  ̃ , A, B, A  B }. Then, we can see that ( X, τ )  is a Tich. space, but not  Haus. space.  References Atanassov K., 1986, " Intuitionistic fuzzy sets", Fuzzy sets and System, 20 (1), 87-96. Chang  C. L., 1968, "Fuzzy topological spaces", J. Math. Anal. Appl., 24, 182-190. Coker,  D., 1997, "An introduction to intuitionistic fuzzy topological spaces",Fuzzy sets and System,  88: 81-89. Zadeh, L. A.,1965," Fuzzy Sets",  Information and Control , 8 (3): 338-353.    

Generalization of Tichonov and Hausdorff ‎Separation Axiomes in Intuitionistic Fuzzy ‎Special Topological Spaces

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Generalization of Tichonov and Hausdorff ‎Separation Axiomes in Intuitionistic Fuzzy ‎Special Topological Spaces

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