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On semi separation axioms in L-fuzzifying bitopological spaces
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Journal of the Egyptian Mathematical Society (2016) 000, 1–8
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Original Article
On semi separation axioms in L-fuzzifying bitopological spaces Ahmed M. Zahran, Mohammed M. Khalaf1,∗ Department of Mathematics, Faculty of Science, Al-Azhar University, Higher Institute of Engineering and Technology King Mariout, Assiut 71524, Egypt Received 24 May 2015; revised 27 March 2016; accepted 29 March 2016 Available online xxx
Keywords L-fuzzifying bitopological spaces; (i, j) ST0 (X, τ1 , τ2 ); (i, j)
STRi
(X, τ1 , τ2 )
Abstract The main purpose of this paper is to introduce a concept of semi separation axioms in L-fuzzifying bitopological spaces (here L is a complete residuated lattice some times used double negation law) and discuss some of their basic properties and the structures. We shown that the category of semi-separation axioms in L- fuzzifying bitopological. Finlay we prove that its some a relation between a semi-separation axioms in L-fuzzifying bitopological spaces. 2010 MSC:
03E72; 18B30; 54A40 Copyright 2016, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The concept of L-fuzzifying topology appeared in Höhle [1] under the name „ L-fuzzy topology„ (cf. Definition 4.6, Proposition 4.11 in Höhle [1] where L is a completely distributive complete lattice. In the case of L = [0, 1] this terminology traces
∗
Corresponding author. Tel.: 00201022024989. E-mail address: [email protected] (M.M. Khalaf). 1 Current Address: Mathematics Department, Faculty of Science in Zulfi, Majmaah University, Zulfi 11932, P.O. Box 1712, Saudi Arabia. Peer review under responsibility of Egyptian Mathematical Society.
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back to (Ying [2–4]) who studied the fuzzifying topology and elementarily developed fuzzy topology from a new direction with semantic method of continuous valued logic. Fuzzifying topology (resp. L-Fuzzifying topology) in the sense of Ying (resp. U. Höhle) was introduced as a fuzzy subset (resp. an L-Fuzzy subset) of the power set of an ordinary set. Höhle [5] introduced and studied a characterization of stratified and transitive L- topology by stratified and transitive Linterior operator K, where L is a complete MV-algebra. A characterization of L-fuzzifying topology by L-fuzzifying neighborhood system, where L is a completely distributive, was given also in Höhle [5]. Finally, Höhle [5] introduced a characterization of stratified and transitive L- topology by Lcontiguity and L -fuzzifying topology, where L is a completely distributive complete MV-algebra. Many separation axioms in fuzzy to pological spaces in the sense of Chang [6] or in the sense of Lowen [7] are introduced and studied by many authors. Ying
S1110-256X(16)30021-9 Copyright 2016, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/j.joems.2016.03.006
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A.M. Zahran, M.M. Khalaf
[2], introduced and characterized the concept of T2 (Hausdorff)separation axiom as a fuzzy subset on the family of all [0, 1]− fuzzifying topological spaces. Shen [8], introduced and studied T0 , T1 , T3 (regular), T4 (normal)-separation axioms. Kheder, et al. [9], introduced R0 , R1 -separation axioms in fuzzifying topology and studied their relations with T1 and T2 -separation axioms. In 2003 Zhang et al. [10], studied the concept of fuzzy (i; j) -closed, (i; j)-open sets in fuzzifying bitopological spaces. In this paper, we introduce and study the concepts of semi − R0 and semi − R1 -separation axioms in fuzzifying bitopology and study their relations with semi − T1 -and semi − T2 -separation axioms . Furthermore, we discuss semi − T0 , semi − T3 (semiregularity) and semi − T4 (semi-normality)-separation axioms in fuzzifying bitopological spaces and give some of their characterizations as well as the relations of these axioms and other semi separation axioms in fuzzifying bitopology introduced. In Section 1, we recall some notions and results in separation axioms in L- fuzzifying bitopology, Also, in Section 2, semi separation axioms in L- fuzzifying bitopology are introduced and studied. Finally In Section 3, relations among semi separation axioms are discussed. Definition 1.1 (Höhle [5]). The double negation law in a complete residuated lattice L is given as follows: ∀a, b ∈ L, (a → ⊥ ) → ⊥ = a. Definition 1.2 (Höhle [5], Rosenthal [11]). A structure (L, ∨, ∧, ∗, →, ⊥, ) is called a strictly two-sided commutative quantale iff (1) (L, ∨, ∧, ⊥, ) is a complete lattice whose greatest and least element are , ⊥ respectively, (2) (L, ∗, ) is a commutative monoid, (3) (a) ∗ is distributive over arbitrary joins, i.e., a ∗ j∈J b j = j∈J (a ∗ b j ) ∀a ∈ L, ∀{bj |j ∈ J}⊆L, . (b) → is a binary operation on L defined by : a → b = λ∗a≤b λ ∀a, b ∈ L. Definition 1.3 (Ying [4]). A structure (L, ∨, ∧, ∗, →, ⊥, ) is called a complete residuated lattice iff (1) (L, ∨, ∧, ⊥, ) is a complete lattice whose greatest and least element are , ⊥ respectively, (2) (L, ∗, ) is a commutative monoid, i.e., (a) ∗ is a commutative and associative binary operation on L, and (b) ∀ a ∈ L, a ∗ = ∗ a = a, (3) (a) ∗ is isotone, (b) → is a binary operation on L which is antitone in the first and isotone in the second variable, (c) → is couple with ∗ as: a∗b ≤ c iff a ≤ b → c ∀ a, b, c ∈ L. Definition 1.4 [4]. Let (X, τ ) be an L-fuzzifying topological space, let x ∈ X. (1) The fuzzifying neighborhood system of x, denoted by Nx ∈ IP(X) , is defined as follows: Nx (A ) =
x∈B⊆A
τ (B).
2- The family of all fuzzifying closed sets is denoted by F ∈ (P(X )) and defined as follows: A ∈ F := X ∼ A ∈ τ 3- The closure cl(A) of A⊆X is defined as follows: cl (A )(x ) = 1 − Nx (X ∼ A ). Definition 1.9 [12]. Let (X, τ 1 ) and (X, τ 2 ) be two an Lfuzzifying topological space. Then a system (X, τ 1 , τ 2 ) consisting of a universe of discourse X with two L-fuzzifying topologies τ 1 and τ 2 on X is called an L-fuzzifying bitopological space. Definition 1.10 [12]. Let (X, τ 1 , τ 2 ) be an L-fuzzifying bitopological space. (1) A set A is said to be a pairwise open if and only if A ∈ τ 1 ∩ τ 2 . i.e., P (A ) = min(τ1 (A ), τ2 (A )). (1) A set B is said to be a pairwise closed if and only if X − B ∈ P . i.e., B ∈ F P = X − B ∈ P . Lema 1.1 [12]. Let (X, τ 1 , τ 2 ) be an L-fuzzifying bitopological space. Then F P (B) = min(F1 (B), F2 (B)). Lema 1.1 [12]. Let (X, τ 1 , τ 2 ) be an L-fuzzifying bitopological space. Then 1- P ⊆ τi , i = 1, 2 2- F P ⊆ Fi i = 1, 2. 2. Semi separation axioms in L-fuzzifying bitopological space Definition 2.1. Let (X, τ 1 , τ 2 ) be an L -fuzzifying bitopological space. (1) The family of all L-fuzzifying (i, j)-semiopen sets, denoted by sτ (i, j) ∈ (P(X)), is defined as follows: sτ(i, j) (A ) =
cl j (inti (A )(x ).
x∈A
(2) The family of all fuzzifying (i, j)-semiclosed sets, denoted by sF(i, j) ∈ (P(X)), is defined as follows: sF(i, j) (A ) = sτ(i, j) (A ) → ⊥. Example 2.1. If L = [0, 1] and Let X = {a, b}, τ 1 , τ 2 be two fuzzifying topologies defined as follows:
τ1 (A ) =
⎧ ⎨1 1 ⎩2
0
if if if
⎧ A ∈ {φ, X } ⎨1 A = {a} , τ2 (A ) = 14 ⎩ A = {b} 0
if if if
A ∈ {φ, X } A = {b} A = {a}
if if if
⎧ A ∈ {φ, X } ⎨1 A = {b} , F2 (A ) = 14 ⎩ A = {a} 0
if if if
A ∈ {φ, X } A = {a} A = {b}
and F1 (A ) =
⎧ ⎨1 1 ⎩2
0
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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On semi separation axioms in L-fuzzifying bitopological spaces
3 STR(i,j j) (X , τ1 , τ2 )
Note that 1 P (A ) = 0 ⎧ ⎨1 Sτ(i, j) (A ) = 14 ⎩ 0
A ∈ {φ, X } , o.w.
if if
=
(1) The (i, j)-semi neighborhood system of x is denoted by SNx(i, j) ∈ (P(X )) and defined as follows:
τ1 (A )
⎧ ⎨1
1 = 5 ⎩1 2
P (A ) = (2) The (i, j)-semi derived set sd(i, j) (A) of A is defined as follows: sd(i, j) (A )(x ) =
(
Sτ(1,2) (A ) =
→ ⊥ ).
B∩ (A−{x} ) = φ
(3) The (i, j)-semi closure of A⊆X, is scl(i, j) (A)(x) and defined as follows:
scl(i, j) (A )(x ) =
denoted
by
denoted
by
(sF(i, j) (B) → ⊥ ).
x∈ / B⊇A
(4) The (i, j)-semi interior of A⊆X, is scl(i, j) (A)(x) and defined as follows:
sint(i, j) (A )(x ) =
⎧ A ∈ {φ, X } ⎨1 A = {a} , τ2 (A ) = 14 ⎩1 A = {b} 8
if if if ⎧ ⎨1 1 ⎩ 51 8
if if if
A ∈ {φ, X } A = {a} , A = {b}
1 ⎩ 51 2
if if if
A ∈ {φ, X } A = {a} A = {b}
⎧ ⎨1
(SNx(i, j) (A )).
ST3(i, j) (X , τ1 , τ2 )
=
=
(SNx(i, j) (B) ∧ SNy(i, j) (C)),
min(Sτ(i, j) (C), Sτ(i, j) (D )),
Definition 2.3. Let be the class of all L -fuzzifying bitopological spaces. The unary L-predicates STR(i,i j) , STR(i,j j) and STR(i, j) ∈ L are defined as follows:
=
x∈ /u
((Fτi (u ) →
A∩B=φ, u⊆B
min
SNx(i, j) (A ),
(i, j) min(Fτ(i, j) (A ), Fτ(i, j) (B)) → SWA,B ),
=
(i, j) (i, j) SKx,y , → SHx,y
SR1(i, j) (X , τ1 , τ2 ) =
(i, j) (i, j) SKx,y . → SMx,y
τ1 (A ) =
where x, y ∈ X, A, B, C, D ∈ P(X) and (X, τ 1 , τ 2 ) is an Lfuzzifying bitopological space.
=
Example 2.3. If L = [0, 1] and Let X = {a, b}, τ 1 , τ 2 be two fuzzifying topologies defined as follows:
A⊆C, B⊆D, C∩D=φ
Fτ(i, j) (D ) → SVx,(i,Dj) ,
x=y
(SNx(i, j) (A ) ∧ Sτ(i, j) (B)),
STR(i,i j) (X , τ1 , τ2 )
x=y
x∈ /C
(i, j) SMx,y ,
A∩B=φ
SR0(i, j) (X , τ1 , τ2 )
A∩B=φ,D⊆B (i, j) SWA,B
=
x∈ /A
(i, j) SHx,y ,
x∈ /D
ST4(i, j) (X , τ1 , τ2 )
C∩B=φ (i, j) SVx,D =
A ∈ {φ, X } A = {a} A = {b}
x=y
ST2(i, j) (X , τ1 , τ2 ) =
⎛ ⎞ (i, j) (i, j) ⎝ ⎠ = SNx (B) ∧ SNy (C) ,
(i, j) SMx,y =
if if if
7 10 Definition 2.4. Let be the class of all L-fuzzifying bitopological spaces. The unary L-predicates STn(i, j) ∈ L , n = 0, 1, 2, 3, 4 and SRn(i, j) ∈ L , n = 0, 1 are defined as follows: (i, j) ST0(i, j) (X , τ1 , τ2 ) = SKx,y ,
x=y
y∈ /B
,
x=y
For simplicity we put the following notations: ⎛ ⎞ (i, j) SKx,y = ⎝ SNx(i, j) (A )⎠ ∨ SNy(i, j) (A ) ,
(i, j) SHx,y
Sτ(i, j) (B)
y∈u
STR(i, j) (X , τ1 , τ2 ) = STR(i,i j) (X , τ1 , τ2 ) ∧ STR(i,j j) (X , τ1 , τ2 ) =
ST1(i, j) (X , τ1 , τ2 )
x∈ / B⊇A
y∈ /A
Example 2.2. If L = [0, 1] and Let X = {a, b}, τ 1 , τ 2 be two fuzzifying topologies defined as follows:
Note that
sτ(i, j) (B).
SNx(i, j) (B)
A∩B=φ, u⊆B
x∈B⊆A
min
SNx(i, j) (A ),
STR(i, j) (X , τ1 , τ2 ) = STR(i,i j) (X , τ1 , τ2 ) ∧ STR(i,j j) (X , τ1 , τ2 ).
Definition 2.2. Let (X, τ 1 , τ 2 ) be an L -fuzzifying bitopological space and x ∈ A
SNx(i, j) (A ) =
((Fτ j (u ) →
x∈ /u
A ∈ {φ, X } A = {a} . A = {b}
if if if
Sτ(i, j) (B)
⎧ ⎨1 1 ⎩51 2
Note that P (A ) =
Sτ(1,2) (A ) =
if if if ⎧ ⎨1
⎧ A ∈ {φ, X } ⎨1 A = {a} , τ2 (A ) = 14 ⎩1 A = {b} 8
1 ⎩ 51 8
if if if
A ∈ {φ, X } A = {a} , A = {b}
1 ⎩ 51 2
if if if
A ∈ {φ, X } A = {a} A = {b}
⎧ ⎨1
if if if
A ∈ {φ, X } A = {a} A = {b}
y∈u
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ST3(1 ,2) (X , τ1 , τ2 )
=
Fτ(1 ,2) (D ) →
SVx,(1 D,2)
x∈ /D
STR(11 ,2) (X , τ1 , τ2 ) = min
A∩B=φ, u⊆B
STR(12 ,2) (X , τ1 , τ2 )
=
SNx(1,2) (A ),
min
Sτ(1,2) (B)
=
y∈u
7 10
x∈ /u
SNx(1,2) (A ),
Sτ(1,2) (B)
=
y∈u
A∩B=φ, u⊆B
7 10
Remark 2.1. Let (X, τ 1 , τ 2 ) be an L-fuzzifying bitopological space. From remark 3.1 in [15], we have STi (1 ,2) (X , τ1 , τ2 ) = STi (2 ,1) (X , τ1 , τ2 ), i = 0, 1, 2, 3, 4 Lemma 2.1. Let (X, τ ) ∈ . Then for any x, y ∈ X, (i, j) SMx,y (i, j) SHx,y (i, j) SMx,y
(1) (2) (3)
(i, j) min(SNx(i, j) (A ), SNy(i, j) (B)) = SHx,y
y∈ / A, x∈ /B
3- It is obtained from (1) and (2).
Proof. The proof is similar to the proof of Theorem 2.2.
SR1(i, j) (X , τ1 , τ2 ) ≤
ST1(i, j) (X , τ1 , τ2 ) =
SF(i, j) ({x} ).
Proof. Let x1 , x2 ∈ X s.t. x1 = x2 . Then SF(i, j) ({x} ) = Sτ(i, j) (X − {x} ) x∈X
=
⎛
≤
⎝
x∈X
=
⎞ SNy(i, j) (X − {x} )⎠
y∈X −{x}
y∈X −{x2 }
(i, j) (i, j) (SKx,y → SHx,y ) = SR0(i, j) (X , τ1 , τ2 ).
SNy(i, j) (X − {x2 } ) ≤ SNx(i,1 j) (X − {x2 } )
SNx(i,1 j) (A )
x2 ∈ /A
Similary, we have
Theorem 2.2. For any (X, τ 1 , τ 2 ) ∈ , the following statements are satisfied: ST1(i, j) (X , τ1 , τ2 ) ≤ SR0(i, j) (X , τ1 , τ2 ), ST1(i, j) (X , τ1 , τ2 ) ≤ ST0(i, j) (X , τ1 , τ2 ), ST1(i, j) (X , τ1 , τ2 ) ≤ SR0(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ), If ST0(i, j) (X , τ1 , τ2 ) = , then ST1(i, j) (X , τ1 , τ2 ) = SR0(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ).
SF(i, j) ({x} ) ≤
x∈X
x1 ∈ /B
SNx(i,2 j) (B)
Then SF(i, j) ({x} ) ≤ x∈X
x1 =x2 x1 ∈ / B,x2 ∈ /A
(i, j) SHx,y
ST1(i, j) (X , τ1 ,
(1) Since → so that τ2 ) ≤ (2) From Lemma 2.1 (2) one can deduce that ST1(i, j) (X , τ1 , τ2 ) ≤ ST0(i, j) (X , τ1 , τ2 ). (3) The proof follows from (1) and (2) .
(SNx(i,1 j) (A ) ∧ SNx(i,2 j) (B))
= ST1 (X , τ ). For the other hand,
Proof. (i, j) (i, j) SHx,y ≤ SKx,y SR0(i, j) (X , τ1 , τ2 ).
ST2(i, j) (X , τ1 , τ2 ) ≤
x∈X
x∈X
x=y
(1) (2) (3) (4)
Theorem 2.5. If L satisfies the completely distributive law, then for any (X, τ 1 , τ 2 ) ∈ ,
Theorem 2.1. For any (X, τ1 , τ2 ) ∈ , SR1(i, j) (X , τ1 , τ2 ) ≤ SR0(i, j) (X , τ1 , τ2 ). (i, j) (i, j) Proof. SR1(i, j) (X , τ1 , τ2 ) = x=y (SKx,y → SMx,y ). Since → is isotone in the second, then
ST2(i, j) (X , τ1 , τ2 ) ≤ SR1(i, j) (X , τ1 , τ2 ), ST2(i, j) (X , τ1 , τ2 ) ≤ ST1(i, j) (X , τ1 , τ2 ), ST2(i, j) (X , τ1 , τ2 ) ≤ SR1(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) If ST0(i, j) (X , τ1 , τ2 ), then ST2(i, j) (X , τ1 , τ2 ) = SR1(i, j) (X , τ1 , τ2 ).
Proof. The conclusion is obtained from Theorem 2.2(2) and Theorem 2.3(2).
(i, j) (B), SNy(i, j) (C)) ≤ C∩B=φ min (S Nx (i, j) (i, j) (B), SNy(i, j) (C)) = SHx,y , y∈ / B, x∈ /C min (SNx (i, j) (i, j) (i, j) 2- SKx,y = max( y∈/ A SNx (A )), ( x∈/ A SNy (A )) ≥ (i, j) (A ) y∈ / A SNx
≥
(1) (2) (3) (4)
Theorem 2.4. For any (X, τ1 , τ2 ) ∈ , ST0(i, j) (X , τ1 , τ2 ).
(i, j) ≤ SHx,y , (i, j) ≤ SKx,y , (i, j) ≤ SKx,y .
Proof. (i, j) 1- SMx,y =
(4) Since → α = α ∀α ∈ L (Indeed → α = λ∗ ≤α λ = (i, j) λ = α. ) then ST1(i, j) (X , τ1 , τ2 ) = x=y SHx,y = λ≤α (i, j) (i, j) (i, j) ( x=y (SKx,y → SHx,y )) ∧ = SR0 (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) because ∀x, y ∈ X s.t. x = y, if ST0(i, j) (X , τ1 , τ2 ) = , then (i, j) SKx,y = . Theorem 2.3. For any (X, τ 1 , τ 2 ) ∈ , the following statements are satisfied:
((Fτ2 (u )
→
((Fτ1 (u )
x∈ /u
→
7 = 10
ST1(i, j) (X , τ1 , τ2 )
=
x1 ,x2 ∈X,x1 =x2
⎛ ⎝
x1 ∈ /B
⎛⎛ ⎝⎝
⎞ SNx(i,1 j) (A )⎠∧
x2 ∈ /A
⎞⎞
SNx(i,2 j) (B)⎠⎠
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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=
x1 =x2
x1 =x2
x2 ∈X x1 ∈X −{x2 }
=
=
x∈X
=
=
SF(i, j) ({x} )
A∈P(X )
Sτ(i, j) (X − {x} ) =
x∈X
→
x∈X x∈X
y∈u
min
STR(i,j j) (X , τ1 , τ2 ) =
→
(Scl(i, j) (A )(y ) → ⊥ ) ,
y∈u
((Fτ j (u )
SNx(i, j) (A ),
(Sτ(i, j) (A )(y ) → ⊥ ) ,
STR(i, j) (X , τ1 , τ2 ) = 1 STR(i,i j) (X , τ1 , τ2 ) ∧1 STR(i,j j) (X , τ1 , τ2 ) 2 STR(i,i j) (X , τ1 , τ2 ) = ((τi (u ) 1
→
B∈P(X ) 2
STR(i,j j) (X , τ1 , τ2 ) =
→
((τ j (u )
x∈u
B∈P(X ) 2
STR(i, j) (X , τ1 , τ2 )
=2 STR(i,i j) (X , τ1 , τ2 ) ∧2 STR(i,j j) (X , τ1 , τ2 ).
Theorem 2.6. Let (X, τ 1 , τ 2 ) ∈ , then, 1- STR(i,i j) (X , τ1 , τ2 ) = 2- STR(i,j j) (X , τ1 , τ2 ) = 3- STR(i, j) (X , τ1 , τ2 ) =
n n n
STR(i,i j) (X , τ1 , τ2 ), n = 1, 2 STR(i,j j) (X , τ1 , τ2 ), n = 1, 2
STR(i, j) (X , τ1 , τ2 ), n = 1, 2
y
τ
∃ K = ∪{ f (y )| f (y ) ∈ ℘y(i, j) , y ∈
℘y
Theorem 2.7. If L satisfies the completely distributive law, for any (X, τ 1 , τ 2 ) ∈ . Then 2 STR(i,i j) (X , τ1 , τ2 ) = STR(i,i j) (X , τ1 , τ2 ) Proof. STR(i,i j) (X , τ1 , τ2 ) = 1 STR(i,i j) (X , τ1 , τ2 ) =
min(SNx(i, j) (A ),
=
⎛
min ⎝SNx(i, j) (A ),
((Fτi (u ) →
x∈ /u
(Scl(i, j) (A )(y ) → ⊥ ))
((Fτi (X − B) →
x∈ / X −B
=
y∈u
A∈P(X )
Proof. (1) 1 STR(i,i j) (X , τ1 , τ2 ) = x∈/ u ((Fτi (u ) → A∈P(X ) min (SNx(i, j) (A ), y∈u (Scl(i, j) (A )(y ) → ⊥ )) = x∈/ u ((SFτi (u ) → A∈P(X ) min(SNx(i, j) (A ), ( y∈u SNy(i, j) (X − A )))) and STR(i,i j) (X , τ1 , τ2 ) = x∈/ u ((Fτi (u ) → A∩B=φ,u⊆B min(SNx(i, j) (A ), ( y∈u Sτ(i, j) (B))). So, the result holds if we prove that
y∈D
y∈D
f∈
∧[[Scl(i, j) (B), u[[)),
min(SNx(i, j) (B), ∧[[Scl(i, j) (B), u[[ ),
f ∈
τ ℘y(i, j) ,
y∈D A∩B=φ, D⊆B Sτ(i, j) (B). So shoud prove that min(SNx(i, j) (A ), (Scl(i, j) (A )(y ) → ⊥ ) = A∈P(X ) y∈u (i, j) min (SN (A ) , ( x A∩B=φ, u⊆B y∈u Sτ(i, j) (B)) (2) It is similar to (1) (3) It is clear.
x∈u
min(SNx(i, j) (B),
D} s.t. D⊆ K ⊆ X − A τ(i, j) and , y ∈ D }) = y∈D f (y ) ≤ τ (∪{ f (y ) f (y ) ∈ ℘y Sτ (K ) so that (i, j) (y ) ≤ Sτ(i, j) (K ) ≤ A∩ B=φ, D⊆B Sτ(i, j) y∈D f (B) so that Sτ (B) = Sτ (i, j) (i, j) y∈D y∈B⊆X −A y∈D f (y ) ≤
y∈u
A∈P(X )
Sτ(i, j) (B)⎠⎠.
y∈B⊆X −A
we
for each
x∈ /u
min
⎞⎞
prove that y∈u y∈B⊆X −A Sτ(i, j) (B) = A∩B=φ,u⊆B Sτ(i, j) (B). Let y ∈ u. Assume S = {H ∈ P(X )|H ∩ A = φ and u ⊆ H } and ℘y = {M ∈ P(X )|y ∈ M ⊆ X − A}. Then S ⊆℘y so that B∈℘y Sτ(i, j) (B) ≥ B∈S Sτ(i, j) (B) so that y∈D y∈B⊆X −A Sτ(i, j) (B) ≥ A∩B=φ, D⊆B Sτ(i, j) (B). τ(i, j) τ(i, j) Let ℘ ℘y }. Then y∈D ℘y = y = {τ (M) M ∈ τ(i, j) f (y ) . Then y∈D B∈℘y Sτ(i, j) (B) = y ∈ D f∈ ℘ Now
((Fτi (u )
x∈ /u
SNx(i, j) (A ),
A∈P(X ) 1
min(SNx(i, j) (A ),
⎛ ⎛ ⎝ ⎝
SF(i, j) ({x} ).
Definition 2.3. Let be the class of all L -fuzzifying bitopological spaces. We define STR(i,i j) (X , τ1 , τ2 ) =
y∈u
A∩u=φ, A∈P(X )
1
Sτ(i, j) (B) .(∗)
y∈u
=
τ(i, j) (X − {x2 } )
In the left / X −A side of (∗) if A ∩ u = φ, ∃ y ∈ u s.t. y ∈ so that y∈u SNy(i, j) (X − A ) = ⊥. Second,
min(SNx(i, j) (A ), SNy(i, j) (X − A )
x∈X x∈X
x2 ∈X x∈X
SNy(i, j) (X − A )
min(SNx(i, j) (A ),
A∩B=φ, u⊆B
τ(i, j) (X − {x2 } )
τ(i, j) (X − {x} ) =
y∈u
=
SNx(i,1 j) (X − {x2 } )
x2 ∈X x∈X
min(SNx(i, j) (A ),
A∈P(X )
SNx(i,1 j) (X − {x2 } )
=
((SNx(i,1 j) (X −{x2 } )∧(SNx(i,2 j) (X −{x1 } ))
≤
5
⎛ ⎝
A∈P(X )
⎞⎞
(Scl(i, j) (A )(y ) → ⊥ )⎠⎠
y∈X −B
(Sτ(i, j) (B) →
x∈B
min(SNx(i, j) (A ),
A∈P(X )
⎛ ⎝
⎞ (Scl(i, j) (A )(y ) → ⊥ )⎠
y∈X −B
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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6 =
A.M. Zahran, M.M. Khalaf
((τi (u ) →
x∈u
min(SNx(i, j) (B),
B∈P(X )
∧[[Scl(i, j) (A ), B[[)) =2 STR(i,i j) (X , τ1 , τ2 ) that Note y∈X (Scl(i, j) (A )(y ) → B(y )) = ( y∈ B (Scl (i, j) (A )(y ) → ) ∧ ( y∈X −B (Scl(i, j) (A )(y ) → ⊥ )) = ∧ y∈X −B (Scl(i, j) (A )(y ) → ⊥ ) = y∈X −B (Scl(i, j) (A )(y ) → ⊥) Theorem 2.8. Let (X, τ 1 , τ 2 ) ∈ . If L satisfies the double negation law, then, ST0(i, j) (X , τ1 , τ2 ) = ((cl(i, j) ({y} )(x ) → ⊥ ) x=y
ST0(i, j) (X , τ1 , τ2 ) =
∨
⎛⎛ ⎝⎝
x=y
=
SNx(i, j) (A )⎠
y∈X
((SNx(i, j) (X
x=y
Theorem 2.9. Let (X, τ 1 , τ 2 ) ∈ , and let A be a finite subset of X. If L satisfies the completely distributive, then,
SNy(i, j) ((X − A ) ∪ {y} ).
y∈X
=
SNy(i, j) ( X −
y∈X −A
=
y∈X −A
{x} =
x∈A
y∈X −A
SNy(i, j)
y∈X −A
SNy(i, j) (X − {x} ) ≥
and SNy(i, j) ((X − A ) ∪ {y} ) = SNy(i, j) (X − (A − {y} ))
=
=
⎛
⎛
SNy(i, j) ⎝X − ⎝
y∈A
=
y∈A
⎛
y∈A
⎝
x∈A−{y}
⎞
SNy(i, j) (X − {x} )⎠ ≥
→ =
=
B⊆A
→ SNy(i, j) (X − {x} ).
x=y
Then SNy(i, j) ((X − A ) ∪ {y} ) ≥ x=y SNy(i, j) (X − y∈X {x} ) = x∈X ( y∈X −{x} SNy(i, j) (X − {x} )) = x∈X Sτ(i, j) (X − {x} ) = x∈X SF τ(i, j) ({x} ) = ST1(i, j) (X , τ1 , τ2 ).
⎞ (min(Sτ(i, j) (u ), Sτ(i, j) (X − v )))⎠
(u∩X −v )=φu⊆X −v, A⊆X −v, B⊆u
→
− {x} )⎠
min(Fτ(i, j) (X − A ), Fτ(i, j) (B))
(X −A∩B)=φ
B⊆A
⎞
=
{x}⎠⎠
x∈A−{y}
SNy(i, j) ⎝ (X y∈A x∈A−{y} ⎛
⎞⎞
(min(Sτ(i, j) (u ), Sτ(i, j) (v )))).
u∩v=φ, A⊆v, B⊆u
x∈A
SNy(i, j) (X − {x} ) ,
A∩B=φ
→
(X − {x} )
x=y
x∈A
y∈A
Theorem 2.11. Let (X, τ ) ∈ , and let (1) 1 STN(i, j) (X , τ1 , τ2 ) = A⊆B ((Fi (A ) ∧ τ j (B)) → A⊆u⊆v⊆B (min (Sτ(i, j) (u ), SF(i, j) (v ))) (2) 2 STN(i, j) (X , τ1 , τ2 ) = A∩B=φ ((Fi (A ) ∧ τ j (B)) → A⊆u (min (Sτ(i, j) (u ), Scl (i, j) (u ))) (3) 3 STN(i, j) (X , τ1 , τ2 ) = A⊆B ((Fi (A ) ∧ τ j (B)) → A⊆B (min (Sτ(i, j) (u ), Scl (i, j) (v ))) (4) STN(i, j) (X , τ1 , τ2 ) = A∩B=φ (min(Fτi (A ), Fτ j (B)) → U ∩V =φ, A⊆V, B⊆U (min (Sτ(i, j) (u ), Sτ(i, j) (v )))). Then STN(i, j) (X , τ1 , τ2 ) = n STN(i, j) (X , τ1 , τ2 ), n = 1, 2, 3 Proof. (1) From Lema 1.1 STN(i, j) (X , τ1 , τ2 ) = (min(Fτi (A ), Fτ j (B))
Proof. Now, SNy(i, j) ((X − A ) ∪ {y} ) = SNy(i, j) (X − A ) y∈X −A
y∈X
− {y} )
∨(SNy(i, j) (X − {x} )) = ((cl(i, j) ({y} )(x ) → ⊥ ) ∨ (cl(i, j) ({x} )(y ) → ⊥ )).
ST1(i, j) (X , τ1 , τ2 ) ≤
ST1(i, j) (X , τ1 , τ2 ) ≤ [[d(i, j) (A ), 1φ ]].
d(i, j) (y ))) = (d(i, j) (A )(y ) → 1φ (y )) = SNy(i, j) ((X − A ) ∪ {y}.
x=y
x∈ /A
Theorem 2.10. Let (X, τ ) ∈ and let A be a finite subset of X. If L satisfies the completely distributive and the double negation law, then
y∈X
⎞
y∈ /A
SNy(i, j) (A )
d(i, j) (A )(x ) = SNx(i, j) ((X − A ) ∪ {x} ) → ⊥ ∀x ∈ X .
Proof. It follows from Theorem 2.7 and since [[d(i, j) (A ), 1φ ]] = (d(i, j) (A )(y ) → 1φ (y )) ∧ (1φ (y ) →
∨(cl(i, j) ({x} )(y ) → ⊥ )). Proof.
Definition 2.4. Let (X, τ 1 , τ 2 ) ∈ and let A ⊆ X. The Lfuzzifying derived set of A is denoted by dτ (A) ∈ LX and defined as follows:
(min(Fτ(i, j) (A ), Fτ(i, j) (B))
(min(Sτ(i, j) (u ), SF(i, j) (v )))
u⊆v, v⊆A, B⊆u
(min(Fτ(i, j) (A ), Fτ(i, j) (B))
(min(Sτ(i, j) (u ), SF(i, j) (v ))) =1 STN(i, j) (X , τ1 , τ2 )
B⊆u⊆v⊆A
(2) and (3) are similar
Theorem 2.12. Let (X, τ 1 , τ 2 ) ∈ , Then STN(i, j) (X , τ1 , τ2 ) = ST4(i, j) (X , τ1 , τ2 ). Proof. From Lema 1.1 (2) . It is obtained.
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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On semi separation axioms in L-fuzzifying bitopological spaces The following example shows that generally the reverse of the Theorem 2.12 need not be true. Example 2.4. [12] If L = [0, 1] and Let X = {a, b}, τ 1 , τ 2 be two fuzzifying topologies defined as follows: ⎧ ⎨ 1 if A ∈ {φ, X } τ1 (A ) = 12 if A = {a} , ⎩ 0 if A = {b} ⎧ 1 if A ∈ {φ, X } ⎨ τ2 (A ) = 14 if A = {b} ⎩ 0 if A = {a} and
⎧ ⎨1
F1 (A ) =
1 ⎩2
if if if
A ∈ {φ, X } A = {b} , A = {a}
1 ⎩4
if if if
A ∈ {φ, X } A = {a} A = {b}
0 ⎧ ⎨1
F2 (A ) =
0
Note that 1 P (A ) = 0 ⎧ ⎨1 Sτ(i, j) (A ) = 14 ⎩ 0
A ∈ {φ, X } , o.w.
if if if
A ∈ {φ, X } A = {a} . A = {b} 3 4
= STN(i, j) (X , τ1 , τ2 )
Proof. Now, (Sτ(i, j) (X − {y} ))
→
⎛ ⎞⎞ (i, j) (i, j) ⎝ (SNx (A ) ∧ SNy (B))⎠⎠ y∈B
A∩B=φ, y∈B
≤
x=y
⎛ ⎝
=
y∈B
ST1(i, j) (X , τ1 , τ2 ))
x=y
y∈B
= ST1(i, j) (X , τ1 , τ2 ) ⎛ ⎛ ⎞⎞ (i, j) (i, j) ⎝ ⎝ (SNx (A ) ∧ SNy (B))⎠⎠ → x=y
A∩B=φ,
y∈B
(SNx(i, j) (A ) ∧ SNy(i, j) (B))⎠
A∩B=φ
ST2(i, j) (X , τ1 , τ2 ).
→
Since
Theorem 3.3. For any (X, τ ) ∈ , 1- ST2(i, j) (X , τ1 , τ2 ) ≤ SR1(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) 2- If ST2(i, j) (X , τ1 , τ2 ) = , then ST2(i, j) (X , τ1 , τ2 ) = (i, j) (i, j) SR1 (X , τ1 , τ2 ) ∧ ST0 (X , τ1 , τ2 ) 3- SR1(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ) 4- If ST0(i, j) (X , τ1 , τ2 ) = , then ST2(i, j) (X , τ1 , τ2 ) = (i, j) (i, j) SR1 (X , τ1 , τ2 ) ∧ ST0 (X , τ1 , τ2 )
Proof. It obvious
Theorem 3.4. If L satisfies the completely distributive law, then for any (X , τ ) ∈ , ST4(i, j) (X , τ1 , τ2 ) ∗ ST1(i, j) (X , τ1 , τ2 ) ≤ ST3(i, j) (X , τ1 , τ2 ). Proof. ST4(i, j) (X , τ1 , τ2 )
=
⎛ ⎝ min(SFτ
(i, j)
(E ), SFτ(i, j) (D ))
E∩D=φ
→
⎞
min(Sτ(i, j) (A ) ∧ Sτ(i, j) (B))⎠
E⊆A, D⊆B, A∩B=φ
≤
⎛
⎝ min(SFτ
(i, j)
({x} ), SFτ(i, j) (D ))
x∈ /D
⎛ ⎞⎞ (i, j) (i, j) ⎝ (SNx (A ) ∧ SNy (B))⎠⎠
A∩B=φ, y∈B
ST3(i, j) (X , τ1 , τ2 ) = x∈/ D (Sτ(i, j) (X − D ) → (i, j) (A ) ∧ τ(i, j) (B))) A∩B=φ,D⊆B (SNx = x∈/ D (Sτ(i, j) (X − D ) → A∩B=φ,D⊆B (SNx(i, j) (A ) ∧ ( y∈B (SNy(i, j) (B)))) ≤ x∈/ {y} (Sτ(i, j) (X − {y} ) → A∩B=φ, y∈B ( y∈B (SNx(i, j) (A ) ∧ SNy(i, j) (B)))) = x=y (Sτ(i, j) (X − {y} ) → A∩B=φ, y∈B ( y∈B (SNx(i, j) (A ) ∧ SNy(i, j) (B)))), then from above, ST3(i, j) (X , τ1 , τ2 ) ≤ (i, j) (i, j) ST1 (X , τ1 , τ2 ) → ST2 (X , τ1 , τ2 ), so that ST3(i, j) (X , τ1 , τ2 ) ∗ ST1(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ).
⎛ ⎞⎞ ⎝ (SNx(i, j) (A ) ∧ SNy(i, j) (B))⎠⎠
A∩B=φ, y∈B
→
=
Sτ(i, j) (X − {y} )
y∈X
→
x=y
ST1(i, j) (X , τ1 , τ2 )
Proof. It obvious
Theorem 3.1. If L satisfies the completely distributive law, then for any (X , τ ) ∈ , ST3(i, j) (X , τ1 , τ2 ) ∗ ST1(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ).
⎝
1- ST1(i, j) (X , τ1 , τ2 ) ≤ SR0(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) 2- If ST1(i, j) (X , τ1 , τ2 ) = , then ST1(i, j) (X , τ1 , τ2 ) ≤ (i, j) (i, j) SR0 (X , τ1 , τ2 ) ∧ ST0 (X , τ1 , τ2 ) 3- SR1(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ) 4- If ST0(i, j) (X , τ1 , τ2 ) = , then ST2(i, j) (X , τ1 , τ2 ) = (i, j) (i, j) SR0 (X , τ1 , τ2 ) ∧ ST0 (X , τ1 , τ2 )
3. Relations among semi-separation axioms in fuzzifying bitopological spaces
x=y
≤ ST1(i, j) (X , τ1 , τ2 ) →
⎞
Theorem 3.2. For any (X, τ ) ∈ ,
if if
Hence ST4(i, j) (X , τ1 , τ2 ) = 1
7 ⎛
⎞
→
min(Sτ(i, j) (A ) ∧ Sτ(i, j) (B))⎠
x∈A, D⊆B, A∩B=φ
=
min(SFτ(i, j) ({x} ), SFτ(i, j) (D ))
x∈ /D
→
D⊆B, A∩B=φ
⎞ Sτ(i, j) (A ) ∧ Sτ(i, j) (B) ⎠
x∈A
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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8 ≤
min(SFτ(i, j) ({x} ), SFτ(i, j) (D ))
x∈ /D
→
D⊆B, A∩B=φ
=
(i, j)
Theorem 3.5. If L satisfies the completely distributive law, then for any (X , τ1 , τ2 ) ∈ , ST4(i, j) (X , τ1 , τ2 ) = SRi(i, j) (X , τ1 , τ2 ).
({x} ), SFτ(i, j) (D ))
→
⎞
x∈ /D
→ =
x∈ /D
→ ≤
x∈ /D
→
SFτ(i, j) ({x} ) ∧ SFτ(i, j) (D )
x∈X
⎞
(SNx(i, j) (A ) ∧ Sτ(i, j) (B))⎠
Acknowledgments The authors are highly grateful to the referees for their valuable comments and suggestions.
(((ST1i, j (X , τ1 , τ2 ) ∧ SFτ(i, j) (D ))
(SNx(i, j) (A ) ∧ Sτ(i, j) (B)))
References
D⊆B, A∩B=φ
(((ST1(i, j) (X , τ1 , τ2 ) ∗ SFτ(i, j) (D ))
(SNx(i, j) (A ) ∧ Sτ(i, j) (B)))
→
((SFτ(i, j) (D )
x∈ /D
(SNx(i, j) (A ) ∧ Sτ(i, j) (B)))
D⊆B, A∩B=φ
=
1- SRi(i, j) (X , τ1 , τ2 ) ∧ ST1(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ) j) 2- SR(i, (X , τ1 , τ2 ) ∧ ST1(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ) j
D⊆B, A∩B=φ
ST1(i, j) (X , τ1 , τ2 ) →
Theorem 3.7. For any (X , τ1 , τ2 ) ∈ ,
D⊆B, A∩B=φ
≤
Theorem 3.6. If L satisfies the completely distributive law, then j) for any (X , τ1 , τ2 ) ∈ , ST4(i, j) (X , τ1 , τ2 ) = SR(i, (X , τ1 , τ2 ). j
(SNxi, j (A ) ∧ Sτ(i, j) (B))⎠
D⊆B, A∩B=φ
K f ∗ α = ( j∈J f ( j)) ∗ α ≤ j∈J ( f ( j) ∗α) ≤ j∈J (B j → M j ). Then ≤ j∈J ((α ∗ B j ) → M j ) ≤ f∈ Aj Kf j∈J λ∗α≤ j∈J (B j →M j ) λ = α → j∈J (B j → M j ). ). We have the following results which their proof are obvious.
x∈K⊆A
⎛
⎝ min(SFτ
⎞ Sτ(i, j) (K ) ∧ Sτ(i, j) (B) ⎠
x∈ /D
≤
[m;May 11, 2016;18:16]
A.M. Zahran, M.M. Khalaf
ST1(i, j) (X , τ1 , τ2 ) ST4(i, j) (X , τ1 , τ2 )
→ ST3(i, j) (X , τ1 , τ2 )
so that
∗ ST1(i, j) (X , τ1 , τ2 ) ≤ ST3(i, j) (X , τ1 , τ2 ).
(Indeed, put ST1(i, j) (X , τ1 , τ2 ) = α, j = (x, D ), J= {(x, D )|x ∈ X , D ∈ P(X ), x ∈ / D }, B(x,D ) = (Fτi (D ), M(x,D ) = D⊆B, A∩B=φ (SNx(i, j) (A ) ∧ Sτ(i, j) (B)) and λ ∗ α ≤ B A = { λ → M } . Then ((α ∗ B ) → M j j j j) = j∈J j λ j∈J λ∗(α∗Bj )≤M j = j∈J( λ∗α)∗B j ≤M j λ = j∈J λ∗α≤(B j →M j ) λ = j∈J A j = f ∈ A j j∈J f ( j). Now, ∀ f ∈ A j , there exists K f = j∈J j∈J j∈J f ( j) s.t.
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Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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[m;May 11, 2016;18:16]
Journal of the Egyptian Mathematical Society (2016) 000, 1–8
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
Original Article
On semi separation axioms in L-fuzzifying bitopological spaces Ahmed M. Zahran, Mohammed M. Khalaf1,∗ Department of Mathematics, Faculty of Science, Al-Azhar University, Higher Institute of Engineering and Technology King Mariout, Assiut 71524, Egypt Received 24 May 2015; revised 27 March 2016; accepted 29 March 2016 Available online xxx
Keywords L-fuzzifying bitopological spaces; (i, j) ST0 (X, τ1 , τ2 ); (i, j)
STRi
(X, τ1 , τ2 )
Abstract The main purpose of this paper is to introduce a concept of semi separation axioms in L-fuzzifying bitopological spaces (here L is a complete residuated lattice some times used double negation law) and discuss some of their basic properties and the structures. We shown that the category of semi-separation axioms in L- fuzzifying bitopological. Finlay we prove that its some a relation between a semi-separation axioms in L-fuzzifying bitopological spaces. 2010 MSC:
03E72; 18B30; 54A40 Copyright 2016, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The concept of L-fuzzifying topology appeared in Höhle [1] under the name „ L-fuzzy topology„ (cf. Definition 4.6, Proposition 4.11 in Höhle [1] where L is a completely distributive complete lattice. In the case of L = [0, 1] this terminology traces
∗
Corresponding author. Tel.: 00201022024989. E-mail address: [email protected] (M.M. Khalaf). 1 Current Address: Mathematics Department, Faculty of Science in Zulfi, Majmaah University, Zulfi 11932, P.O. Box 1712, Saudi Arabia. Peer review under responsibility of Egyptian Mathematical Society.
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back to (Ying [2–4]) who studied the fuzzifying topology and elementarily developed fuzzy topology from a new direction with semantic method of continuous valued logic. Fuzzifying topology (resp. L-Fuzzifying topology) in the sense of Ying (resp. U. Höhle) was introduced as a fuzzy subset (resp. an L-Fuzzy subset) of the power set of an ordinary set. Höhle [5] introduced and studied a characterization of stratified and transitive L- topology by stratified and transitive Linterior operator K, where L is a complete MV-algebra. A characterization of L-fuzzifying topology by L-fuzzifying neighborhood system, where L is a completely distributive, was given also in Höhle [5]. Finally, Höhle [5] introduced a characterization of stratified and transitive L- topology by Lcontiguity and L -fuzzifying topology, where L is a completely distributive complete MV-algebra. Many separation axioms in fuzzy to pological spaces in the sense of Chang [6] or in the sense of Lowen [7] are introduced and studied by many authors. Ying
S1110-256X(16)30021-9 Copyright 2016, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/j.joems.2016.03.006
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A.M. Zahran, M.M. Khalaf
[2], introduced and characterized the concept of T2 (Hausdorff)separation axiom as a fuzzy subset on the family of all [0, 1]− fuzzifying topological spaces. Shen [8], introduced and studied T0 , T1 , T3 (regular), T4 (normal)-separation axioms. Kheder, et al. [9], introduced R0 , R1 -separation axioms in fuzzifying topology and studied their relations with T1 and T2 -separation axioms. In 2003 Zhang et al. [10], studied the concept of fuzzy (i; j) -closed, (i; j)-open sets in fuzzifying bitopological spaces. In this paper, we introduce and study the concepts of semi − R0 and semi − R1 -separation axioms in fuzzifying bitopology and study their relations with semi − T1 -and semi − T2 -separation axioms . Furthermore, we discuss semi − T0 , semi − T3 (semiregularity) and semi − T4 (semi-normality)-separation axioms in fuzzifying bitopological spaces and give some of their characterizations as well as the relations of these axioms and other semi separation axioms in fuzzifying bitopology introduced. In Section 1, we recall some notions and results in separation axioms in L- fuzzifying bitopology, Also, in Section 2, semi separation axioms in L- fuzzifying bitopology are introduced and studied. Finally In Section 3, relations among semi separation axioms are discussed. Definition 1.1 (Höhle [5]). The double negation law in a complete residuated lattice L is given as follows: ∀a, b ∈ L, (a → ⊥ ) → ⊥ = a. Definition 1.2 (Höhle [5], Rosenthal [11]). A structure (L, ∨, ∧, ∗, →, ⊥, ) is called a strictly two-sided commutative quantale iff (1) (L, ∨, ∧, ⊥, ) is a complete lattice whose greatest and least element are , ⊥ respectively, (2) (L, ∗, ) is a commutative monoid, (3) (a) ∗ is distributive over arbitrary joins, i.e., a ∗ j∈J b j = j∈J (a ∗ b j ) ∀a ∈ L, ∀{bj |j ∈ J}⊆L, . (b) → is a binary operation on L defined by : a → b = λ∗a≤b λ ∀a, b ∈ L. Definition 1.3 (Ying [4]). A structure (L, ∨, ∧, ∗, →, ⊥, ) is called a complete residuated lattice iff (1) (L, ∨, ∧, ⊥, ) is a complete lattice whose greatest and least element are , ⊥ respectively, (2) (L, ∗, ) is a commutative monoid, i.e., (a) ∗ is a commutative and associative binary operation on L, and (b) ∀ a ∈ L, a ∗ = ∗ a = a, (3) (a) ∗ is isotone, (b) → is a binary operation on L which is antitone in the first and isotone in the second variable, (c) → is couple with ∗ as: a∗b ≤ c iff a ≤ b → c ∀ a, b, c ∈ L. Definition 1.4 [4]. Let (X, τ ) be an L-fuzzifying topological space, let x ∈ X. (1) The fuzzifying neighborhood system of x, denoted by Nx ∈ IP(X) , is defined as follows: Nx (A ) =
x∈B⊆A
τ (B).
2- The family of all fuzzifying closed sets is denoted by F ∈ (P(X )) and defined as follows: A ∈ F := X ∼ A ∈ τ 3- The closure cl(A) of A⊆X is defined as follows: cl (A )(x ) = 1 − Nx (X ∼ A ). Definition 1.9 [12]. Let (X, τ 1 ) and (X, τ 2 ) be two an Lfuzzifying topological space. Then a system (X, τ 1 , τ 2 ) consisting of a universe of discourse X with two L-fuzzifying topologies τ 1 and τ 2 on X is called an L-fuzzifying bitopological space. Definition 1.10 [12]. Let (X, τ 1 , τ 2 ) be an L-fuzzifying bitopological space. (1) A set A is said to be a pairwise open if and only if A ∈ τ 1 ∩ τ 2 . i.e., P (A ) = min(τ1 (A ), τ2 (A )). (1) A set B is said to be a pairwise closed if and only if X − B ∈ P . i.e., B ∈ F P = X − B ∈ P . Lema 1.1 [12]. Let (X, τ 1 , τ 2 ) be an L-fuzzifying bitopological space. Then F P (B) = min(F1 (B), F2 (B)). Lema 1.1 [12]. Let (X, τ 1 , τ 2 ) be an L-fuzzifying bitopological space. Then 1- P ⊆ τi , i = 1, 2 2- F P ⊆ Fi i = 1, 2. 2. Semi separation axioms in L-fuzzifying bitopological space Definition 2.1. Let (X, τ 1 , τ 2 ) be an L -fuzzifying bitopological space. (1) The family of all L-fuzzifying (i, j)-semiopen sets, denoted by sτ (i, j) ∈ (P(X)), is defined as follows: sτ(i, j) (A ) =
cl j (inti (A )(x ).
x∈A
(2) The family of all fuzzifying (i, j)-semiclosed sets, denoted by sF(i, j) ∈ (P(X)), is defined as follows: sF(i, j) (A ) = sτ(i, j) (A ) → ⊥. Example 2.1. If L = [0, 1] and Let X = {a, b}, τ 1 , τ 2 be two fuzzifying topologies defined as follows:
τ1 (A ) =
⎧ ⎨1 1 ⎩2
0
if if if
⎧ A ∈ {φ, X } ⎨1 A = {a} , τ2 (A ) = 14 ⎩ A = {b} 0
if if if
A ∈ {φ, X } A = {b} A = {a}
if if if
⎧ A ∈ {φ, X } ⎨1 A = {b} , F2 (A ) = 14 ⎩ A = {a} 0
if if if
A ∈ {φ, X } A = {a} A = {b}
and F1 (A ) =
⎧ ⎨1 1 ⎩2
0
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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On semi separation axioms in L-fuzzifying bitopological spaces
3 STR(i,j j) (X , τ1 , τ2 )
Note that 1 P (A ) = 0 ⎧ ⎨1 Sτ(i, j) (A ) = 14 ⎩ 0
A ∈ {φ, X } , o.w.
if if
=
(1) The (i, j)-semi neighborhood system of x is denoted by SNx(i, j) ∈ (P(X )) and defined as follows:
τ1 (A )
⎧ ⎨1
1 = 5 ⎩1 2
P (A ) = (2) The (i, j)-semi derived set sd(i, j) (A) of A is defined as follows: sd(i, j) (A )(x ) =
(
Sτ(1,2) (A ) =
→ ⊥ ).
B∩ (A−{x} ) = φ
(3) The (i, j)-semi closure of A⊆X, is scl(i, j) (A)(x) and defined as follows:
scl(i, j) (A )(x ) =
denoted
by
denoted
by
(sF(i, j) (B) → ⊥ ).
x∈ / B⊇A
(4) The (i, j)-semi interior of A⊆X, is scl(i, j) (A)(x) and defined as follows:
sint(i, j) (A )(x ) =
⎧ A ∈ {φ, X } ⎨1 A = {a} , τ2 (A ) = 14 ⎩1 A = {b} 8
if if if ⎧ ⎨1 1 ⎩ 51 8
if if if
A ∈ {φ, X } A = {a} , A = {b}
1 ⎩ 51 2
if if if
A ∈ {φ, X } A = {a} A = {b}
⎧ ⎨1
(SNx(i, j) (A )).
ST3(i, j) (X , τ1 , τ2 )
=
=
(SNx(i, j) (B) ∧ SNy(i, j) (C)),
min(Sτ(i, j) (C), Sτ(i, j) (D )),
Definition 2.3. Let be the class of all L -fuzzifying bitopological spaces. The unary L-predicates STR(i,i j) , STR(i,j j) and STR(i, j) ∈ L are defined as follows:
=
x∈ /u
((Fτi (u ) →
A∩B=φ, u⊆B
min
SNx(i, j) (A ),
(i, j) min(Fτ(i, j) (A ), Fτ(i, j) (B)) → SWA,B ),
=
(i, j) (i, j) SKx,y , → SHx,y
SR1(i, j) (X , τ1 , τ2 ) =
(i, j) (i, j) SKx,y . → SMx,y
τ1 (A ) =
where x, y ∈ X, A, B, C, D ∈ P(X) and (X, τ 1 , τ 2 ) is an Lfuzzifying bitopological space.
=
Example 2.3. If L = [0, 1] and Let X = {a, b}, τ 1 , τ 2 be two fuzzifying topologies defined as follows:
A⊆C, B⊆D, C∩D=φ
Fτ(i, j) (D ) → SVx,(i,Dj) ,
x=y
(SNx(i, j) (A ) ∧ Sτ(i, j) (B)),
STR(i,i j) (X , τ1 , τ2 )
x=y
x∈ /C
(i, j) SMx,y ,
A∩B=φ
SR0(i, j) (X , τ1 , τ2 )
A∩B=φ,D⊆B (i, j) SWA,B
=
x∈ /A
(i, j) SHx,y ,
x∈ /D
ST4(i, j) (X , τ1 , τ2 )
C∩B=φ (i, j) SVx,D =
A ∈ {φ, X } A = {a} A = {b}
x=y
ST2(i, j) (X , τ1 , τ2 ) =
⎛ ⎞ (i, j) (i, j) ⎝ ⎠ = SNx (B) ∧ SNy (C) ,
(i, j) SMx,y =
if if if
7 10 Definition 2.4. Let be the class of all L-fuzzifying bitopological spaces. The unary L-predicates STn(i, j) ∈ L , n = 0, 1, 2, 3, 4 and SRn(i, j) ∈ L , n = 0, 1 are defined as follows: (i, j) ST0(i, j) (X , τ1 , τ2 ) = SKx,y ,
x=y
y∈ /B
,
x=y
For simplicity we put the following notations: ⎛ ⎞ (i, j) SKx,y = ⎝ SNx(i, j) (A )⎠ ∨ SNy(i, j) (A ) ,
(i, j) SHx,y
Sτ(i, j) (B)
y∈u
STR(i, j) (X , τ1 , τ2 ) = STR(i,i j) (X , τ1 , τ2 ) ∧ STR(i,j j) (X , τ1 , τ2 ) =
ST1(i, j) (X , τ1 , τ2 )
x∈ / B⊇A
y∈ /A
Example 2.2. If L = [0, 1] and Let X = {a, b}, τ 1 , τ 2 be two fuzzifying topologies defined as follows:
Note that
sτ(i, j) (B).
SNx(i, j) (B)
A∩B=φ, u⊆B
x∈B⊆A
min
SNx(i, j) (A ),
STR(i, j) (X , τ1 , τ2 ) = STR(i,i j) (X , τ1 , τ2 ) ∧ STR(i,j j) (X , τ1 , τ2 ).
Definition 2.2. Let (X, τ 1 , τ 2 ) be an L -fuzzifying bitopological space and x ∈ A
SNx(i, j) (A ) =
((Fτ j (u ) →
x∈ /u
A ∈ {φ, X } A = {a} . A = {b}
if if if
Sτ(i, j) (B)
⎧ ⎨1 1 ⎩51 2
Note that P (A ) =
Sτ(1,2) (A ) =
if if if ⎧ ⎨1
⎧ A ∈ {φ, X } ⎨1 A = {a} , τ2 (A ) = 14 ⎩1 A = {b} 8
1 ⎩ 51 8
if if if
A ∈ {φ, X } A = {a} , A = {b}
1 ⎩ 51 2
if if if
A ∈ {φ, X } A = {a} A = {b}
⎧ ⎨1
if if if
A ∈ {φ, X } A = {a} A = {b}
y∈u
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ST3(1 ,2) (X , τ1 , τ2 )
=
Fτ(1 ,2) (D ) →
SVx,(1 D,2)
x∈ /D
STR(11 ,2) (X , τ1 , τ2 ) = min
A∩B=φ, u⊆B
STR(12 ,2) (X , τ1 , τ2 )
=
SNx(1,2) (A ),
min
Sτ(1,2) (B)
=
y∈u
7 10
x∈ /u
SNx(1,2) (A ),
Sτ(1,2) (B)
=
y∈u
A∩B=φ, u⊆B
7 10
Remark 2.1. Let (X, τ 1 , τ 2 ) be an L-fuzzifying bitopological space. From remark 3.1 in [15], we have STi (1 ,2) (X , τ1 , τ2 ) = STi (2 ,1) (X , τ1 , τ2 ), i = 0, 1, 2, 3, 4 Lemma 2.1. Let (X, τ ) ∈ . Then for any x, y ∈ X, (i, j) SMx,y (i, j) SHx,y (i, j) SMx,y
(1) (2) (3)
(i, j) min(SNx(i, j) (A ), SNy(i, j) (B)) = SHx,y
y∈ / A, x∈ /B
3- It is obtained from (1) and (2).
Proof. The proof is similar to the proof of Theorem 2.2.
SR1(i, j) (X , τ1 , τ2 ) ≤
ST1(i, j) (X , τ1 , τ2 ) =
SF(i, j) ({x} ).
Proof. Let x1 , x2 ∈ X s.t. x1 = x2 . Then SF(i, j) ({x} ) = Sτ(i, j) (X − {x} ) x∈X
=
⎛
≤
⎝
x∈X
=
⎞ SNy(i, j) (X − {x} )⎠
y∈X −{x}
y∈X −{x2 }
(i, j) (i, j) (SKx,y → SHx,y ) = SR0(i, j) (X , τ1 , τ2 ).
SNy(i, j) (X − {x2 } ) ≤ SNx(i,1 j) (X − {x2 } )
SNx(i,1 j) (A )
x2 ∈ /A
Similary, we have
Theorem 2.2. For any (X, τ 1 , τ 2 ) ∈ , the following statements are satisfied: ST1(i, j) (X , τ1 , τ2 ) ≤ SR0(i, j) (X , τ1 , τ2 ), ST1(i, j) (X , τ1 , τ2 ) ≤ ST0(i, j) (X , τ1 , τ2 ), ST1(i, j) (X , τ1 , τ2 ) ≤ SR0(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ), If ST0(i, j) (X , τ1 , τ2 ) = , then ST1(i, j) (X , τ1 , τ2 ) = SR0(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ).
SF(i, j) ({x} ) ≤
x∈X
x1 ∈ /B
SNx(i,2 j) (B)
Then SF(i, j) ({x} ) ≤ x∈X
x1 =x2 x1 ∈ / B,x2 ∈ /A
(i, j) SHx,y
ST1(i, j) (X , τ1 ,
(1) Since → so that τ2 ) ≤ (2) From Lemma 2.1 (2) one can deduce that ST1(i, j) (X , τ1 , τ2 ) ≤ ST0(i, j) (X , τ1 , τ2 ). (3) The proof follows from (1) and (2) .
(SNx(i,1 j) (A ) ∧ SNx(i,2 j) (B))
= ST1 (X , τ ). For the other hand,
Proof. (i, j) (i, j) SHx,y ≤ SKx,y SR0(i, j) (X , τ1 , τ2 ).
ST2(i, j) (X , τ1 , τ2 ) ≤
x∈X
x∈X
x=y
(1) (2) (3) (4)
Theorem 2.5. If L satisfies the completely distributive law, then for any (X, τ 1 , τ 2 ) ∈ ,
Theorem 2.1. For any (X, τ1 , τ2 ) ∈ , SR1(i, j) (X , τ1 , τ2 ) ≤ SR0(i, j) (X , τ1 , τ2 ). (i, j) (i, j) Proof. SR1(i, j) (X , τ1 , τ2 ) = x=y (SKx,y → SMx,y ). Since → is isotone in the second, then
ST2(i, j) (X , τ1 , τ2 ) ≤ SR1(i, j) (X , τ1 , τ2 ), ST2(i, j) (X , τ1 , τ2 ) ≤ ST1(i, j) (X , τ1 , τ2 ), ST2(i, j) (X , τ1 , τ2 ) ≤ SR1(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) If ST0(i, j) (X , τ1 , τ2 ), then ST2(i, j) (X , τ1 , τ2 ) = SR1(i, j) (X , τ1 , τ2 ).
Proof. The conclusion is obtained from Theorem 2.2(2) and Theorem 2.3(2).
(i, j) (B), SNy(i, j) (C)) ≤ C∩B=φ min (S Nx (i, j) (i, j) (B), SNy(i, j) (C)) = SHx,y , y∈ / B, x∈ /C min (SNx (i, j) (i, j) (i, j) 2- SKx,y = max( y∈/ A SNx (A )), ( x∈/ A SNy (A )) ≥ (i, j) (A ) y∈ / A SNx
≥
(1) (2) (3) (4)
Theorem 2.4. For any (X, τ1 , τ2 ) ∈ , ST0(i, j) (X , τ1 , τ2 ).
(i, j) ≤ SHx,y , (i, j) ≤ SKx,y , (i, j) ≤ SKx,y .
Proof. (i, j) 1- SMx,y =
(4) Since → α = α ∀α ∈ L (Indeed → α = λ∗ ≤α λ = (i, j) λ = α. ) then ST1(i, j) (X , τ1 , τ2 ) = x=y SHx,y = λ≤α (i, j) (i, j) (i, j) ( x=y (SKx,y → SHx,y )) ∧ = SR0 (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) because ∀x, y ∈ X s.t. x = y, if ST0(i, j) (X , τ1 , τ2 ) = , then (i, j) SKx,y = . Theorem 2.3. For any (X, τ 1 , τ 2 ) ∈ , the following statements are satisfied:
((Fτ2 (u )
→
((Fτ1 (u )
x∈ /u
→
7 = 10
ST1(i, j) (X , τ1 , τ2 )
=
x1 ,x2 ∈X,x1 =x2
⎛ ⎝
x1 ∈ /B
⎛⎛ ⎝⎝
⎞ SNx(i,1 j) (A )⎠∧
x2 ∈ /A
⎞⎞
SNx(i,2 j) (B)⎠⎠
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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On semi separation axioms in L-fuzzifying bitopological spaces
=
x1 =x2
x1 =x2
x2 ∈X x1 ∈X −{x2 }
=
=
x∈X
=
=
SF(i, j) ({x} )
A∈P(X )
Sτ(i, j) (X − {x} ) =
x∈X
→
x∈X x∈X
y∈u
min
STR(i,j j) (X , τ1 , τ2 ) =
→
(Scl(i, j) (A )(y ) → ⊥ ) ,
y∈u
((Fτ j (u )
SNx(i, j) (A ),
(Sτ(i, j) (A )(y ) → ⊥ ) ,
STR(i, j) (X , τ1 , τ2 ) = 1 STR(i,i j) (X , τ1 , τ2 ) ∧1 STR(i,j j) (X , τ1 , τ2 ) 2 STR(i,i j) (X , τ1 , τ2 ) = ((τi (u ) 1
→
B∈P(X ) 2
STR(i,j j) (X , τ1 , τ2 ) =
→
((τ j (u )
x∈u
B∈P(X ) 2
STR(i, j) (X , τ1 , τ2 )
=2 STR(i,i j) (X , τ1 , τ2 ) ∧2 STR(i,j j) (X , τ1 , τ2 ).
Theorem 2.6. Let (X, τ 1 , τ 2 ) ∈ , then, 1- STR(i,i j) (X , τ1 , τ2 ) = 2- STR(i,j j) (X , τ1 , τ2 ) = 3- STR(i, j) (X , τ1 , τ2 ) =
n n n
STR(i,i j) (X , τ1 , τ2 ), n = 1, 2 STR(i,j j) (X , τ1 , τ2 ), n = 1, 2
STR(i, j) (X , τ1 , τ2 ), n = 1, 2
y
τ
∃ K = ∪{ f (y )| f (y ) ∈ ℘y(i, j) , y ∈
℘y
Theorem 2.7. If L satisfies the completely distributive law, for any (X, τ 1 , τ 2 ) ∈ . Then 2 STR(i,i j) (X , τ1 , τ2 ) = STR(i,i j) (X , τ1 , τ2 ) Proof. STR(i,i j) (X , τ1 , τ2 ) = 1 STR(i,i j) (X , τ1 , τ2 ) =
min(SNx(i, j) (A ),
=
⎛
min ⎝SNx(i, j) (A ),
((Fτi (u ) →
x∈ /u
(Scl(i, j) (A )(y ) → ⊥ ))
((Fτi (X − B) →
x∈ / X −B
=
y∈u
A∈P(X )
Proof. (1) 1 STR(i,i j) (X , τ1 , τ2 ) = x∈/ u ((Fτi (u ) → A∈P(X ) min (SNx(i, j) (A ), y∈u (Scl(i, j) (A )(y ) → ⊥ )) = x∈/ u ((SFτi (u ) → A∈P(X ) min(SNx(i, j) (A ), ( y∈u SNy(i, j) (X − A )))) and STR(i,i j) (X , τ1 , τ2 ) = x∈/ u ((Fτi (u ) → A∩B=φ,u⊆B min(SNx(i, j) (A ), ( y∈u Sτ(i, j) (B))). So, the result holds if we prove that
y∈D
y∈D
f∈
∧[[Scl(i, j) (B), u[[)),
min(SNx(i, j) (B), ∧[[Scl(i, j) (B), u[[ ),
f ∈
τ ℘y(i, j) ,
y∈D A∩B=φ, D⊆B Sτ(i, j) (B). So shoud prove that min(SNx(i, j) (A ), (Scl(i, j) (A )(y ) → ⊥ ) = A∈P(X ) y∈u (i, j) min (SN (A ) , ( x A∩B=φ, u⊆B y∈u Sτ(i, j) (B)) (2) It is similar to (1) (3) It is clear.
x∈u
min(SNx(i, j) (B),
D} s.t. D⊆ K ⊆ X − A τ(i, j) and , y ∈ D }) = y∈D f (y ) ≤ τ (∪{ f (y ) f (y ) ∈ ℘y Sτ (K ) so that (i, j) (y ) ≤ Sτ(i, j) (K ) ≤ A∩ B=φ, D⊆B Sτ(i, j) y∈D f (B) so that Sτ (B) = Sτ (i, j) (i, j) y∈D y∈B⊆X −A y∈D f (y ) ≤
y∈u
A∈P(X )
Sτ(i, j) (B)⎠⎠.
y∈B⊆X −A
we
for each
x∈ /u
min
⎞⎞
prove that y∈u y∈B⊆X −A Sτ(i, j) (B) = A∩B=φ,u⊆B Sτ(i, j) (B). Let y ∈ u. Assume S = {H ∈ P(X )|H ∩ A = φ and u ⊆ H } and ℘y = {M ∈ P(X )|y ∈ M ⊆ X − A}. Then S ⊆℘y so that B∈℘y Sτ(i, j) (B) ≥ B∈S Sτ(i, j) (B) so that y∈D y∈B⊆X −A Sτ(i, j) (B) ≥ A∩B=φ, D⊆B Sτ(i, j) (B). τ(i, j) τ(i, j) Let ℘ ℘y }. Then y∈D ℘y = y = {τ (M) M ∈ τ(i, j) f (y ) . Then y∈D B∈℘y Sτ(i, j) (B) = y ∈ D f∈ ℘ Now
((Fτi (u )
x∈ /u
SNx(i, j) (A ),
A∈P(X ) 1
min(SNx(i, j) (A ),
⎛ ⎛ ⎝ ⎝
SF(i, j) ({x} ).
Definition 2.3. Let be the class of all L -fuzzifying bitopological spaces. We define STR(i,i j) (X , τ1 , τ2 ) =
y∈u
A∩u=φ, A∈P(X )
1
Sτ(i, j) (B) .(∗)
y∈u
=
τ(i, j) (X − {x2 } )
In the left / X −A side of (∗) if A ∩ u = φ, ∃ y ∈ u s.t. y ∈ so that y∈u SNy(i, j) (X − A ) = ⊥. Second,
min(SNx(i, j) (A ), SNy(i, j) (X − A )
x∈X x∈X
x2 ∈X x∈X
SNy(i, j) (X − A )
min(SNx(i, j) (A ),
A∩B=φ, u⊆B
τ(i, j) (X − {x2 } )
τ(i, j) (X − {x} ) =
y∈u
=
SNx(i,1 j) (X − {x2 } )
x2 ∈X x∈X
min(SNx(i, j) (A ),
A∈P(X )
SNx(i,1 j) (X − {x2 } )
=
((SNx(i,1 j) (X −{x2 } )∧(SNx(i,2 j) (X −{x1 } ))
≤
5
⎛ ⎝
A∈P(X )
⎞⎞
(Scl(i, j) (A )(y ) → ⊥ )⎠⎠
y∈X −B
(Sτ(i, j) (B) →
x∈B
min(SNx(i, j) (A ),
A∈P(X )
⎛ ⎝
⎞ (Scl(i, j) (A )(y ) → ⊥ )⎠
y∈X −B
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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6 =
A.M. Zahran, M.M. Khalaf
((τi (u ) →
x∈u
min(SNx(i, j) (B),
B∈P(X )
∧[[Scl(i, j) (A ), B[[)) =2 STR(i,i j) (X , τ1 , τ2 ) that Note y∈X (Scl(i, j) (A )(y ) → B(y )) = ( y∈ B (Scl (i, j) (A )(y ) → ) ∧ ( y∈X −B (Scl(i, j) (A )(y ) → ⊥ )) = ∧ y∈X −B (Scl(i, j) (A )(y ) → ⊥ ) = y∈X −B (Scl(i, j) (A )(y ) → ⊥) Theorem 2.8. Let (X, τ 1 , τ 2 ) ∈ . If L satisfies the double negation law, then, ST0(i, j) (X , τ1 , τ2 ) = ((cl(i, j) ({y} )(x ) → ⊥ ) x=y
ST0(i, j) (X , τ1 , τ2 ) =
∨
⎛⎛ ⎝⎝
x=y
=
SNx(i, j) (A )⎠
y∈X
((SNx(i, j) (X
x=y
Theorem 2.9. Let (X, τ 1 , τ 2 ) ∈ , and let A be a finite subset of X. If L satisfies the completely distributive, then,
SNy(i, j) ((X − A ) ∪ {y} ).
y∈X
=
SNy(i, j) ( X −
y∈X −A
=
y∈X −A
{x} =
x∈A
y∈X −A
SNy(i, j)
y∈X −A
SNy(i, j) (X − {x} ) ≥
and SNy(i, j) ((X − A ) ∪ {y} ) = SNy(i, j) (X − (A − {y} ))
=
=
⎛
⎛
SNy(i, j) ⎝X − ⎝
y∈A
=
y∈A
⎛
y∈A
⎝
x∈A−{y}
⎞
SNy(i, j) (X − {x} )⎠ ≥
→ =
=
B⊆A
→ SNy(i, j) (X − {x} ).
x=y
Then SNy(i, j) ((X − A ) ∪ {y} ) ≥ x=y SNy(i, j) (X − y∈X {x} ) = x∈X ( y∈X −{x} SNy(i, j) (X − {x} )) = x∈X Sτ(i, j) (X − {x} ) = x∈X SF τ(i, j) ({x} ) = ST1(i, j) (X , τ1 , τ2 ).
⎞ (min(Sτ(i, j) (u ), Sτ(i, j) (X − v )))⎠
(u∩X −v )=φu⊆X −v, A⊆X −v, B⊆u
→
− {x} )⎠
min(Fτ(i, j) (X − A ), Fτ(i, j) (B))
(X −A∩B)=φ
B⊆A
⎞
=
{x}⎠⎠
x∈A−{y}
SNy(i, j) ⎝ (X y∈A x∈A−{y} ⎛
⎞⎞
(min(Sτ(i, j) (u ), Sτ(i, j) (v )))).
u∩v=φ, A⊆v, B⊆u
x∈A
SNy(i, j) (X − {x} ) ,
A∩B=φ
→
(X − {x} )
x=y
x∈A
y∈A
Theorem 2.11. Let (X, τ ) ∈ , and let (1) 1 STN(i, j) (X , τ1 , τ2 ) = A⊆B ((Fi (A ) ∧ τ j (B)) → A⊆u⊆v⊆B (min (Sτ(i, j) (u ), SF(i, j) (v ))) (2) 2 STN(i, j) (X , τ1 , τ2 ) = A∩B=φ ((Fi (A ) ∧ τ j (B)) → A⊆u (min (Sτ(i, j) (u ), Scl (i, j) (u ))) (3) 3 STN(i, j) (X , τ1 , τ2 ) = A⊆B ((Fi (A ) ∧ τ j (B)) → A⊆B (min (Sτ(i, j) (u ), Scl (i, j) (v ))) (4) STN(i, j) (X , τ1 , τ2 ) = A∩B=φ (min(Fτi (A ), Fτ j (B)) → U ∩V =φ, A⊆V, B⊆U (min (Sτ(i, j) (u ), Sτ(i, j) (v )))). Then STN(i, j) (X , τ1 , τ2 ) = n STN(i, j) (X , τ1 , τ2 ), n = 1, 2, 3 Proof. (1) From Lema 1.1 STN(i, j) (X , τ1 , τ2 ) = (min(Fτi (A ), Fτ j (B))
Proof. Now, SNy(i, j) ((X − A ) ∪ {y} ) = SNy(i, j) (X − A ) y∈X −A
y∈X
− {y} )
∨(SNy(i, j) (X − {x} )) = ((cl(i, j) ({y} )(x ) → ⊥ ) ∨ (cl(i, j) ({x} )(y ) → ⊥ )).
ST1(i, j) (X , τ1 , τ2 ) ≤
ST1(i, j) (X , τ1 , τ2 ) ≤ [[d(i, j) (A ), 1φ ]].
d(i, j) (y ))) = (d(i, j) (A )(y ) → 1φ (y )) = SNy(i, j) ((X − A ) ∪ {y}.
x=y
x∈ /A
Theorem 2.10. Let (X, τ ) ∈ and let A be a finite subset of X. If L satisfies the completely distributive and the double negation law, then
y∈X
⎞
y∈ /A
SNy(i, j) (A )
d(i, j) (A )(x ) = SNx(i, j) ((X − A ) ∪ {x} ) → ⊥ ∀x ∈ X .
Proof. It follows from Theorem 2.7 and since [[d(i, j) (A ), 1φ ]] = (d(i, j) (A )(y ) → 1φ (y )) ∧ (1φ (y ) →
∨(cl(i, j) ({x} )(y ) → ⊥ )). Proof.
Definition 2.4. Let (X, τ 1 , τ 2 ) ∈ and let A ⊆ X. The Lfuzzifying derived set of A is denoted by dτ (A) ∈ LX and defined as follows:
(min(Fτ(i, j) (A ), Fτ(i, j) (B))
(min(Sτ(i, j) (u ), SF(i, j) (v )))
u⊆v, v⊆A, B⊆u
(min(Fτ(i, j) (A ), Fτ(i, j) (B))
(min(Sτ(i, j) (u ), SF(i, j) (v ))) =1 STN(i, j) (X , τ1 , τ2 )
B⊆u⊆v⊆A
(2) and (3) are similar
Theorem 2.12. Let (X, τ 1 , τ 2 ) ∈ , Then STN(i, j) (X , τ1 , τ2 ) = ST4(i, j) (X , τ1 , τ2 ). Proof. From Lema 1.1 (2) . It is obtained.
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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On semi separation axioms in L-fuzzifying bitopological spaces The following example shows that generally the reverse of the Theorem 2.12 need not be true. Example 2.4. [12] If L = [0, 1] and Let X = {a, b}, τ 1 , τ 2 be two fuzzifying topologies defined as follows: ⎧ ⎨ 1 if A ∈ {φ, X } τ1 (A ) = 12 if A = {a} , ⎩ 0 if A = {b} ⎧ 1 if A ∈ {φ, X } ⎨ τ2 (A ) = 14 if A = {b} ⎩ 0 if A = {a} and
⎧ ⎨1
F1 (A ) =
1 ⎩2
if if if
A ∈ {φ, X } A = {b} , A = {a}
1 ⎩4
if if if
A ∈ {φ, X } A = {a} A = {b}
0 ⎧ ⎨1
F2 (A ) =
0
Note that 1 P (A ) = 0 ⎧ ⎨1 Sτ(i, j) (A ) = 14 ⎩ 0
A ∈ {φ, X } , o.w.
if if if
A ∈ {φ, X } A = {a} . A = {b} 3 4
= STN(i, j) (X , τ1 , τ2 )
Proof. Now, (Sτ(i, j) (X − {y} ))
→
⎛ ⎞⎞ (i, j) (i, j) ⎝ (SNx (A ) ∧ SNy (B))⎠⎠ y∈B
A∩B=φ, y∈B
≤
x=y
⎛ ⎝
=
y∈B
ST1(i, j) (X , τ1 , τ2 ))
x=y
y∈B
= ST1(i, j) (X , τ1 , τ2 ) ⎛ ⎛ ⎞⎞ (i, j) (i, j) ⎝ ⎝ (SNx (A ) ∧ SNy (B))⎠⎠ → x=y
A∩B=φ,
y∈B
(SNx(i, j) (A ) ∧ SNy(i, j) (B))⎠
A∩B=φ
ST2(i, j) (X , τ1 , τ2 ).
→
Since
Theorem 3.3. For any (X, τ ) ∈ , 1- ST2(i, j) (X , τ1 , τ2 ) ≤ SR1(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) 2- If ST2(i, j) (X , τ1 , τ2 ) = , then ST2(i, j) (X , τ1 , τ2 ) = (i, j) (i, j) SR1 (X , τ1 , τ2 ) ∧ ST0 (X , τ1 , τ2 ) 3- SR1(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ) 4- If ST0(i, j) (X , τ1 , τ2 ) = , then ST2(i, j) (X , τ1 , τ2 ) = (i, j) (i, j) SR1 (X , τ1 , τ2 ) ∧ ST0 (X , τ1 , τ2 )
Proof. It obvious
Theorem 3.4. If L satisfies the completely distributive law, then for any (X , τ ) ∈ , ST4(i, j) (X , τ1 , τ2 ) ∗ ST1(i, j) (X , τ1 , τ2 ) ≤ ST3(i, j) (X , τ1 , τ2 ). Proof. ST4(i, j) (X , τ1 , τ2 )
=
⎛ ⎝ min(SFτ
(i, j)
(E ), SFτ(i, j) (D ))
E∩D=φ
→
⎞
min(Sτ(i, j) (A ) ∧ Sτ(i, j) (B))⎠
E⊆A, D⊆B, A∩B=φ
≤
⎛
⎝ min(SFτ
(i, j)
({x} ), SFτ(i, j) (D ))
x∈ /D
⎛ ⎞⎞ (i, j) (i, j) ⎝ (SNx (A ) ∧ SNy (B))⎠⎠
A∩B=φ, y∈B
ST3(i, j) (X , τ1 , τ2 ) = x∈/ D (Sτ(i, j) (X − D ) → (i, j) (A ) ∧ τ(i, j) (B))) A∩B=φ,D⊆B (SNx = x∈/ D (Sτ(i, j) (X − D ) → A∩B=φ,D⊆B (SNx(i, j) (A ) ∧ ( y∈B (SNy(i, j) (B)))) ≤ x∈/ {y} (Sτ(i, j) (X − {y} ) → A∩B=φ, y∈B ( y∈B (SNx(i, j) (A ) ∧ SNy(i, j) (B)))) = x=y (Sτ(i, j) (X − {y} ) → A∩B=φ, y∈B ( y∈B (SNx(i, j) (A ) ∧ SNy(i, j) (B)))), then from above, ST3(i, j) (X , τ1 , τ2 ) ≤ (i, j) (i, j) ST1 (X , τ1 , τ2 ) → ST2 (X , τ1 , τ2 ), so that ST3(i, j) (X , τ1 , τ2 ) ∗ ST1(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ).
⎛ ⎞⎞ ⎝ (SNx(i, j) (A ) ∧ SNy(i, j) (B))⎠⎠
A∩B=φ, y∈B
→
=
Sτ(i, j) (X − {y} )
y∈X
→
x=y
ST1(i, j) (X , τ1 , τ2 )
Proof. It obvious
Theorem 3.1. If L satisfies the completely distributive law, then for any (X , τ ) ∈ , ST3(i, j) (X , τ1 , τ2 ) ∗ ST1(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ).
⎝
1- ST1(i, j) (X , τ1 , τ2 ) ≤ SR0(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) 2- If ST1(i, j) (X , τ1 , τ2 ) = , then ST1(i, j) (X , τ1 , τ2 ) ≤ (i, j) (i, j) SR0 (X , τ1 , τ2 ) ∧ ST0 (X , τ1 , τ2 ) 3- SR1(i, j) (X , τ1 , τ2 ) ∧ ST0(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ) 4- If ST0(i, j) (X , τ1 , τ2 ) = , then ST2(i, j) (X , τ1 , τ2 ) = (i, j) (i, j) SR0 (X , τ1 , τ2 ) ∧ ST0 (X , τ1 , τ2 )
3. Relations among semi-separation axioms in fuzzifying bitopological spaces
x=y
≤ ST1(i, j) (X , τ1 , τ2 ) →
⎞
Theorem 3.2. For any (X, τ ) ∈ ,
if if
Hence ST4(i, j) (X , τ1 , τ2 ) = 1
7 ⎛
⎞
→
min(Sτ(i, j) (A ) ∧ Sτ(i, j) (B))⎠
x∈A, D⊆B, A∩B=φ
=
min(SFτ(i, j) ({x} ), SFτ(i, j) (D ))
x∈ /D
→
D⊆B, A∩B=φ
⎞ Sτ(i, j) (A ) ∧ Sτ(i, j) (B) ⎠
x∈A
Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006
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8 ≤
min(SFτ(i, j) ({x} ), SFτ(i, j) (D ))
x∈ /D
→
D⊆B, A∩B=φ
=
(i, j)
Theorem 3.5. If L satisfies the completely distributive law, then for any (X , τ1 , τ2 ) ∈ , ST4(i, j) (X , τ1 , τ2 ) = SRi(i, j) (X , τ1 , τ2 ).
({x} ), SFτ(i, j) (D ))
→
⎞
x∈ /D
→ =
x∈ /D
→ ≤
x∈ /D
→
SFτ(i, j) ({x} ) ∧ SFτ(i, j) (D )
x∈X
⎞
(SNx(i, j) (A ) ∧ Sτ(i, j) (B))⎠
Acknowledgments The authors are highly grateful to the referees for their valuable comments and suggestions.
(((ST1i, j (X , τ1 , τ2 ) ∧ SFτ(i, j) (D ))
(SNx(i, j) (A ) ∧ Sτ(i, j) (B)))
References
D⊆B, A∩B=φ
(((ST1(i, j) (X , τ1 , τ2 ) ∗ SFτ(i, j) (D ))
(SNx(i, j) (A ) ∧ Sτ(i, j) (B)))
→
((SFτ(i, j) (D )
x∈ /D
(SNx(i, j) (A ) ∧ Sτ(i, j) (B)))
D⊆B, A∩B=φ
=
1- SRi(i, j) (X , τ1 , τ2 ) ∧ ST1(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ) j) 2- SR(i, (X , τ1 , τ2 ) ∧ ST1(i, j) (X , τ1 , τ2 ) ≤ ST2(i, j) (X , τ1 , τ2 ) j
D⊆B, A∩B=φ
ST1(i, j) (X , τ1 , τ2 ) →
Theorem 3.7. For any (X , τ1 , τ2 ) ∈ ,
D⊆B, A∩B=φ
≤
Theorem 3.6. If L satisfies the completely distributive law, then j) for any (X , τ1 , τ2 ) ∈ , ST4(i, j) (X , τ1 , τ2 ) = SR(i, (X , τ1 , τ2 ). j
(SNxi, j (A ) ∧ Sτ(i, j) (B))⎠
D⊆B, A∩B=φ
K f ∗ α = ( j∈J f ( j)) ∗ α ≤ j∈J ( f ( j) ∗α) ≤ j∈J (B j → M j ). Then ≤ j∈J ((α ∗ B j ) → M j ) ≤ f∈ Aj Kf j∈J λ∗α≤ j∈J (B j →M j ) λ = α → j∈J (B j → M j ). ). We have the following results which their proof are obvious.
x∈K⊆A
⎛
⎝ min(SFτ
⎞ Sτ(i, j) (K ) ∧ Sτ(i, j) (B) ⎠
x∈ /D
≤
[m;May 11, 2016;18:16]
A.M. Zahran, M.M. Khalaf
ST1(i, j) (X , τ1 , τ2 ) ST4(i, j) (X , τ1 , τ2 )
→ ST3(i, j) (X , τ1 , τ2 )
so that
∗ ST1(i, j) (X , τ1 , τ2 ) ≤ ST3(i, j) (X , τ1 , τ2 ).
(Indeed, put ST1(i, j) (X , τ1 , τ2 ) = α, j = (x, D ), J= {(x, D )|x ∈ X , D ∈ P(X ), x ∈ / D }, B(x,D ) = (Fτi (D ), M(x,D ) = D⊆B, A∩B=φ (SNx(i, j) (A ) ∧ Sτ(i, j) (B)) and λ ∗ α ≤ B A = { λ → M } . Then ((α ∗ B ) → M j j j j) = j∈J j λ j∈J λ∗(α∗Bj )≤M j = j∈J( λ∗α)∗B j ≤M j λ = j∈J λ∗α≤(B j →M j ) λ = j∈J A j = f ∈ A j j∈J f ( j). Now, ∀ f ∈ A j , there exists K f = j∈J j∈J j∈J f ( j) s.t.
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Please cite this article as: A.M. Zahran, M.M. Khalaf, On semi separation axioms in L-fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society (2016), http://dx.doi.org/10.1016/j.joems.2016.03.006