Generalized Tychonoff theorem in L -fuzzy supratopological spaces 1

1 Introduction and preliminaries

As the generalization of topological spaces, Mashhour firstly introduced the notion of supratopological spaces [6]. In 1987, El-Monsef and Ramadan [1] introduced the concept of fuzzy supratopological spaces by using the notion of fuzzy supraopen sets. In 2000, Ghanim, Tantawy and Selim [3] introduced the notion of I-fuzzy supratopological spaces. In 2014, Shi and Liang [7] gave the notions of the degrees of compactness in L-fuzzy supratopological spaces by means of implication operations. It is worth mentioning that this concept can not be extended to (L, M)-fuzzy supratopological spaces. Subsequently, Liang and Shi [4] gave the new notions of the degrees of compactness in (L, M)-fuzzy supratopological spaces, they proved the notions of precompactness, semi-compactness and sp-compactness in (L, M)-fuzzy topological spaces can be viewd as special cases of compactness in (L, M)-fuzzy supratopological spaces. However, some results on compactness in (L, M)-fuzzy product supratopological spaces were not given, including some similar results on precompactness, semi-compactness and sp-compactness in (L, M)-fuzzy topological product spaces were not obtained, especially when M is equal to L. Follow this idea, Hu Zhao and Gui-xiu Chen [12] studied the degrees of compactness in (L, M)-fuzzy product supratopological spaces. Significantly, the degrees of compactness in (L, L)-fuzzy supratopological spaces [4] isn’t the degrees of compactness in L-fuzzy supratopological spaces by means of implication operations [7]. Therefore, the purpose of this paper is to study the generalized Tychonoff theorem in L-fuzzy supratopological spaces by means of implication operations.

Throughout this paper, both L and M always denote Hutton algebras. A Hutton algebra (L, ∨ , ∧ , ′), is a completely distributive lattice with order-reversing involution with the smallest element 0 and the largest element 1. An element c ∈ L is called prime if a ∧ b ⩽ c implies a ⩽ c or b ⩽ c . c ∈ L is called coprime if a′ is a prime element. The set of all nonunit prime elements in L is denoted by P (L). The set of all nonzero coprime elements in L is denoted by J (L). We say that a is wedge below b in L, denoted by a ⪡ b, if for every subset C ⊆ L, ⋁C ⩾ b implies c ⩾ a for some c ∈ C (see [5,10-11, 5,10-11]). A complete lattice L is completely distributive lattice iff a = ⋁ {b ∈ L ∣ b ⪡ a}. β (b) is the greatest minimal family of b, β^∗ (b) = β (b) ∩ J (L) is a minimal family of b. α (b) is the greatest maximal family of b, α^∗ (b) = α (b) ∩ P (L) is a maximal family of b. When L is completely distributive lattice, each element b in L has the greatest minimal family(resp., the greatest maximal family) (see[10-11]).

In a completely distributive lattice L, there exists a binary operation →, and the implication is defined by a → b = ⋁ { d ∈ L ∣ a ∧ d ⩽ a } . It is easy to verify the following properties of →.

(1) (a → b) ⩾ c ⇔ a ∧ c ⩽ b.

(2) a → b = ⊤ ⇔ a ⩽ b.

(3) a → ⋀  _i b _i  = ⋀  _i  (a → b _i ).

(4) (⋁  _i a _i ) → b = ⋀  _i  (a _i  → b).

(5) a ⩽ b ⇒ a → c ⩾ b → c, c → a ⩽ c → b.

(6) (a ∧ b) → c = a → (b → c).

For a subfamily G ⊆ L X , 2 ( G ) denotes the set of all finite subfamilies of G , and for every index set I, 2^(I) denotes the set of all finite subsets of I.

Definition 1.1. [1, 4] Let X be a nonempty set, L ^X the set of all L-subsets on X, 0 _X the smallest element of L ^X and 1 _X the largest element of L ^X . An (L, M)-fuzzy supratopology on a set X is a map T : L X ⟶ M such that

(1) T ( 1 X ) = T ( 0 X ) = 1 ;

(2) T ( ⋁ k ∈ K A k ) ⩾ ⋀ k ∈ K T ( A k ) , ∀ { A k } k ∈ K ⊆ L X , T ( A ) can be interpreted as the gradation of supraopenness of A, and T ★ ( A ) = T ( A ′ ) will be called the gradation of supraclosedness of A. The pair ( X , T ) is called an (L, M)-fuzzy supratopological space. Obviously, if (L, M) = (L, L), then ( X , T ) is an L-fuzzy supratopological space.

A mapping f : X ⟶ Y from an (L, M)-fuzzy supratopological space ( X , T X ) to another (L, M)-fuzzy supratopological space ( Y , T Y ) is said to be continuous if T X ( f ← ( G ) ) ⩾ T Y ( G ) for each G ∈ L ^Y . The category of all (L, M)-fuzzy supratopological spaces and their continuous mappings is denoted by (L, M)-FsupraTOP.

Next, we give some examples on (L, M)-fuzzy supratopology.

Example 1.2. (1) Let L = {(0, 0) , (1, 1)} ∪ (0, 1) ² be a subset of the square [0, 1] ² and M = [0, 1], "⩽" and "′" are defined as follows: ( a , b ) ⩽ ( c , d ) iff a ⩽ c and b ⩽ d , and ( x , y ) ′ = { ( 1 - x , 1 - y ) , ( x , y ) ∈ ( 0 , 1 ) 2 , ( 0 , 0 ) , ( x , y ) = ( 1 , 1 ) , ( 1 , 1 ) , ( x , y ) = ( 0 , 0 ) . Then (L, ⩽ , ′) is a Hutton algebra. Let X = {x, y}, define T : L X ⟶ M as follows: T ( A ) = { 1 , A = 0 X , 1 X , 1 4 , A ( z ) = { ( 1 4 , 1 2 ) , z = x , ( 1 2 , 1 4 ) , z = y , 1 2 , others . It is easy to verify that T ( ⋁ k ∈ K A k ) ⩾ ⋀ k ∈ K T ( A k ) , ∀ { A k } k ∈ K ⊆ L X , This implies T is an (L, M)-fuzzy supratopology on X. However, T is not an (L, M)-fuzzy topology on X. In fact, set C ( z ) = { ( 1 4 , 3 4 ) , z = x , ( 1 2 , 1 4 ) , z = y , and D ( z ) = { ( 1 4 , 1 2 ) , z = x , ( 3 4 , 1 4 ) , z = y , then ( C ∧ D ) ( z ) = { ( 1 4 , 1 2 ) , z = x , ( 1 2 , 1 4 ) , z = y , Hence, T ( C ∧ D ) = 1 4 1 2 = T ( C ) ∧ T ( D ) , it follows that T is not an (L, M)-fuzzy topology on X.

(2) (i) Let τ be an (L, M)-fuzzy topology on X. For any E, define a mapping τ s : L X ⟶ M as follows: τ s ( E ) = ⋁ F ⩽ E { τ ( F ) ∧ ⋀ x λ ⪡ E ⋀ x λ G , F ⩽ G [ τ ( G ′ ) ] ′ } . Then τ s is called the (L, M)-fuzzy semiopen operator induced by τ (see [4,7, 4,7]).

(ii) Let τ be an (L, M)-fuzzy topology on X. For any E, define a mapping τ p : L X ⟶ M as follows: τ p ( E ) = ⋀ x λ ⪡ E ⋁ x λ ⪡ F { τ ( F ) ∧ ⋀ y μ ⪡ F ⋀ y μ G , E ⩽ G [ τ ( G ′ ) ] ′ } . Then τ p is called the (L, M)-fuzzy preopen operator induced by τ (see [4,7, 4,7]).

(iii) Let τ be an (L, M)-fuzzy topology on X. For any E, define a mapping τ sp : L X ⟶ M as follows: τ sp ( E ) = ⋁ F ⩽ E { τ p ( F ) ∧ ⋀ x λ ⪡ E ⋀ x λ G , F ⩽ G [ τ ( G ′ ) ] ′ } . Then τ sp is called the (L, M)-fuzzy semi-pre open operator induced by τ (see[2,4,7, 2,4,7]).

It is easy to check that the above three operators and (L, M)-fuzzy topology are special (L, M)-fuzzy supratopology.

Definition 1.3. [12] Let { ( X i , T i ) } i ∈ I be a family of (L, M)-FsupraTOP-objects, let X = ∏ i ∈ I X i , and let p _i  : X ⟶ X _i be the i-th projection. The (L, M)-fuzzy product supratopology on X is denoted ∏ i ∈ I T i , is the weakest (L, M)-fuzzy supratopology on X such that p _i is continuous. The pair ( X , ∏ i ∈ I T i ) is called the product supratopological space of { ( X i , T i ) } i ∈ I .

Theorem 1.4. [12] Let { ( X i , T i ) } i ∈ I be a family of (L, M)-FsupraTOP-objects, let X be a nonempty set, and let f _i  : X ⟶ X _i be a mapping for each i ∈ I. Define T : L X ⟶ M by T ( A ) = ⋀ x λ ⪡ A ⋁ J ∈ 2 ( I ) { ⋀ j ∈ J ⋁ ( f j ( x ) ) λ ⩽ B j ⩽ A j T j ( B j ) } , where I is an index set, and f j ← ( A j ) ⩽ A for each j ∈ J. Then

(1) T is the unique (L, M)-fuzzy supratopological space on X such that each f _i is continuous for each i ∈ I.

(2) If ( Y , T Y ) is an (L, M)-fuzzy supratopological space and g : ( Y , T Y ) ⟶ ( X , T ) a mapping, then g is continuous iff f _i  ∘ g is continuous for each i ∈ I (see Fig.1).

OPEN IN VIEWER

Fig.1

Commutative diagram.

By Theorem 1.4, we can obtain Corollary 1.5 as follows:

Corollary 1.5. [12] Let { ( X i , T i ) } i ∈ I be a family of (L, M)-FsupraTOP-objects, let X = ∏ i ∈ I X i , and let p _i  : X ⟶ X _i be the i-th projection. Define T : L X ⟶ M by T ( A ) = ⋀ x λ ⪡ A ⋁ J ∈ 2 ( I ) { ⋀ j ∈ J ⋁ ( p j ( x ) ) λ ⩽ B j ⩽ A j T j ( B j ) } , where I is an index set, and p j ← ( A j ) ⩽ A for each j ∈ J. Then ( X , ∏ i ∈ I T i ) is the product supratopological space of { ( X i , T i ) } i ∈ I .

Definition 1.6. ([8, 9]) An L-fuzzy inclusion on X is a mapping ⊂ ˜ : L X × L X ⟶ L defined by the equality ⊂ ˜ ( E , F ) = ⋀ x ∈ X ( E ′ ( x ) ∨ F ( x ) ) .

In the sequel, we will write [ E ⊂ ˜ F ] instead of ⊂ ˜ ( E , F ) .

Definition 1.7. ([7]) Let ( X , T ) be an L-fuzzy supratopological space. For any A ∈ L ^X , define COM T : L X ⟶ L as follows: COM T ( A ) = ⋀ G ⊆ L X { T ( G ) ∧ [ A ⊂ ˜ ⋁ G ] → ⋁ H ∈ 2 ( G ) [ A ⊂ ˜ ⋁ H ] } . Then COM T ( A ) is called the degree of fuzzy compactness of A with respect to T .

Lemma 1.8. ([7]) Let ( X , T ) be an L-fuzzy supratopological space and A ∈ L ^X . For any G ⊆ L X , if T ( G ) ∧ [ A ⊂ ˜ ⋁ G ] ∧ a ⩽ ⋁ H ∈ 2 ( G ) [ A ⊂ ˜ ⋁ H ] . Then COM T ( A ) ⩾ a .

Lemma 1.9. ([7]) Let f : X ⟶ Y be a set mapping. T X be an L-fuzzy supratopology on X, T Y be an L-fuzzy supratopology on Y, and f : ( X , T X ) ⟶ ( Y , T Y ) be continuous. Then ∀A ∈ L ^X , COM T Y ( f → ( A ) ) ⩾ COM T X ( A ) .

The main results are as follows:

Theorem 1.10. (Generalized Tychonoff theorem)

(1) Let ( X , T ) be the L-fuzzy product supratopological space of { ( X i , T i ) } i ∈ I , then ∀ A = ∏ i ∈ I A i ∈ L ∏ i ∈ I X i , COM T ( A ) ⩾ ⋀ i ∈ I COM T i ( A i ) , where A _i  ∈ L ^{X _i} for any i ∈ I.

(2) Let ( X , T ) be the L-fuzzy product supratopological space of { ( X i , T i ) } i ∈ I , then COM T ( 1 X ) = ⋀ i ∈ I COM T i ( 1 X j ) .

In this section, using the structures of (L, L)-fuzzy product supratopological spaces, we will give a proof of Theorem 1.10.

Proof. (1) Suppose that a ⪡ ⋀ i ∈ I COM T i ( A i ) , then a ⪡ COM T i ( A i ) for any i ∈ I .

Thus ∀i ∈ I, ∀ G i ⊆ L X i , we have that T i ( G i ) ∧ [ A i ⊂ ˜ ⋁ G i ] ∧ a ⩽ ⋁ H i ∈ 2 ( G i ) [ A i ⊂ ˜ ⋁ H i ] . By Lemma 1.8, we only need to prove T ( G ) ∧ [ A ⊂ ˜ ⋁ G ] ∧ a ⩽ ⋁ H ∈ 2 ( G ) [ A ⊂ ˜ ⋁ H ] for any G ⊆ L X .

Suppose that b ∈ β ( T ( G ) ∧ [ A ⊂ ˜ ⋁ G ] ∧ a ) , then b ∈ β ( T ( G ) ) , b ∈ β ( [ A ⊂ ˜ ⋁ G ] ) , and b ∈ β (a).

Notice that, for any C ∈ G , T ( C ) = ⋀ x λ ⪡ C ⋁ J ∈ 2 ( I ) { ⋀ j ∈ J ⋁ ( p j ( x ) ) λ ⩽ V j ⩽ C j T j ( V j ) } , where I is an index set, and p j ← ( C j ) ⩽ C for each j ∈ J.

Thus ∀x _λ  ⪡ C, there exists a finite J₀ of I and C _j  ∈ L ^{X _j} such that p j ← ( C j ) ⩽ C for each j ∈ J₀, and b ⩽ ⋀ j ∈ J 0 ⋁ ( p j ( x ) ) λ ⩽ V j ⩽ C j T j ( V j ) .

Furthermore, there exists V j x λ ∈ L X j such that p j → ( x λ ) ⩽ V j x λ ⩽ C j and b ⩽ T j ( V j x λ ) for each j ∈ J₀,

From the above proved, we can obtain the following result:

∀ G ⊆ L X , if b ⩽ T ( C ) for each C ∈ G , then there exists a finite J₀ of I and V _j  ∈ L ^{X _j} such that p j ← ( V j ) = C and b ⩽ T j ( V j ) for each j ∈ J₀ .

Let U j = { V j ∈ L X j ∣ p j ← ( V j ) = C , b ⩽ T j ( V j ) , C ∈ G } , and R j = { p j ← ( V j ) ∈ L X ∣ V j ∈ L X j , p j ← ( V j ) = C , b ⩽ T j ( V j ) , C ∈ G } , where ∀j ∈ J₀ ⊆ I .

So, ∀j ∈ J₀, ∀ U j ⊆ L X j , we can obtain T j ( U j ) ∧ [ A j ⊂ ˜ ⋁ U j ] ∧ a ⩽ ⋁ V j ∈ 2 ( U j ) [ A j ⊂ ˜ ⋁ V j ] , and G = ⋃ j ∈ J 0 R j .

Meanwhile, ∀x ∈ X, we can obtain b ⪡ A ′ ( x ) ∨ ⋁ C ∈ G C ( x ) = ⋁ i ∈ I A i ′ ( x i ) ∨ ⋁ j ∈ J 0 ⋁ B j ∈ U j p j ← ( B j ) ( x ) = ⋁ i ∈ I A i ′ ( x i ) ∨ ⋁ j ∈ J 0 ⋁ B j ∈ U j B j ( x j ) = ⋁ j ∉ J 0 A j ′ ( x j ) ∨ ⋁ j ∈ J 0 ( A j ′ ( x j ) ∨ ⋁ B j ∈ U j B j ( x j ) ) .

Case 1:  If b ∈ β ( ⋁ i ∈ I A i ′ ( x i ) ) for any x ∈ X, then b ⩽ ⋁ H ∈ 2 ( G ) [ A ⊂ ˜ ⋁ H ] . In this case, we have that T ( G ) ∧ [ A ⊂ ˜ ⋁ G ] ∧ a ⩽ ⋁ H ∈ 2 ( G ) [ A ⊂ ˜ ⋁ H ] ( ∀ G ⊆ L X ) .

Case 2:  If there exists x = {x _i } _i∈I ∈ X such that b ∉ β ( ⋁ i ∈ I A i ′ ( x i ) ) , then b ∉ β ( A i ′ ( x i ) ) for each i ∈ I.

Now, we show that there exists j ∈ J₀ such that b ⪡ A j ′ ( y j ) ∨ ⋁ B j ∈ U j B j ( y j ) for any y _j  ∈ X _j . If not, ∀j ∈ J₀, there exists y _j  ∈ X _j such that b ∉ β ( A j ′ ( y j ) ∨ ⋁ B j ∈ U j B j ( y j ) ) . Hence, b ∉ β ( A j ′ ( y j ) ) and b ∉ β ( ⋁ B j ∈ U j B j ( y j ) ) . We take z = {z _j } _j∈I such that z _j  = y _j when j ∈ J₀, z _j  = x _j otherwise. Obviously, A ′ ( z ) = ( ⋁ j ∈ J 0 A j ′ ( y j ) ) ∨ ( ⋁ j ∉ J 0 A j ′ ( x j ) ) , and b ∉ β (A′ (z)) .

By b ∉ β ( ⋁ B j ∈ U j B j ( y j ) ) for any j ∈ J₀, we know that b ∉ β ( ⋁ j ∈ J 0 ⋁ B j ∈ U j B j ( y j ) ) = β ( ⋁ j ∈ J 0 ⋁ B j ∈ U j p j ← ( B j ) ( z ) ) = β ( ⋁ j ∈ J 0 ⋁ C ∈ R j C ( z ) ) = β ( ⋁ C ∈ G C ( z ) ) . Hence b ∉ β ( A ′ ( z ) ∨ ⋁ C ∈ G C ( z ) ) . This yields a contradiction.

Thus we obtain the proof that there exists j ∈ J₀ such that b ⪡ A j ′ ( y j ) ∨ ⋁ B j ∈ U j B j ( y j ) for any y _j  ∈ X _j . This shows b ⩽ [ A j ⊂ ˜ ⋁ U j ] .

Therefore, b ⩽ T j ( U j ) ∧ [ A j ⊂ ˜ ⋁ U j ] ∧ a ⩽ ⋁ V j ∈ 2 ( U j ) [ A j ⊂ ˜ ⋁ V j ] = ⋁ V j ∈ 2 ( U j ) ⋀ y j ∈ X j ( A j ′ ∨ ⋁ V j ) ( y j ) ⩽ ⋁ V j ∈ 2 ( U j ) ⋀ y ∈ X ( p j ← ( A j ′ ) ∨ ⋁ D ∈ V j p j ← ( D ) ) ( y ) ⩽ ⋁ V j ∈ 2 ( U j ) ⋀ y ∈ X ( A ′ ∨ ⋁ D ∈ V j p j ← ( D ) ) ( y ) ⩽ ⋁ D j ∈ 2 ( R j ) ⋀ y ∈ X ( A ′ ∨ ⋁ D j ) ( y ) ⩽ ⋁ H ∈ 2 ( G ) [ A ⊂ ˜ ⋁ H ] .

In this case, we also have that T ( G ) ∧ [ A ⊂ ˜ ⋁ G ] ∧ a ⩽ ⋁ H ∈ 2 ( G ) [ A ⊂ ˜ ⋁ H ] ( ∀ G ⊆ L X ) .

By cases 1 and 2, we know that COM T ( G ) ⩾ a , Hence, COM T ( G ) ⩾ ⋀ j ∈ J COM T j ( A j ) .

(2) By (1) and Lemma 1.9, we can easily obtained the result, so we omit it.□