Characterizations of L -topologies

1 Introduction

In 1968, the concept of fuzzy topologies was introduced by Chang [1]. Later it was generalized to L-fuzzy setting [7, 11] by many authors.

In order to characterize L-topologies, the notions of Q-neighborhood systems, L-interior operators and L-closure operators are introduced in [2, 15]. Then a natural problem is that can L-topologies be characterized by some order relations? In this paper, we shall answer this problem.

In Section 3 and Section 4, the notions of L-internal relations and L-enclosed relations are introduced which can be used to characterize L-topologies. Also, we introduce the notions of L-internal relation dual-preserving mappings (L-enclosed relation dual-preserving mappings) between two L-internal relation spaces (L-enclosed relation spaces). They can form two categories, called the category of L-internal relation spaces and the category ofL-enclosed relation spaces, respectively. It is proved that the category of L-internal relation spaces, the category of L-enclosed relation spaces and the category of L-topological spaces are isomorphic.

2 Preliminaries

Throughout this paper, (L, ∨ , ∧ , ′) is a completely distributive lattice with an order-reversing involution ^′. X is a nonempty set. L ^X is the set of all L-fuzzy sets [6] (L-sets for short) on X. The smallest element and the largest element in L ^X are denoted by 0 _ and 1 _ respectively.

The set of all non-zero co-prime elements in L is denoted by J (L). The set of all non-zero co-prime elements in L ^X is denoted by J (L ^X ). By Gierz [5] and Wang [24] we know that each element in L is a supremum of some elements in J (L). It is easy to check that J (L ^X ) ={ x _λ  ∣ x ∈ X, λ ∈ J (L) } where x _λ denotes an L-fuzzy point.

Let L be a complete lattice and let a, b ∈ L. We say that a is way below (wedge below) b in symbols a ≪ b (a ⊲ b) if and only if for every directed (arbitrary) subset D ⊆ L, the relation b ⩽ ⋁ D always implies the existence of d ∈ D with a ⩽ d [3]. If a is a co-prime, we have a ⊲ b if and only if a ≪ b . L is called continuous if b = ⋁ {a ∈ L ∣ a ≪ b} for all b ∈ L [5]. L is completely distributive if and only if L is a continuous lattice and each element is the supremum of co-primes.

According to [23], we know that in a completely distributive lattice L, each element has a minimal family, hence each element a has the greatest minimal family, denoted by β (a). It is easy to see that β^* (a) = β (a) ∩ J (L) is also a minimal family of a. In fact, for each b ∈ J (L), b ∈ β^* (a) if and only if b ⊲ a. Some properties of the mapping β can be found in [14, 23].

An L-topological space is a pair (X, τ), where τ is a subfamily of L ^X which contains 0 _ , 1 _ and is closed for any suprema and finite infima. τ is called anL-topology on X. Each member of τ is called an open L-set and its quasi-complement is called a closedL-set. A closed L-set P is called a remote-neighborhood (R-neighborhood for short) of x _λ  ∈ J (L ^X ) if x _λ ≰P. The set of all closed R-neighborhood of x _λ is denoted by η^- (x _λ ) [23].

A mapping f : X ⟶ Y induces a mapping f L → : L X ⟶ L Y (called an L-valued Zadeh function or an L-fuzzy mapping or an L-forward powerset operator [16]), which is defined by ∀A ∈ L ^X , y ∈ Y, f L → ( A ) ( y ) = ⋁ { A ( x ) ∣ f ( x ) = y } .

The right adjoint of f L → (called an L-backward powerset operator [16]), is denoted f L ← and given by f L ← ( B ) = B ∘ f , ∀ B ∈ L Y .

It is known that f L → preserves arbitrary suprema and f L ← preserves arbitrary suprema, arbitrary infima and quasi-complements [16].

Definition 2.1. [23] Let (X, τ₁) and (Y, τ₂) be two L-topological spaces. A mapping f : (X, τ₁) ⟶ (Y, τ₂) is called

an L-continuous mapping if f L ← ( V ) ∈ τ 1 for each V ∈ τ₂;

The category whose objects are L-topological spaces and whose morphisms are L-continuous mappings will be denoted by L-TOP.

Definition 2.2. [21] A mapping φ : J (L ^X ) ⟶ L ^X is called a remote-neighborhood mapping (R-mapping for short) on L ^X , if for all x _λ  ∈ J (L ^X ), x _λ ≰φ (x _λ ). The set of all R-mappings on L ^X is denoted by R ( L X ) .

Definition 2.3. [18] A pointwise quasi-uniformity on X is a nonempty subset U of R ( L X ) satisfying the following conditions:

φ ∈ R ( L X ) , ψ ∈ U , φ ⩽ ψ ⇒ φ ∈ U ;

A ⊆ U is called a basis of U , if for each φ ∈ U , there exists a ψ ∈ A such that φ ⩽ ψ. B ⊆ U is called a sub-basis of U , if the set of all finite supremum of elements in B is a basis of U .

If U is a pointwise quasi-uniformity on X, then we call ( X , U ) a pointwise quasi-uniform space.

Theorem 2.4. [20] Let ( X , U ) be a pointwise quasi-uniform space. For each A ∈ L ^X , let int U ( A ) = ⋁ φ ∈ U ⋀ x λ ≰ A φ ( x λ ) .

Then int U : L X ⟶ L X is an L-interior operator. Hence it induces an L-topology on X, denoted by τ ((U)) .

Theorem 2.5. [18, 19] Let ( X , U ) be a pointwise quasi-uniform space. For each A ∈ L ^X , let cl U ( A ) = ⋁ { x λ ∈ J ( L X ) ∣ ∀ φ ∈ U , A ≰ φ ( x λ ) } .

Then cl U : L X ⟶ L X is an L-closure operator. Hence it induces an L-topology on X, denoted by η ((U)).

Definition 2.6. [22] pointwise S-quasi-proximity on X is a mapping δ : J (L ^X ) × L ^X  ⟶ {0, 1} satisfying the following conditions:

For all x _λ  ∈ J (L ^X ), δ ( x λ , 0 _ ) = 0 .

In this case, δ is a pointwise S-quasi-proximity on X, (X, δ) is called a pointwise S-quasi-proximity space.

Theorem 2.7. [22] Let (X, δ) be a pointwise S-quasi-proximity space. For each A ∈ L ^X , let int δ ( A ) = ⋁ { B ∈ L X ∣ ∀ x λ ≰ A , δ ( x λ , B ) = 0 } .

Then int _δ  : L ^X  ⟶ L ^X is an L-interior operator. Hence it induces an L-topology on X, denoted by τ (δ) .

Theorem 2.8. [22] Let (X, δ) be a pointwise S-quasi-proximity space. For each A ∈ L ^X , let cl δ ( A ) = ⋁ { x λ ∈ J ( L X ) ∣ δ ( x λ , A ) = 1 } .

Then cl _δ  : L ^X  ⟶ L ^X is an L-closure operator. Hence it induces an L-topology on X, denoted by η (δ) .

Definition 2.9. [4] A binary relation ≺ on a complete lattice L is called an auxiliary relation, or an auxiliary order, if it satisfies the following conditions for all u, x, y, z:

if the smallest element 0 exists, then 0 ≺ x;

Definition 2.10. [4] We call an auxiliary relation ≺ on L ^X an approximating, if A = ⋁ {B ∈ L ^X  ∣ B ≺ A} for all A ∈ L ^X .

Definition 2.11. [4] We say that an auxiliary relation ≺ on L ^X satisfies the interpolation property, if A ≺ B implies that there exists a C ∈ L ^X such that A ≺ C ≺ B for all A, B ∈ L ^X .

Definition 2.12. [14] An L-topological space (X, τ) is called L-T₀ (sub-T₀), if for all x, y ∈ X with x ≠ y, there exists a U ∈ τ such thatU (x) ≠ U (y) .

Definition 2.13. [13] An L-topological space (X, τ) is called L-T₁, if for all x, y ∈ X with x ≠ y, there exist U, V ∈ τ such that U (x) ≰U (y) andV (y) ≰V (x) .

Definition 2.14. [21] An L-topological space (X, τ) is called T₂, if for all x _λ , y _μ  ∈ J (L ^X ) with x _λ ≰y _μ , there exist a closed L-set P and an open L-set Q such that x _λ ≰P ⩾ Q ⩾ y _μ  .

Definition 2.15. [11] An L-topological space (X, τ) is regular if each open L-set A is the suprema of some open L-sets B with B^- ⩽ A.

Definition 2.16. [10] An L-topological space (X, τ) is normal if for each closed L-set A and each open L-set B with A ⩽ B, there exists an L-set D suchthat A ⩽ D ∘ ⩽ D - ⩽ B .

3 L-internal relation spaces

In this section, our aim is to introduce the notion of L-internal relations on L ^X which can be used to characterize L-topologies on X.

Definition 3.1. A binary relation ≼ on L ^X is called an L-internal relation or an L-internal order on L ^X if it satisfies the following conditions:

1 _ ≼ 1 _ ;

In this case, ≼ is an L-internal relation on L ^X , (X, ≼) is called an L-internal relation space or an L-internal order space.

Remark 3.2. It is easy to see that an L-internal order is different from auxiliary order on L ^X .

From (LIO3) and (LIO4) in Definition 3.1 we can easily obtain the following lemma.

Lemma 3.3. Let ≼ be an L-internal relation on L ^X . If A ⩽ B≼C ⩽ D, then A≼D.

Lemma 3.4. Let ≼ be an L-internal relation on L ^X . If A≼B, then there exists a C ∈ L ^X such that A ⩽ C≼C ⩽ B.

Proof. Let D = { D ∈ L X ∣ D ≼ B } and let C = ⋁ { D ∈ L X ∣ D ∈ D } . Obviously we have A ⩽ C since A is an element of D . Moreover from (LIO3) in Definition 3.1 we can obtain C≼B, this implies that C is the maximum L-set in D . By (LIO5) in Definition 3.1 we know that there exists an E ∈ L ^X such that C≼E≼B. By the definition of C we know E ⩽ C. From (LIO2) in Definition 3.1 and C≼E we have C ⩽ E. Hence E = C. A ⩽ C≼C ⩽ B is proved. □

The following two theorems show that we can obtain two L-internal relations from a pointwise quasi-uniformity and a pointwise S-quasi-proximity on a set.

Theorem 3.5. Let ( X , U ) be a pointwise quasi-uniform space. For all A, B ∈ L ^X , we define A ≼ B ⇔ ∃ φ ∈ U such that B ′ ⩽ ⋀ x λ ≰ A ′ φ ( x λ ) .

Then ≼ is an L-internal relation.

Proof. It is trivial that ≼ satisfies (LIO1) and (LIO4).

(LIO2) can be obtained by the following fact: x _λ ≰φ (x _λ ) for all φ ∈ U .

(LIO3) holds by the following fact.

Suppose A = ⋁ { A _i  ∣ A _i  ∈ L ^X , i ∈ Ω } . In order to prove that A _i ≼B implies A≼B, we need to prove that there exists a φ ∈ U such that B′ ⩽ ⋀ { φ (z _ν ) ∣ z _ν ≰A′ } . For all z _ν ≰A′, it is obvious that there exists i₀ ∈ Ω such that z ν ≰ A i 0 ′ . Since A _i ≼B, we know that there exist φ i ∈ U such that B ′ ⩽ ⋀ { φ i ( x λ ) ∣ x λ ≰ A i ′ } for each i ∈ Ω. This implies B ′ ⩽ ⋀ { φ i 0 ( x λ ) ∣ x λ ≰ A i 0 ′ } ⩽ φ i 0 ( z ν ) .

Hence we have B′ ⩽ ⋀ {φ _{i 0}  (z _ν ) ∣ z _ν ≰A′} . This shows A≼B .

Conversely, by A≼B we know that there exists a φ ∈ U such that B′ ⩽ ⋀ {φ (x _λ ) ∣ x _λ ≰A′} . For all x λ ≰ A i ′ , it is obvious that x _λ ≰A′. This implies B ′ ⩽ ⋀ { φ ( x λ ) ∣ x λ ≰ A ′ } ⩽ ⋀ { φ ( x λ ) ∣ x λ ≰ A i ′ } .

Thus we can obtain A _i ≼B for each i ∈ Ω .

In order to prove that ≼ satisfies (LIO5), suppose that A≼B holds. Then there exists a φ ∈ U such that B′ ⩽ ⋀ {φ (x _λ ) ∣ x _λ ≰A′} . Take ψ ∈ U such that ψ ⊙ ψ ⩾ φ. Let C = ⋁ {ψ (x _λ ) ′ ∣ x _λ ≰A′}. Obviously A≼C holds. From the following fact, B ′ ⩽ ⋀ x λ ≰ A ′ φ ( x λ ) ⩽ ⋀ x λ ≰ A ′ ( ψ ⊙ ψ ) ( x λ ) = ⋀ x λ ≰ A ′ ⋀ { ψ ( y μ ) ∣ y μ ≰ ψ ( x λ ) } ⩽ ⋀ { ψ ( z ν ) ∣ z ν ≰ C ′ } , we obtain C≼B. (LIO5) is proved. □

Theorem 3.6. Let (X, δ) be a pointwise S-quasi-proximity space. For all A, B ∈ L ^X , we define A ≼ B ⇔ ∀ x λ ≰ A ′ , δ ( x λ , B ′ ) = 0 .

Then ≼ is an L-internal relation.

Proof. (LIO1) and (LIO2) can respectively obtain from (SP1) and (SP4) in Definition 2.6.

(LIO3) holds by the following fact.

Suppose A = ⋁ {A _i  ∣ A _i  ∈ L ^X , i ∈ Ω} . In order to prove that for A _i ≼B implies A≼B, we need to prove that for all z _ν ≰A′, δ (z _ν , B′) =0 . Since for all z _ν ≰A′, there exits i₀ ∈ Ω such that z ν ≰ A i 0 ′ , by A _i ≼B we know that for all x λ ≰ A i ′ , δ ( x λ , B ′ ) = 0 for each i ∈ Ω . This implies for all z ν ≰ A i 0 ′ , δ (z _ν , B′) =0 . Hence for all z _ν ≰A′, δ (z _ν , B′) =0 . This shows A≼B .

Conversely, we need to prove that for all x λ ≰ A i ′ , δ ( x λ , B ′ ) = 0 . For all x λ ≰ A i ′ , it is obvious that x _λ ≰A′. This is A≼B. Hence A _i ≼B for each i ∈ Ω .

(LIO4) holds by the following fact.

Let A≼B ∧ C. Then for all x _λ ≰A′, δ (x _λ , (B ∧ C) ′) =0 . Hence δ ( x λ , ( B ∧ C ) ′ ) = δ ( x λ , B ′ ) ∨ δ ( x λ , C ′ ) = 0 .

Therefore, we have δ (x _λ , B′) = δ (x _λ , C′) =0, i.e., A≼B and A≼C.

Conversely, we need to prove that A≼B and A≼C imply A≼B ∧ C. By the definitions of A≼B and A≼C, we know that for all x _λ ≰A′, δ (x _λ , B′) =0 and δ (x _λ , C′) =0 . Hence δ ( x λ , ( B ∧ C ) ′ ) = δ ( x λ , B ′ ) ∨ δ ( x λ , C ′ ) = 0 .

This shows A≼B ∧ C .

In order to prove that ≼ satisfies (LIO5), suppose that A≼B holds. Then for each x _λ ≰A′, δ (x _λ , B′) =0 . From (SP3) in Definition 2.6, we know that there exist C _λ  ∈ L ^X such that δ (x _λ , C _λ ) =0 and δ (x _μ , B′) =0 for all x _μ ≰C _λ  . Let C = ⋁ { C λ ′ ∣ x λ ≰ A ′ } . Obviously C≼B holds. From the following fact, for all x _λ ≰A′, δ ( x λ , C ′ ) = δ ( x λ , ⋀ x λ ≰ A ′ C λ ) ⩽ ⋀ x λ ≰ A ′ δ ( x λ , C λ ) = 0 . we obtain A≼C. (LIO5) is proved. □

The following theorem shows that we can obtain an L-internal relation from an L-topology. It also shows that an L-topological space can be regarded as an L-internal relation space.

Theorem 3.7. Let (X, τ) be an L-topological space. For any A, B ∈ L ^X , define A_≼τB if and only if A ⩽ B°. Then _≼τ is an L-internal relation.

Proof. In order to prove _≼τ being an L-internal relation, we need to prove that ≼ satisfies the conditions (LIO1)–(LIO5). (LIO1)–(LIO4) are obvious. Now we show (LIO5). Suppose A, B ∈ L ^X with A≼B. Then A ⩽ B°. Let C = B°. It is easy to see that A≼C≼B holds. □

The following theorem shows that we can obtain an L-topology from an L-internal relation.

Theorem 3.8. Let ≼ be an L-internal relation on L ^X . For all A ∈ L ^X , we define int ≼ ( A ) = ⋁ { G ∈ L X ∣ G ≼ A } .

Then the following assertions hold.

For each G ∈ L ^X , G ⩽ int_≼ (A) holds iff G≼A holds.

Proof. (1) By (LIO3) we can prove that for any G ∈ L ^X , G ⩽ int_≼ (A) holds if and only if G≼A holds.

(2) From (LIO1) and (LIO2) we know that int ≼ ( 1 _ ) = 1 _ , and for any A ∈ L ^X , int (A) ⩽ A. By (LIO4) we can obtain int_≼ (A ∧ B) ⩽ int_≼ (A) ∧ int_≼ (B). To prove int_≼ (A) ∧ int_≼ (B) ⩽ int_≼ (A ∧ B), let G ⩽ int_≼ (A) ∧ int_≼ (B). Then G⩽int_≼ (A) and G ⩽ int_≼ (B). Hence G≼A and G≼B. By (LIO4) we obtain G≼A ∧ B, this implies G ⩽ int_≼ (A ∧ B). Thus we have int_≼ (A) ∧ int_≼ (B) ⩽ int_≼ (A ∧ B). Therefore int_≼ (A) ∧int_≼ (B) = int_≼ (A ∧ B). In order to prove int_≼ (int_≼ (A)) = int_≼ (A), we only need to prove int_≼ (int_≼ (A)) ⩾ int_≼ (A). Suppose G ⩽ int_≼ (A). Then G≼A. By (LIO5) we know that there exists a C ∈ L ^X such that G≼C≼A. This implies G ⩽ int_≼ (C) and C ⩽ int_≼ (A). Thus we obtain G ⩽ int_≼ (C) ⩽ int_≼ (int_≼ (A)). Therefore int_≼ (int_≼ (A)) ⩾ int_≼ (A) holds. □

Corollary 3.9. For an L-internal relation ≼ on L ^X , τ_≼ is the set of all fixed L-sets of ≼, i.e., τ_≼ ={ A ∈ L ^X  ∣ A≼A }.

From Theorems 3.7 and 3.8 we obtain the following two theorems.

Theorem 3.10. For an L-internal relation ≼ on L ^X , we have _≼τ≼ = ≼.

Theorem 3.11. For an L-topological space (X, τ), we have τ _≼τ  = τ.

It is easy to see that an L-internal relation ≼ is an auxiliary order on L ^X satisfying the interpolation property, but in general, it is not approximating. If it is approximating, then what will happen?

By Corollary 3.9, Theorems 3.10 and 3.11, we can easily obtain the following result.

Corollary 3.12. An L-internal relation ≼ on L ^X is approximating if and only if τ_≼ is discrete.

Now we consider mappings between L-internal relation spaces.

Definition 3.13. Let (X, _≼1) and (Y, _≼2) be two L-internal relation spaces. A mapping f : (X, _≼1) ⟶ (Y, _≼2) is said to be:

an L-internal relation dual-preserving mapping (L-IRDP mapping for short), if for any U, V ∈ L ^Y , U_≼2V implies f L ← ( U ) ≼ 1 f L ← ( V ) ;

The following theorem can be easily proved.

Theorem 3.14. (1) If f : (X, _≼1) ⟶ (Y, _≼2) and g : (Y, _≼2) → (Z, _≼3) are two L-IRDP mappings, then g ∘ f : (X, _≼1) ⟶ (Z, _≼3) is also an L-IRDP mapping.

(2) If f : (X, _≼1) ⟶ (Y, _≼2) and g : (Y, _≼2) ⟶ (Z, _≼3) are two L-IRP mappings, then g ∘ f : (X, _≼1) ⟶ (Z, _≼3) is also an L-IRP mapping.

It is easy to prove that all L-internal relation spaces and their L-IRDP mappings form a category, which is called the category of L-internal relation spaces, denoted by L-IRS. Now we consider the relation between the categories L-TOP and L-IRS.

Theorem 3.15. (1) If f : (X, _≼1) ⟶ (Y, _≼2) is an L-IRDP mapping, then f : (X, τ _≼1 ) ⟶ (Y, τ _≼2 ) is an L-continuous mapping.

(2) If f : (X, _≼1) ⟶ (Y, _≼2) is an L-IRP mapping, then f : (X, τ _≼1 ) ⟶ (Y, τ _≼2 ) is an L-open mapping.

Proof. (1) Let U be an open L-set in (Y, τ _≼2 ). Then by Theorem 3.8 we know U = int _≼2  (U) =⋁ { G ∈ L ^Y  ∣ G_≼2U }. Hence f L ← ( U ) = ⋁ { f L ← ( G ) ∣ G ∈ L Y , G ≼ 2 U } . Since G_≼2U implies f L ← ( G ) ≼ 1 f L ← ( U ) , we have f L ← ( U ) = ⋁ { f L ← ( G ) ∣ G ∈ L Y , G ≼ 2 U } ⩽ ⋁ { E ∈ L X ∣ E ≼ 1 f L ← ( U ) } ⩽ f L ← ( U ) .

Therefore f L ← ( U ) = ⋁ { E ∈ L X ∣ E ≼ 1 f L ← ( U ) } = int ≼ 1 ( f L ← ( U ) ) , i.e., f L ← ( U ) is an open L-set in (X, τ _≼1 ). The proof of (1) is completed.

The proof of (2) is dually analogous to (1). □

Theorem 3.16. (1) If f : (X, τ₁) ⟶ (Y, τ₂) is an L-continuous mapping, then f : (X, _≼τ1) ⟶ (Y, _≼τ2) is an L-IRDP mapping.

(2) If f : (X, τ₁) ⟶ (Y, τ₂) is an L-open mapping, then f : (X, _≼τ1) ⟶ (Y, _≼τ2) is an L-IRP mapping.

Proof. (1) Let U, V ∈ L ^Y with U_≼τ2V. Then U ⩽ V°. Hence f L ← ( U ) ⩽ f L ← ( V ∘ ) . Since f is an L-continuous mapping, we have f L ← ( V ∘ ) ⩽ f L ← ( V ) ∘ . Thus we obtain f L ← ( U ) ⩽ f L ← ( V ∘ ) ⩽ f L ← ( V ) ∘ . This shows f L ← ( U ) ≼ τ 1 f L ← ( V ) . Therefore f : (X, _≼τ1) ⟶ (Y, _≼τ2) is an L-IRDP mapping.

The proof of (2) is dually analogous to (1). □

Now we define a functor 𝕋 : L-IRS ⟶ L-TOP such that 𝕋 ( X , ≼ ) = ( X , τ ≼ ) , 𝕋 ( f ) = f .

By Theorems 3.10, 3.11, 3.15 and 3.16, we can obtain the following theorem.

Theorem 3.17. 𝕋 : L-IRS ⟶ L-TOP is an isomorphism, that is, the category of L-internal relation spaces is isomorphic to the category of L-topological spaces.

4 L-enclosed relation spaces

In this section, our aim is to introduce the notion of L-enclosed relations on L ^X which can be used to characterize L-topologies on X.

Definition 4.1. A binary relation ⋞ on L ^X is called an L-enclosed relation or an L-enclosed order if it satisfies the following conditions:

0 _ ⋞ 0 _ ;

In this case, ⋞ is an L-enclosed relation on L ^X , (X, ⋞) is called an L-enclosed relation space.

From (LEO3) and (LEO4) in Definition 4.1 we easily obtain the following lemma.

Lemma 4.2. Let ⋞ be an L-enclosed relation on L ^X . If A ⩽ B⋞C ⩽ D, then A⋞D.

Lemma 4.3. Let ⋞ be an L-enclosed relation on L ^X . If A⋞B, then there exists a C ∈ L ^X such that A ⩽ C⋞C ⩽ B.

Proof. Let D = { D ∈ L X ∣ A ⋞ D } and let C = ⋀ { D ∈ L X ∣ D ∈ D } . Obviously we have C ⩽ B since B ∈ D . Moreover from (LEO3) in Definition 4.1 we can obtain A⋞C, this implies that C is the minimum L-set in D . By (LEO5) in Definition 4.1 we know that there exists an E ∈ L ^X such that A⋞E⋞C. By the definition of C we know E ⩾ C. From (LEO2) in Definition 4.1 and E⋞C we have E ⩽ C. Hence E = C. A ⩽ C⋞C ⩽ B is proved. □

Remark 4.4. From Definition 4.1 and Lemma 4.2 we easily check that an L-enclosed relation is an auxiliary order satisfying the interpolation property on L ^X . In general, however it is not approximating. Thiscan be seen from the following example.

Example 4.5. Let X be an arbitrary set containing more than one point and let x₀ be a point in X. For any B ∈ 2 ^X , we define a binary relation ⋞ on 2 ^X as follows. { A ∣ A ⋞ B } = { { ∅ } , if x 0 ∉ B ; 2 B , if x 0 ∈ B .

It is easy to check that ⋞ is an L-enclosed relation, but it is not approximating.

The following two theorems show that a pointwise quasi-uniform space and a pointwise S-quasi-proximity space can be regarded as L-enclosed relation spaces.

Theorem 4.6. Let ( X , U ) be a pointwise quasi-uniform space. For all A, B ∈ L ^X , we define A ⋞ B ⇔ ∃ φ ∈ U such that A ⩽ ⋀ y μ ≰ B φ ( y μ ) .

Then ⋞ is an L-enclosed relation.

Proof. It is straightforward that ⋞ satisfies (LEO1) and (LEO4).

(LEO2) can be obtained by the following fact: y _μ ≰φ (y _μ ) for all φ ∈ U .

(LEO3) holds by the following fact.

Suppose B = ⋀ { B _i  ∣ B _i  ∈ L ^X , i ∈ Ω } . In order to prove that A⋞B _i implies A⋞B, we need to prove that there exists a φ ∈ U such that A ⩽ ⋀ {φ (y _μ ) ∣ y _μ ≰B} . For all y _μ ≰B, there exists i₀ ∈ Ω such that y _μ ≰B _{i 0} . Since A⋞B _i , we can obtain that there exist φ i ∈ U such that A ⩽ ⋀ {φ _i  (z _ν ) ∣ z _ν ≰B _i } for each i ∈ Ω. This implies A ⩽ ⋀ { φ i 0 ( z ν ) ∣ z ν ≰ B i 0 } ⩽ φ i 0 ( y μ ) .

Hence we have A ⩽ ⋀ {φ _{i 0}  (y _μ ) ∣ y _μ ≰B} . This shows A⋞B .

Conversely, by A⋞B we know that there exists a φ ∈ U such that A ⩽ ⋀ {φ (y _μ ) ∣ y _μ ≰B} . For all y _μ ≰B _i , it is obvious that y _μ ≰B. This implies A ⩽ ⋀ { φ ( y μ ) ∣ y μ ≰ B } ⩽ ⋀ { φ ( y μ ) ∣ y μ ≰ B i } .

Thus we can obtain A⋞B _i for each i ∈ Ω .

In order to prove that ≼ satisfies (LIO5), suppose that A≼B holds. Then there exists a φ ∈ U such that A ⩽ ⋀ _{y μ ≰B}φ (y _μ ) . Take ψ ∈ U such that ψ ⊙ ψ ⩾ φ. Let C = ⋀ {ψ (y _μ ) ∣ y _μ ≰B}. Obviously C≼B holds. From the following fact, A ⩽ ⋀ y μ ≰ B φ ( y μ ) ⩽ ⋀ y μ ≰ B ( ψ ⊙ ψ ) ( y μ ) = ⋀ y μ ≰ B ⋀ { ψ ( z ν ) ∣ z ν ≰ ψ ( y μ ) } ⩽ ⋀ { ψ ( x λ ) ∣ x λ ≰ C } , we obtain A≼C. (LIO5) is proved. □

Theorem 4.7. Let (X, δ) be a pointwise S-quasi-proximity space. For all A, B ∈ L ^X , we define A ⋞ B ⇔ ∀ y μ ≰ B , δ ( y μ , A ) = 0 .

Then ⋞ is an L-enclosed relation.

Proof. It is easy to check that ⋞ satisfies (LEO1) and (LEO2) by (SP1) and (SP4) in Definition 2.6, respectively.

(LEO3) holds by the following fact. Suppose B = ⋀ { B _i  ∣ B _i  ∈ L ^X , i ∈ Ω } . In order to prove that for A⋞B _i implies A⋞B, we need to prove that for all y _μ ≰B, δ (y _μ , A) =0 . For all y _μ ≰B, there exits some i₀ ∈ Ω such that y _μ ≰B _{i 0} . Since A⋞B _i , it follows that for all z _ν ≰B _i , δ (z _ν , A) =0 for each i ∈ Ω . This implies for all y _μ ≰B _{i 0} , δ (y _μ , A) =0 . Hence for all y _μ ≰B, δ (y _μ , A) =0 . This shows A⋞B .

Conversely, we need to prove that for all z _ν ≰B _i , δ (z _ν , A) =0 . For all z _ν ≰B _i , it is obvious that z _ν ≰B. This is A⋞B . Hence A⋞B _i for each i ∈ Ω .

(LEO4) holds by the following fact.

Let A ∨ B⋞C, then for all y _μ ≰C, δ (y _μ , A ∨ B) =0 . Hence δ ( y μ , A ) ∨ ( y μ , B ) = δ ( y μ , A ∨ B ) = 0 .

Therefore, we have δ (y _μ , A) = δ (y _μ , B) =0, i.e., A⋞C and B⋞C.

Conversely, we need to prove that A⋞C and B⋞C imply A ∨ B⋞C. By the definitions of A⋞C and B⋞C,we know that for all y _μ ≰C, δ (y _μ , A) =0 and δ (y _μ , B) =0 . Hence we have δ (y _μ , A ∨ B) = δ (y _μ , A) ∨ δ (y _μ , B) =0 . This shows A ∨ B⋞C .

In order to prove that ⋞ satisfies (LEO5), suppose that A⋞B holds. Then for all y _μ ≰B, δ (y _μ , A) =0 . From (SP3) in definition 2.6, we know that there exists a C _μ  ∈ L ^X such that δ (y _μ , C _μ ) =0 and δ (y _λ , A) =0 for all y _λ ≰C _μ  . Let C = ⋀ {C _μ  ∣ y _μ ≰B}. Obviously, A≼C holds. From the following fact, for all y _μ ≰B, δ ( y μ , C ) = δ ( y μ , ⋀ y μ ≰ B C μ ) ⩽ ⋀ y μ ≰ B δ ( y μ , C μ ) = 0 , we obtain C≼B. (LEO5) is proved. □

The following theorem shows that we can obtain an L-enclosed relation from an L-topology. Its proof is easy.

Theorem 4.8. Let (X, τ) be an L-topological space. For all A, B ∈ L ^X , we define A⋞B if and only if A^- ⩽ B. Then ⋞ is an L-enclosed relation, written as _⋞τ.

The following theorem gives the links between L-internal relations and L-enclosed relations.

Theorem 4.9. For an L-internal relation ≼ on L ^X , we define a relation ⋞ on L ^X such that for all A, B ∈ L ^X , A⋞B if and only if B′≼A′. Then ⋞ is an L-enclosed relation on L ^X . Conversely, for an L-enclosed relation ⋞ on L ^X , we define an L-relation ≼ on L ^X such that for any A, B ∈ L ^X , A≼B if and only if B′⋞A′. Then ≼ is an L-internal relation on L ^X . In sequel, ⋞ is called the associate L-enclosed relation of the L-internal relation ≼.

Proof. This can easily be obtained. □

From an L-enclosed relation on X we can obtain an L-topology on X as follows.

Theorem 4.10. Let ⋞ be an L-enclosed relation on X. For all A ∈ L ^X , we define cl ⋞ ( A ) = ⋀ { G ∈ L X ∣ A ⋞ G } .

Then the following assertions hold.

For each G ∈ L ^X , cl_⋞ (A) ⩽ G holds if and only if A⋞G holds.

Proof. (1) Obviously we have A⋞G ⇒ cl_⋞ (A) ⩽ G. From (LEO3) of Definition 4.1 we know that A⋞cl_⋞ (A). Hence it follows that cl_⋞ (A) ⩽ G ⇒ A⋞G.

(2) From (LEO1) and (LEO2) we know that cl ⋞ ( 0 _ ) = 0 _ , and for any A ∈ L ^X , cl_⋞ (A) ⩾ A. By (LEO4) we can obtain that cl_⋞ (A) ∨ cl_⋞ (B) ⩽ cl_⋞ (A ∨ B). Additionally, let G ⩾ cl_⋞ (A) ∨ cl_⋞ (B), then G ⩾ cl_⋞ (A) and G ⩾ cl_⋞ (B). Hence A⋞G and B⋞G. By (LEO4) we obtain A ∨ B⋞G. This implies cl_⋞ (A ∨ B) ⩽ G. Hence cl_⋞ (A) ∨ cl_⋞ (B) ⩾ cl_⋞ (A ∨ B). Therefore cl_⋞ (A) ∨ cl_⋞ (B) = cl_⋞ (A ∨ B).

With the purpose of proving cl_⋞ (cl_⋞ (A)) = cl_⋞ (A), we only need to prove cl_⋞ (cl_⋞ (A)) ⩽ cl_⋞ (A). Let cl_⋞ (A) ⩽ G. Then A⋞G. By (LEO5) we know that there exists a C ∈ L ^X such that A⋞C⋞G. This implies cl_⋞ (C) ⩽ G and cl_⋞ (A) ⩽ C. Thus we have cl_⋞ (cl_⋞ (A)) ⩽ cl_⋞ (C) ⩽ G. Therefore cl_⋞ (cl_⋞ (A)) ⩽ cl_⋞ (A) holds. □

Corollary 4.11. For an L-enclosed relation ⋞ on L ^X , every closed L-set A in (X, τ_⋞) is a fixed L-set of ⋞, i.e., A⋞A.

From Theorems 4.8 and 4.10 we directly obtain the following two theorems.

Theorem 4.12. For an L-enclosed relation ⋞ on L ^X , we have _⋞τ⋞ = ⋞.

Theorem 4.13. For an L-topological space (X, τ), we have τ _⋞τ  = τ.

Now we consider mappings between two L-enclosed relation spaces.

Definition 4.14. Let (X, _⋞1) and (Y, _⋞2) be two L-enclosed relation spaces. A mapping f : (X, _⋞1) ⟶ (Y, _⋞2) is said to be

an L-enclosed relation dual-preserving mapping (L-ERDP mapping for short), if for any U, V ∈ L ^Y , U_⋞2V implies f L ← ( U ) ⋞ 1 f L ← ( V ) ;

The following theorem gives the relation between L-IRDP mappings and L-ERDP mappings.

Theorem 4.15. Let (X, _≼1) and (Y, _≼2) be twoL-internal relation spaces, and let _⋞1 and _⋞2 are respectively the associate L-enclosed relations of _≼1 and _≼2. Then a mapping f : (X, _≼1) ⟶ (Y, _≼2) is an L-IRDP mapping iff f : (X, _⋞1) ⟶ (Y, _⋞2) is an L-ERDP mapping.

Proof. This can easily be obtained. □

The following theorem can be easily proved.

Theorem 4.16. (1) If f : (X, _⋞1) ⟶ (Y, _⋞2) and g : (Y, _⋞2) ⟶ (Z, _⋞3) are two L-ERDP mappings, then g ∘ f : (X, _⋞1) ⟶ (Z, _⋞3) is also an L-ERDP mapping.

(2) If f : (X, _⋞1) ⟶ (Y, _⋞2) and g : (Y, _⋞2) ⟶ (Z, _⋞3) are two L-ERP mappings, then g ∘ f : (X, _⋞1) ⟶ (Z, _⋞3) is also an L-ERP mapping.

It is easy to prove that all L-enclosed relation spaces and their L-ERDP mappings form a category, which is called the category of L-enclosed relation spaces, denoted by L-ERS.

Now we consider the relation between two categories L-TOP and L-ERS.

Theorem 4.17. (1) If f : (X, _⋞1) ⟶ (Y, _⋞2) is an L-ERDP mapping, then f : (X, τ _⋞1 ) ⟶ (Y, τ _⋞2 ) is an L-continuous mapping.

(2) If f : (X, _⋞1) → (Y, _⋞2) is an L-ERP mapping, then f : (X, τ _⋞1 ) → (Y, τ _⋞2 ) is an L-closed mapping.

Proof. (1) Let A be a closed L-set in (Y, τ _⋞2 ). Then A =⋀ { G ∈ L ^Y  ∣ A_⋞2G }. Hence f L ← ( A ) = ⋀ { f L ← ( G ) ∣ G ∈ L Y , A ⋞ 2 G } . Since A_⋞2G implies f L ← ( A ) ⋞ 1 f L ← ( G ) , we obtain f L ← ( A ) = ⋀ { f L ← ( G ) ∣ G ∈ L Y , A ⋞ 2 G } ⩾ ⋀ { E ∈ L X ∣ f L ← ( A ) ⋞ 1 E } ⩾ f L ← ( A ) .

This implies f L ← ( A ) = ⋀ { E ∈ L X ∣ f L ← ( A ) ⋞ 1 E } = cl ⋞ 1 ( f L ← ( A ) ) , i.e., f L ← ( A ) is a closed L-set in (X, τ _⋞1 ). Therefore f : (X, τ _⋞1 ) ⟶ (Y, τ _⋞2 ) is an L-continuous mapping.

The proof of (2) is dually analogous to (1). □

Theorem 4.18. (1) If f : (X, τ₁) ⟶ (Y, τ₂) is an L-continuous mapping, then f : (X, _⋞τ1) ⟶ (Y, _⋞τ2) is an L-ERDP mapping.

(2) If f : (X, τ₁) ⟶ (Y, τ₂) is an L-closed mapping, then f : (X, _⋞τ1) ⟶ (Y, _⋞τ2) is an L-ERP mapping.

Proof. (1) Let A, B ∈ L ^Y with A_⋞τ2B. Then A^- ⩽ B. Hence f L ← ( A - ) ⩽ f L ← ( B ) . Since f is an L-continuous mapping, we have f L ← ( A ) - ⩽ f L ← ( A - ) . Thus we obtain f L ← ( A ) - ⩽ f L ← ( A - ) ⩽ f L ← ( B ) . This shows f L ← ( A ) ⋞ τ 1 f L ← ( B ) . Therefore f : (X, _⋞τ1) ⟶ (Y, _⋞τ2) is an L-ERDP mapping.

The proof of (2) is dually analogous to (1). □

Now we define a functor 𝕊 : L-ERS ⟶ L-TOP such that 𝕊 ( X , ⋞ ) = ( X , τ ⋞ ) , 𝕊 ( f ) = f .

By Theorems 4.12, 4.13, 4.17 and 4.18 we can obtain the following theorem.

Theorem 4.19. 𝕊 : L-ERS ⟶ L-TOP is an isomorphism, that is, the category of L-enclosed relation spaces is isomorphic to the category of L-topological spaces.

5 Applications

In this section, we shall characterize separation axioms by means of L-internal relations andL-enclosed relations.

In [12] (see Chapter III), the notion of “well inside” is introduced in a distributive lattice L. Now we replace L by L ^X . Then the definition of “well inside” can be stated as follows.

Let A, B ∈ L ^X . A is said to be well inside B (and write A ⪕ B) if there exists a C ∈ L ^X with A ∧ C = 0 _ and C ∨ B = 1 _ . However ⪕ is different from an L-enclosed relation ⋞. This can be seen from the following example.

Example 5.1. Let X be an arbitrary set containing more than one point and let x₀ be a point in X. For any B ∈ 2 ^X , we define a binary relation ⋞ on 2 ^X as follows. { A ∣ A ⋞ B } = { { ∅ } , if x 0 ∉ B ; 2 B , if x 0 ∈ B .

As pointed out in Example 4.5, ⋞ is an L-enclosed relation, but it is different from ⪕ since ⪕ is exactly the relation ⩽ on L ^X .

However as pointed out in [12], the relation ⪕ can be used to define regularity of a locale. Analogously we also can present a characterization for a regular L-topological space by means of L-enclosed relation ⋞.

In Remark 4.4, we say that an L-enclosed relation is an auxiliary order satisfying the interpolation property on L ^X . However it is not approximating. If it is approximating, then what will happen?

Theorem 5.2. Let (X, τ) be an L-topological space and let _⋞τ be the L-enclosed relation induced by τ. Then (X, τ) is regular if and only if the restriction of _⋞τ on τ is approximating.

Proof. Sufficiency. Suppose that the restriction of _⋞τ on τ is approximating. Then for any A ∈ τ, A = ⋁ {B ∈ τ ∣ B_⋞τA}. This implies A = ⋁ {B ∈ τ ∣ B^- ⩽ A}. Therefore (X, τ) is regular.

Necessity. Suppose that (X, τ) is regular. Then for all A ∈ τ, we have A = ⋁ {B ∈ τ ∣ B^- ⩽ A} This implies A = ⋁ {B ∈ τ ∣ B_⋞τA}. This shows that the restriction of _⋞τ on τ is approximating. □

Theorem 5.3. Let (X, τ) be an L-topological space and let _≼τ be the L-internal relation induced by τ. Then (X, τ) is L-T₀ if and only if for all x, y ∈ X with x ≠ y, there exists a λ ∈ J (L) such that {U ∣ {x _λ } _≼τU} ≠ {U ∣ {y _λ } _≼τU} .

Proof. Sufficiency. Suppose that for all x, y ∈ X with x ≠ y, there exists a λ ∈ J (L) such that {U ∣ {x _λ } _≼τU} ≠ {U ∣ {y _λ } _≼τU} . This implies there exists a U ∈ τ such that either λ ⩽ U (x), λ≰U (y) or λ ⩽ U (y), λ≰U (x) . In both of the cases, we can obtain U (x) ≠ U (y) . This shows (X, τ) is L-T₀ .

Necessity. Suppose that (X, τ) is L-T₀. Then for all x, y ∈ X with x ≠ y, there exists a U ∈ τ such that U (x) ≠ U (y). Then there exists a λ ∈ J (L) such that λ ⩽ U (x) ≰U (y) or λ ⩽ U (y) ≰U (x) . This shows {U ∣ {x _λ } _≼τU} ≠ {U ∣ {y _λ } _≼τU} . □

Theorem 5.4. Let (X, τ) be an L-topological space and let _≼τ be the L-internal relation induced by τ. Then (X, τ) is L-T₁ if and only if for all x, y ∈ X with x ≠ y, there exist λ, μ ∈ J (L) such that y _μ ≰ ⋀ {U ∣ {x _λ } _≼τU} and x _λ ≰ ⋀ {V ∣ {y _μ } _≼τV} .

Proof. Sufficiency. Suppose that for all x, y ∈ X with x ≠ y, there exist λ, μ ∈ J (L) such that y μ ≰ ⋀ { U ∣ { x λ } ≼ τ U } ,(1) x λ ≰ ⋀ { V ∣ { y μ } ≼ τ V } .(2)

By (1), there exists a U ∈ τ such that λ ⩽ U (x) and μ≰U (y) . Hence U (x) ≰U (y) . By (2), there exists a V ∈ τ such that μ ⩽ V (y) and λ≰V (x) . Hence V (y) ≰V (x) .

Necessity. Suppose that for all x, y ∈ X with x ≠ y, there exist U, V ∈ τ such that U (x) ≰U (y)  and V (y) ≰V (x). Then there exist λ, μ ∈ J (L) such that x _λ  ⩽ U, y _μ ≰U, y _μ  ⩽ V, x _λ ≰V . Hence y _μ ≰ ⋀ {U ∣ {x _λ } _≼τU} and x _λ ≰ ⋀ {V ∣ {y _μ } _≼τV} . □

Theorem 5.5. Let (X, τ) be an L-topological space, let _≼τ and _⋞τ be respectively the L-internal relation and the L-enclosed relation induced by τ. Then (X, τ) is T₂ if and only if every y _μ  ∈ J (L ^X ) satisfies y _μ  = ⋀ {A ∣ {y _μ } _≼τA, A_⋞τA}.

Proof. Sufficiency. Suppose that every L-fuzzy point y _μ  ∈ J (L ^X ) satisfies y _μ  = ⋀ {A ∣ {y _μ } _≼τA, A_⋞τA}. Then for all x _λ , y _μ  ∈ J (L ^X ) with x _λ ≰y _μ , there exists a fixed L-set A of _⋞τ such that {y _μ } _≼τA and x _λ ≰A . This implies y _μ  ⩽ A° and x _λ ≰A^-. Therefore (X, τ) is T₂.

Necessity. Suppose that (X, τ) is T₂. Then for all x _λ , y _μ  ∈ J (L ^X ) with x _λ ≰y _μ , there exist a closedL-set P and an open L-set Q such that x _λ ≰P ⩾ Q ⩾ y _μ  . It is obvious that y _{μ ≼τ}P, P_⋞τP. This implies y μ ⩽ ⋀ { A ∣ { y μ } ≼ τ A , A ⋞ τ A } and x λ ≰ ⋀ { A ∣ { y μ } ≼ τ A , A ⋞ τ A } .

This shows y _μ  = ⋀ {A ∣ {y _μ } _≼τA, A_⋞τA}. □

The following theorem is straightforward.

Theorem 5.6. Let (X, τ) be an L-topological space, let _≼τ and _⋞τ be respectively the L-internal relation and the L-enclosed relation induced by τ. Then (X, τ) is normal if and only if for any L-sets A, B with A ⩽ B, there exists a fixed L-set D of _≼τ such that A_≼τD_⋞τB.

In this paper, the notions of L-internal relation spaces and L-enclosed relation spaces are introduced. It is proved that the category of L-internal relation spaces, the category of L-enclosed relation spaces and the category of topological spaces are isomorphism. Pointwise quasi-uniformities and pointwise S-quasi-proximities are in close relationship withL-internal relations and L-enclosed relations. Also, they can be used to characterize some axioms of separation. They maybe be more useful in many ways.