Generalized L -separations and generalized L -Urysohn spaces

1 Introduction

In 2002, Csaszar introduced the concept of generalized topological spaces in [4]. In [3], we introduced the concept of generalized L-topological spaces, and studied the basic concepts of open sets, closed sets, interior and closure operators, continuous mappings, sub-spaces, quotient spaces and connectedness in generalized L-topological spaces. Since the generalized topological structure is weaker than the topological structure, it has extensive scope.

On the basis of the text [3], in this paper we introduce generalized L - T _i separation axioms (i=−1,0,1,2), generalized L-Urysohn spaces and generalized L-homeomorphism in generalized L-topological spaces and study their basic properties.

2 Preliminaries

Throughout this paper, (L, ∨ , ∧ , ^′) is a completely distributive De Morgan algebra and X is a nonempty set. L ^X is the set of all L-sets on X. The smallest element and the largest element of L ^X will be denoted by 0 and 1 respectively.

An element r in L is called a prime element [6], if a ∧ b ≤ r implies a ≤ r or b ≤ r, where a, b ∈ L. The set of nonunit prime elements in L is denoted by pr (L). An element r in L is called a co-prime element [6], if for arbitrary a, b ∈ L with r ≤ a ∨ b then r ≤ a or r ≤ b. The set of nonzero co-prime elements in L and L ^X is denoted by M (L) and M ^* (L ^X ) respectively. Clearly, r ∈ pr (L) iff r′ ∈ M (L). According to [9, 12], we know that if L is completely distributive, then each element a in L has the greatest minimal family (the greatest maximal family), denoted by β (a)(α (a)). In this case, β ^* (a) = β (a) ∩ M (L) is a minimal family of a and α ^* (a) = α (a) ∩ pr (L) is a maximal family of a. We use χ _Y to denote the characteristic functions of Y, (∅ ¬ = Y ⊂ X). For each ψ ⊂ L, we define ψ′ = {A′ : A ∈ ψ}.

Definition 2.1. [12, 14]. A net in a nonempty ordinary set X is a mapping S : D → X, where D is a directed set. A net in L ^X is a mapping S : D → M ^* (L ^X ), denote by S = {S (n) , n ∈ D} , where D is a directed set.

Definition 2.2. [10]. Each 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), which is defined by

f L → ( A ) = ⋁ { A ( x ) ∣ f ( x ) = y }

(∀ A ∈ L ^X , y ∈ Y) .

The right adjoint to f L → (called L-backward powerset operator) is denoted f L ← and given by

f L ← ( B ) = ⋁ { A ∈ L X ∣ f L → ( A ) ≤ B } = B ∘ f

(∀ B ∈ L ^Y ) .

It is known that f L → (briefly f ^→) preserves arbitrary unions and that f L ← (briefly f ^←) preserves arbitrary unions, arbitrary intersections and complements.

Definition 2.3. [3]. Let L be a completely distributive De Morgan algebra, X be a nonempty set and δ be a collection of subsets of L ^X . Then δ is called a generalized L-topology (briefly GL - t) on X if 0 ∈ δ and G _i  ∈ δ for i∈ I ≠ ∅ implies G = ⋁ _i∈I G _i  ∈ δ. We call the pair (L ^X , δ) a generalized L-topological space (briefly GL - ts) on X. The elements of δ are called generalized L-open sets (briefly GL-open sets) and the complements are called generalized L-closed sets (briefly GL-closed sets). We say δ is strong if 1 ∈ δ.

Definition 2.4. [3]. Let (L ^X , δ) and (L ^Y , τ) be two GL - ts′s and f ^→ : L ^X  → L ^Y an L-fuzzy mapping. f ^→ is called

a generalized L-continuous mapping (briefly GL-continuous mapping) if f ^← (B) ∈ δ for each B ∈ τ.

Definition 2.5. [3]. Let (L ^X , δ) be a GL - ts and x _λ  ∈ M ^* (L ^X ). A ∈ δ′ is called a generalized L-closed remote-neighborhood (briefly GLC-RN) of x _λ , if x _λ  ≰ A. B ∈ L ^X is called a generalized L-remote-neighborhood (briefly GL-RN) of x _λ if there is a GLC-RN A of x _λ such that B ≤ A. The set of all GLC-RNs (GL-RNs) of x _λ is denoted by η ^- (x _λ ) (η (x _λ )).

3 Generalized L - T _i Separation Axioms (i=−1,0,1,2)

Definition 3.1. Let (L ^X , δ) be a GL - ts.

If ∀x _λ , x _μ  ∈ M ^* (L ^X ), λ < μ, there exists a P ∈ η (x _μ ) such that x _λ  ≤ P, then (L ^X , δ) is said to be generalized L - T _-1 (briefly GL - T _-1).

Remark 3.2. Clearly, P ∈ η (x _λ ) if and only if P ^- ∈ η ^- (x _λ ). Therefore P ∈ η (x _λ ) and Q ∈ η (y _μ ) in the Definition 3.1 we may instead by P ∈ η ^- (x _λ ) and Q ∈ η ^- (y _μ ), respectively.

Corollary 3.3. Let (L ^X , δ) be a GL - ts. Then

(GL - T ₂) + (GL - T _-1)   ⇒  GL - T ₁  ⇒  GL - T ₀⇒GL - T _-1.

Proof. GL - T ₁ ⇒ GL - T ₀ and GL - T ₀ ⇒ GL - T _-1 are immediate from the Definition 3.1.

We prove (GL - T ₂) + (GL - T _-1) ⇒ GL - T ₁. Let x _λ , y _μ  ∈ M ^* (L ^X ) and x _λ  ¬ = y _μ .

If x = y, we may assume that μ < λ, then x _λ  ≰ x _μ . Since (L ^X , δ) is GL - T _-1, there exists a P ∈ η (x _λ ) such that x _μ  ≤ P. Thus, (L ^X , δ) is GL - T ₁.

If x ¬ = y, then ∀λ, μ ∈ M (L), we have x _λ  ≰ y _μ and y _μ  ≰ x _λ . Since (L ^X , δ) is GL - T ₂, there exist P ∈ η (x _λ ) and Q ∈ η (y _μ ) such that P ∨ Q = 1. From y _μ  ≰ Q and y _μ  ≤ P ∨ Q = 1, we have y _μ  ≤ P. From x _λ  ≰ P and x _λ  ≤ P ∨ Q = 1, we have x _λ  ≤ Q. Thus, (L ^X , δ) is GL - T ₁.

That the converses of the Corollary 3.3 need not be true be are is shown by Examples 3.4, 3.5 and 3.11.

Example 3.4. Let X be an ordinary set such that |X|≥2, L = [0, 1] and δ be generated by all those subsets in X whose membership functions are constant functions. It is clear that GL - ts (L ^X , δ) is GL - T _-1. But (L ^X , δ) is not GL - T ₀. In fact, ∀x, y ∈ X, x ¬ = y and ∀λ ∈ (0, 1], if P ∈ η (x _λ ), then y _λ  ≰ P, and if Q ∈ η (y _λ ), then x _λ  ≰ Q. Thus, (L ^X , δ) is not GL - T ₀.

Example 3.5. Let X = {x, y}, L = {0, 1} and δ = {0, A, 1}, where A = {1, 0}. It is clear that GL - ts (L ^X , δ) is GL - T ₀. For x ₁ and y ₁, we easily obtain that η (y ₁) = {0}, but x ₁ ≰ 0. Thus, (L ^X , δ) is not GL - T ₁.

Example 3.6. Let X = L = [0, 1] and δ = {χ _A |A ⊂ X}. It is clear that GL - ts (L ^X , δ) is GL - T ₂. But (L ^X , δ) is not GL  -  T _-1, not to say GL  -  T ₀ and GL  -  T ₁. In fact, ∀x ∈ X and ∀P ∈ η (x ₁), we have P (x) =0. So x 1 2 ≰ P. Thus, (L ^X , δ) is not GL - T _-1.

Let X be a singleton. Then ∀L and ∀δ, (L ^X , δ) is GL - T ₂. But ∀L ¬ = {0, 1}, we can choose a δ such that (L ^X , δ) is not GL - T _-1.

Theorem 3.7. Let (L ^X , δ) be a GL - ts. Then it is GL - T _-1 if and only if ∀x _λ  ∈ M ^* (L ^X ), x _λ is a maximal point contained in (x _λ ) ^-.

Proof. Sufficiency. Let x _λ , x _μ  ∈ M ^* (L ^X ) and λ < μ. By the fact that x _λ is maximal point contained in (x _λ ) ^-, we have x _μ  ≰ (x _λ ) ^-. Hence, (x _λ ) ^- ∈ η (x _μ ) and x _λ  ≤ (x _λ ) ^-, so that (L ^X , δ) is GL - T _-1.

Necessity. Suppose that there exists an x _λ  ∈ M ^* (L ^X ), x _λ is not a maximal point contained in (x _λ ) ^-. Then there exists an x _μ  ∈ M ^* (L ^X ) such that x _λ  < x _μ  ≤ (x _λ ) ^-. Let ∀P ∈ η ^- (x _μ ). Then x _λ  ≰ P. In fact, if x _λ  ≤ P, then (x _λ ) ^- ≤ P, hence, x _μ  ≤ P. This is a contradiction. Thus, (L ^X , δ) is not GL - T _-1.

Theorem 3.8. Let (L ^X , δ) be a GL - ts. Then it is GL - T ₀ if and only if ∀x _λ , y _μ  ∈ M ^* (L ^X ), x _λ  ¬ = y _μ , we have η (x _λ ) ¬ = η (y _μ ).

Proof. Let (L ^X , δ) be a GL - T ₀ space, x _λ , y _μ  ∈ M ^* (L ^X ) and x _λ  ¬ = y _μ . Then either there exists a P ∈ η (x _λ ) such that y _μ  ≤ P, so that P ∉ η (y _μ ), or there exists a Q ∈ η (y _μ ) such that x _λ  ≤ Q, so that Q ∉ η (x _λ ). Thus, η (x _λ ) ¬ = η (y _μ ).

Conversely, suppose that (L ^X , δ) is not a GL - T ₀ space. Then there exist x _λ , y _μ  ∈ M ^* (L ^X ) and x _λ  ¬ = y _μ , such that ∀P ∈ η (x _λ ), y _μ  ≰ P, i.e. P ∈ η (y _μ ), and ∀Q ∈ η (y _μ ), x _λ  ≰ Q, i.e. Q ∈ η (x _λ ). Thus, η (x _λ ) = η (y _μ ).

Theorem 3.9. Let (L ^X , δ) be a GL - ts. Then it is GL - T ₀ if and only if ∀x _λ , y _μ  ∈ M ^* (L ^X ), x _λ  ¬ = y _μ , we have x _λ  ≰ (y _μ ) ^- or y _μ  ≰ (x _λ ) ^-.

Proof. Let (L ^X , δ) be a GL - T ₀ space, x _λ , y _μ  ∈ M ^* (L ^X ) and x _λ  ¬ = y _μ . From the Theorem 3.8 we have η (x _λ ) ¬ = η (y _μ ). We may assume that there exists a P ∈ η ^- (x _λ ) such that y _μ  ≤ P. It follows that (y _μ ) ^- ≤ P. Thus, x _λ  ≰ (y _μ ) ^-.

Conversely, suppose that (L ^X , δ) is not a GL - T ₀ space. From the Theorem 3.8 there exist x _λ , y _μ  ∈ M ^* (L ^X ) and x _λ  ¬ = y _μ such that η (x _λ ) = η (y _μ ). Hence, x _λ  ≤ (y _μ ) ^- and y _μ  ≤ (x _λ ) ^-. In fact, if x _λ  ≰ (y _μ ) ^- or y _μ  ≰ (x _λ ) ^-, then (y _μ ) ^- ∈ η (x _λ ) = η (y _μ ) or (x _λ ) ^- ∈ η (y _μ ) = η (x _λ ), a contradiction. Thus sufficiency is proved.

Theorem 3.10. Let (L ^X , δ) be a GL - ts. Then it is GL - T ₁ if and only if ∀x _λ  ∈ M ^* (L ^X ), x _λ is a GL-closed set.

Proof. Sufficiency. Let x _λ , y _μ  ∈ M ^* (L ^X ) and y _μ  ≰ x _λ . By x _λ is a GL-closed set, we have x _λ  ∈ η (y _μ ) and x _λ  ≤ x _λ . Thus, (L ^X , δ) is GL - T ₁.

Necessity. Let x _λ , y _μ  ∈ M ^* (L ^X ) and y _μ  ≤ (x _λ ) ^-. Suppose that y _μ  ≰ x _λ . By (L ^X , δ) is GL - T ₁, then there exists a Q  ∈  η ^- (y _μ ), i.e. y _μ  ¬ ! ≤  Q such that x _λ   ≤  Q.So (x _λ ) ^- ≤ Q, hence y _μ  ≤ Q. This is a contradiction. Thus, y _μ  ≤ x _λ , i.e. x _λ is a GL-closed set.

Example 3.11. Let X = {x ₁, x ₂, ⋯} be a countably infinite set, L = {0, 1} and δ = {0, A _i , 1}, where A _i  = 1 - (x _i ) ₁, (i = 1, 2, ⋯). Since for each (x _i ) ₁ is a GL-fuzzy closed set, (L ^X , δ) is GL - T ₁ by the Theorem 3.10. Let x _i  ¬ = x _j . Then ∀P ∈ η ((x _i ) ₁) and ∀Q ∈ η ((x _j ) ₁) we have P ∨ Q ¬ =1. Thus, (L ^X , δ) is not GL - T ₂.

Theorem 3.12. Let (L ^X , δ) be a GL - ts, where L = [0, 1]. Then it is GL - T ₂ if and only if for every pair x _λ , y _μ  ∈ M ^* (L ^X ) and x ¬ = y, λ < 1, μ < 1 there exist GL-open sets U and V such that x _λ  ∈ U, y _μ  ∈ V and U ∧ V = 0.

Proof. Necessity. Let x _λ , y _μ  ∈ M ^* (L ^X ) and x ¬ = y, λ < 1, μ < 1. Since (L ^X , δ) is GL - T ₂, there exist P ∈ η ^- (x _1-λ) and Q ∈ η ^- (y _1-μ ) such that P ∨ Q = 1. By P ∈ η (x _1-λ), i.e. x _1-λ ≰ P we have P (x) <1 - λ. So that P′ (x) > λ, i.e. x _λ  ∈ P′. Similarly, we have y _μ  ∈ Q′. Clearly, P′ and Q′ are GL-open and P′ ∧ Q′ = 0.

Sufficiency. Let x _λ , y _μ  ∈ M ^* (L ^X ) and x ¬ = y. We take real numbers r and s such that 1 - λ < r < 1 and 1 - μ < s < 1. By hypothesis there exist GL-open sets U and V such that x _r  ∈ U, y _s  ∈ V and U ∧ V = 0. Put P = U′, Q = V′. Then

P (x) = U′ (x) ≤ r′ < λ, i.e. x _λ  ≰ P, and

Q (y) = V′ (y) ≤ s′ < μ, i.e. y _μ  ≰ Q.

Hence P ∈ η (x _λ ), Q ∈ η (y _μ ) and P ∨ Q = 1. Thus, (L ^X , δ) is GL - T ₂.

Definition 3.13. Let (L ^X , δ) be a GL - ts, S = {S (n) , n ∈ D} a net in L ^X and x _λ  ∈ M ^* (L ^X ). Then x _λ is said to be a limit point of S (or S converges to x _λ ), if for each P ∈ η (x _λ ) , S (n) ≰ P is eventually true.

Theorem 3.14. Let (L ^X , δ) be a GL - T ₂ and S a net in L ^X . Then S converges to at most one point.

Proof. Let S = {S (n) , n ∈ D} be a net in L ^X . Suppose that S converge to two points x _λ and y _μ with x ¬ = y. Then for each P ∈ η (x _λ ) there exists a n ₁ ∈ D such that S (n) ≰ P when n ≥ n ₁, and for each Q ∈ η (y _μ ) there exists a n ₂ ∈ D such that S (n) ≰ Q when n ≥ n ₂. Since D is directed, take n ₃ ∈ D such that n ₃ ≥ n ₁ and n ₃ ≥ n ₂. Then when n ≥ n ₃, S (n) ≰ P ∨ Q, and so P ∨ Q ¬ =1. Thus, L ^X , δ) is not GL - T ₂.

Question 3.15. Does the reverse of Theorem 3.14 hold?

We now prove that the GL - T _i (i = -1, 0, 1, 2) property is hereditary.

Theorem 3.16. Let (L ^X , δ) be GL - T _i (i = -1, 0, 1, 2), Y ⊂ X and Y¬ = ∅. Then GL-subspace (L ^Y , δ|Y) is GL - T _i (i = -1, 0, 1, 2).

Proof. We prove only i = -1. The proofs for other cases (i = 0, 1, 2) are similar.

Let x _λ , x _μ  ∈ M ^* (L ^Y ), x ∈ Y and μ < λ. Then x λ * , x μ * ∈ M * ( L X ), where x λ * and x μ * are extensions of x _λ and x _μ on X, respectively. Since (L ^X , δ) is GL - T _-1, there exists a P ∈ η - ( x λ * ) such that x μ * ≤ P. By the Theorem 5.7 in [3], P|Y ∈ η ^- (x _λ ) and x _μ  ≤ P|Y. Thus, (L ^Y , δ|Y) is GL - T _-1.

We now prove that the GL - T _i (i = -1, 0, 1, 2) property is preserved under GL-homeomorphism.

Theorem 3.17. Let f ^→ : (L ^X , δ) → (L ^Y , τ) be a bijective L-fuzzy mapping and f ^← be a GL-continuous mapping. If (L ^X , δ) is GL - T _i (i = -1, 0, 1, 2), then so is (L ^Y , τ).

Proof. We prove i = 0. Let x _λ , y _μ  ∈ M ^* (L ^Y ) and x _λ  ¬ = y _μ . By f ^→ is bijective, we have f ^← (x _λ ) , f ^← (y _μ ) ∈ M ^* (L ^X ) and f ^← (x _λ ) ¬ = f ^← (y _μ ). Since (L ^X , δ) is GL - T ₀, we may assume that there exists a P ∈ η (f ^← (x _λ )) such that f ^← (y _μ ) ≤ P. Form f ^← is GL-continuous and f ^→ is bijective, we have f ^→ (P) = (f ^←) ^← (P) is a GL-closed set in (L ^Y , τ), and x _λ  = f ^→ f ^← (x _λ ) ≰ f ^→ (P) and y _μ  = f ^→ f ^← (y _μ ) ≤ f ^→ (P). Thus, (L ^Y , τ) is GL - T ₀.

i = -1, 2 analogous to the proof of the i = 0.

We prove i = 1. Let y _μ  ∈ M ^* (L ^Y ). By f ^→ is bijective, we have f ^← (y _μ ) ∈ M ^* (L ^X ). Since (L ^X , δ) is GL - T ₁, f ^← (y _μ ) is a GL-closed set in (L ^X , δ). From f ^← is GL-continuous and f ^→ is bijective, we have y _μ  = f ^→ f ^← (y _μ ) is a GL-closed set in (L ^Y , τ). Thus, (L ^Y , τ) is GL - T ₁.

Lemma 3.18. Let (L ^X , δ) and (L ^Y , τ) be two GL - ts′s and f ^→ : L ^X  → L ^Y a bijective L-fuzzy mapping. Then f ^→ is GL-it open if and only if f ^→ is GL-closed.

Proof. Easy to prove.

From the Lemma 3.18 and the Theorem 3.17, we immediately obtain the following corollaries.

Corollary 3.19. Let f ^→ : (L ^X , δ) → (L ^Y , τ) be a bijective GL-open mapping. If (L ^X , δ) is GL - T _i (i = -1, 0, 1, 2), then so is (L ^Y , τ).

Corollary 3.20. Let f ^→ : (L ^X , δ) → (L ^Y , τ) a bijective GL-closed mapping. If (L ^X , δ) is GL - T _i (i = -1, 0, 1, 2), then so is (L ^Y , τ).

Theorem 3.21. Let f ^→ : (L ^X , δ) → (L ^Y , τ) be a bijective GL-continuous mapping. If (L ^Y , τ) is GL - T _i (i = -1, 0, 1, 2), then so is (L ^X , δ).

Proof. We prove only i = 2. The proofs for other cases (i = -1, 0, 1) are similar.

Let x _λ , y _μ  ∈ M ^* (L ^X ) with x ¬ = y. By f ^→ is bijective, we have f ^→ (x _λ ) = (f ^→ (x))  _λ , f ^→ (y _μ ) = (f ^→ (y))  _μ  ∈ M ^* (L ^Y ) and f ^→ (x) ¬ = f ^→ (y). Since (L ^Y , τ) is GL - T ₂, there exist P ∈ η ((f ^→ (x))  _λ ) and Q ∈ η ((f ^→ (y))  _μ ) such that P ∨ Q = 1. Again, since f ^→ is GL-continuous, f ^← (P) and f ^← (Q) are GL-closed sets in (L ^X , δ). From (f ^→ (x))  _λ  ≰ P we have x _λ  ≰ f ^← (P), i.e. f ^← (P) ∈ η (x _λ ). Similarly, f ^← (Q) ∈ η (y _μ ). So we have

f ^← (P) ∨ f ^← (Q) = f ^← (P ∨ Q) = f ^← (1) =1. Thus, (L ^X , δ) is GL - T ₂.

4 Generalized L-Urysohn spaces

Definition 4.1. Let (L ^X , δ) be a strong GL - ts. If ∀x _λ , y _μ  ∈ M ^* (L ^X ), x ¬ = y, there exist P ∈ η (x _λ ) and Q ∈ η (y _μ ) such that P ^o  ∨ Q ^o  = 1, then (L ^X , δ) is said to be a generalized L-Urysohn space (briefly GL-Urysohn space).

Obviously, every GL-Urysohn space is a GL - T ₂ space in the strong GL - ts. But the converse is not true.

Example 4.2. Let X be an ordinary set such that |X|≥2, L = [0, 1] and δ = {0} ∪ {x _i |i ∈ L, i ¬ =1} ∪ {∪ {x _i } |i ∈ L, i ¬ =1}. Then δ is a GL - t on X. Clearly, (L ^X , δ) is GL - T ₂, but it is not GL-Urysohn.

Definition 4.3. A GL - ts (L ^X , δ) is called interior additive if (A ∨ B)  ^o  = A ^o  ∨ B ^o for any two A, B ∈ L ^X .

Theorem 4.4. Let (L ^X , δ) be a strong GL - ts. If (L ^X , δ) is GL - T ₂ and interior additive, then it is a GL-Urysohn space.

Proof. Let x _λ , y _μ  ∈ M ^* (L ^X ) with x ¬ = y. Since (L ^X , δ) is GL - T ₂, there exist P ∈ η (x _λ ) and Q ∈ η (y _μ ) such that P ∨ Q = 1. From (L ^X , δ) are strong and interior additive, we have P ^o  ∨ Q ^o  = (P ∨ Q)  ^o  = 1 ^o  = 1. Thus, (L ^X , δ) is a GL-Urysohn space.

Definition 4.5. Let (L ^X , δ) be a GL - ts, S = {S (n) , n ∈ D} a net in L ^X and x _λ  ∈ M ^* (L ^X ). Then x _λ is said to be a W-limit point of S (or S W-converges to x _λ ), if for each P ∈ η (x _λ ) , S (n) ≰ P ^o is eventually true.

Theorem 4.6. Let (L ^X , δ) be a GL-Urysohn space and S a net in L ^X . Then S W-converges to at most a point.

Proof. Let (L ^X , δ) be a GL-Urysohn space and S = {S (n) , n ∈ D} a net in L ^X which W-converges to a point x _λ  ∈ M ^* (L ^X ). Put y _μ  ∈ M ^* (L ^X ) with x ¬ = y. Then there exist P ∈ η (x _λ ) and Q ∈ η (y _μ ) such that P ^o  ∨ Q ^o  = 1. Since S (n) ≰ P ^o is eventually true, S (n) is eventually in Q ^o . Thus, S does not W-converge to y _μ .

Theorem 4.7. Let (L ^X , δ) be a strong GL - ts, where L = [0, 1]. Then (L ^X , δ) is a GL-Urysohn space if and only if for every pair x _λ , y _μ  ∈ M ^* (L ^X ) with x ¬ = y and λ < 1, μ < 1, there exist GL-open sets U and V such that x _λ  ∈ U, y _μ  ∈ V and U ^- ∧ V ^- = 0.

Proof. Let (X, δ) be a GL-Urysohn space, x _λ , y _μ  ∈ M ^* (L ^X ) with x ¬ = y and λ < 1, μ < 1. Choose two real numbers r and s satisfying 0 < r < 1 - λ and 0 < s < 1 - μ. Then there exist P ∈ η (x _r ) and Q ∈ η (y _s ) such that P ^o  ∨ Q ^o  = 1. Put U = P′ and V = Q′. Then U and V are GL-open and x _λ  ∈ U, y _μ  ∈ V and

U ^- ∧ V ^- = (P′) ^- ∧ (Q′) ^- = (P ^o ) ′ ∧ (Q ^o ) ′

= (P ^o  ∨ Q ^o ) ′ = 0.

Conversely, suppose x _λ , y _μ  ∈ M ^* (L ^X ) with x ¬ = y. Choose two real numbers r and s satisfying 1 - λ < r < 1 and 1 - μ < s < 1. By hypothesis there exist GL-open sets U and V such that x _r  ∈ U, y _s  ∈ V and U ^- ∧ V ^- = 0. Put P = U′ and Q = V′. Then P ∈ η (x _λ ), Q ∈ η (y _μ ) and

P ^o  ∨ Q ^o  = (U′)  ^o  ∨ (V′)  ^o  = (U ^-) ′ ∨ (V ^-) ′

= (U ^- ∧ V ^-) ′ = 1.

Thus, (L ^X , δ) is a GL-Urysohn space.

Theorem 4.8. Let f ^→ : (L ^X , δ) → (L ^Y , τ) be a bijective GL-closed mapping. If (L ^X , δ) is a GL-Urysohn space, then so is (L ^Y , τ).

Proof. Let x _λ , y _μ  ∈ M ^* (L ^X ) and x ¬ = y. By f ^→ is bijective, we have f ^→ (x _λ ) , f ^→ (y _μ ) ∈ M ^* (L ^Y ) and f ^→ (x) ¬ = f ^→ (y). Since (L ^X , δ) is a GL-Urysohn space, there exist P ∈ η (x _λ ) and Q ∈ η (y _μ ) such that P ^o  ∨ Q ^o  = 1. Again, since f ^→ is GL-closed, f ^→ (P) and f ^→ (Q) are GL-closed sets in (L ^Y , τ). From x _λ  ≰ P we have (f ^→ (x))  _λ  ≰ f ^→ (P), i.e. f ^→ (P) ∈ η ((f ^→ (x))  _λ ). Similarly, f ^→ (Q) ∈ η ((f ^→ (y))  _μ ). By the Lemma 3.18, we have f ^→ is also GL-open. Hence,

(f ^→ (P))  ^o  ∨ (f ^→ (Q))  ^o

≥ (f ^→ (P ^o ))  ^o  ∨ (f ^→ (Q ^o ))  ^o

= f ^→ (P ^o ) ∨ f ^→ (Q ^o )

= f ^→ (P ^o  ∨ Q ^o )

= f ^→ (1) =1.

Thus, (L ^Y , τ) is GL-Urysohn space.

Theorem 4.9. Let f ^→ : (L ^X , δ) → (L ^Y , τ) be a bijective GL-continuous mapping. If (L ^Y , τ) is a GL-Urysohn space, then so is (L ^X , δ).

Proof. Analogous to the proof of Theorem 4.8.

5 Generalized L-homeomorphism

Theorem 5.1. Let (L ^X , δ) and (L ^Y , τ) be two GL - ts′s and f ^→ : L ^X  → L ^Y a bijective L-fuzzy mapping. Then f ^→ is GL-continuous if and only if (f ^→ (A))  ^o  ≤ f ^→ (A ^o ) for each A ∈ L ^X .

Proof. Let f ^→ be GL-continuous and A ∈ L ^X . Then f ^← ((f ^→ (A))  ^o ) ∈ δ. By the Theorem 4.3(5) in [3] and the fact that f ^→ is one-to-one, we have

f ^← ((f ^→ (A))  ^o ) ≤ (f ^← (f ^→ (A)))  ^o  = A ^o .

Again, since f ^→ is onto, we have

(f ^→ (A))  ^o  = f ^→ f ^← ((f ^→ (A))  ^o ) ≤ f ^→ (A ^o ).

Conversely, let B ∈ τ. Then B = B ^o . By hypothesis,

f ^→ ((f ^← (B)  ^o ) ≥ (f ^→ f ^← (B))  ^o  = B ^o  = B.

This implies that f ^← f ^→ ((f ^← (B))  ^o ) ≥ f ^← (B). Since f ^→ is one-to-one, we have

(f ^← (B))  ^o  = f ^← f ^→ ((f ^← (B))  ^o ) ≥ f ^← (B).

Hence, f ^← (B) = (f ^← (B))  ^o  ∈ δ. Thus, f ^→ is GL-continuous.

Theorem 5.2. Let (L ^X , δ) and (L ^Y , τ) be two GL - ts′s and f ^→ : L ^X  → L ^Y a bijective L-fuzzy mapping. Then f ^→ is GL-closed if and only if f ^← (B ^-) ≤ (f ^← (B)) ^- for each B ∈ L ^Y .

Proof. Analogous to the proof of Theorem 5.1.

Definition 5.3. Let (L ^X , δ) and (L ^Y , τ) be two GL - ts′s and f ^→ : L ^X  → L ^Y an L-fuzzy mapping. f ^→ is called a generalized L-homeomorphism (briefly GL-homeomorphism) if f ^→ is bijective, f ^→ and f ^← : L ^Y  → L ^X are GL-continuous.

If there exists a GL-homeomorphism of one GL - ts onto another, the two GL - ts′s are said to be GL-homeomorphic and each is a GL-homeomorph of the other.

Two GL - ts′s are generalized topologically equivalent if and only if they are GL-homeomorphic.

Obviously, the identity map of a GL - ts onto itself is always a GL-homeomorphism, and the inverse of a GL-homeomorphism is again a GL-homeomorphism.

Theorem 5.4. Let (L ^X , δ) and (L ^Y , τ) be two GL - ts′s and f ^→ : L ^X  → L ^Y a bijective L-fuzzy mapping. Then the following are equivalent:

f ^→ is a GL-homeomorphism.

Proof. This is immediate from the Theorems 5.1, 5.2 and the Theorems 4.3, 4.4 and 4.5 in [3].

Theorem 5.5. Let (L ^X , δ), (L ^Y , τ) and (Z, η) be three GL - ts′s and f ^→ : L ^X  → L ^Y and g ^→ : L ^Y  → L ^Z L-fuzzy mappings. Then

if f ^→ and g ^→ are GL-continuous, theng ^→ ∘ f ^→ is also GL-continuous.

Proof. This is easy.

Definition 5.6. A property which when possessed by a GL - ts is also possessed by each GL-homeomorph is a generalized topological invariant.

Theorem 5.7. GL - T _i (i = -1, 0, 1, 2) spaces and GL-Urysohn space are all generalized topological invariants.

Proof. This is immediate from the Theorems 3.17 and 4.8.

6 Conclusion

It is known that when L = {0, 1}, then L ^X  ≅ 2 ^X , so ordinary subsets can be equivalently considered as a special case of L-subsets. Besides, a generalized L-topology δ does not necessarily contain 1 and is not necessarily closed for finite intersections. So generalized L-topology is a generalization of generalized topologies, and is also a generalization of L-topologies. Similarly, generalized L-separation axioms and generalized L-Urysohn spaces, introduced in this paper, are generalizations of the corresponding concepts in generalized topological spaces and L-topological spaces respectively. The fact we know is that the weak topology have already had good applications in the computer science and the quantum computing. Thus we believe that these generalized L-topological properties studied hare will have a good and further application in the field above.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 11471202), the Natural Science Foundation of Guangdong Province (No. S2012010008833).

The authors are highly grateful to the referees for their suggestions for improving the paper.