Convergence structures in ( L,M )-DFTML

Abstract

Abd El-latif and Kim in [2], introduced the concepts of (L, M)-double fuzzy ideal and (L, M)-double fuzzy remote neighborhood systems, this paper is devoted to use these concepts to introduce and study the concept of pretopological fuzzy DR-convergence structure, and constructs the category PrFDRCS which contains TFDRCS as a bireflective subcategory. TFDRCS is shown as an isomorphic with (L, M)-TDFRN. Considering the relationship between the categories (L, M)-TDFRN and (L, M)-DFTML, it is proved that TFDRCS is isomorphic with (L, M)-DFTML. Also, pretopological fuzzy DR-convergence spaces have been found to be characterized as a kind of (L, M)-double fuzzy remote neighborhood spaces, which is called an (L, M)- strong double fuzzy remote neighborhood space. Finally, it was proved that the category (L, M)-SDFRN is isomorphic with the category PrFDRCS.

Keywords

isomorphic category

1 Introduction

The concept of topological molecular lattice (briefly, TML) was introduced by Wang [27], also he established the theory of remote neighborhood systems, which played an important role in TML. Since then, TML has been a spectacular topic of study. Many important results on this topic have been obtained [10 , 31]. Yue and Fang [29], studied Wang’s TML in a Kubiak- Šostak sense [14, 23] and they denoted it by FTML. Abd El-latif and Kim [2], introduced the notion of (L, M)-double fuzzy topological molecular lattice and constructed a category (L, M)-DFTML, also they defined the concept of (L, M)-double fuzzy remote neighborhood systems and constructed the category (L, M)-TDFRN and they proved that it is isomorphic with (L, M)-DFTML.

In general topology, by introducing the notion of ideal, Kuratawsik [15], Vaidyanathaswamy [25, 26] and several other authors carried out such analysis. In [13] Jankovic and Hamlett introduced a good study on the importance of the ideal in general topology. Sarkar [22] introduced the notion of fuzzy ideal in fuzzy set theory. In [18], Ramadan et al. investigated some properties of smooth ideals. In [1], Abd El-latif, introduced the concept of (L, M)-fuzzy ideal in FTML and investigated some properties of it. In [2], Abd El-latif and Kim introduced and studied the concept of (L, M)-double fuzzy ideal in (L, M)-DFTML.

Convergence theory of ideals provides a good tool for interpreting topological structures and plays an important role in fuzzy topology. There are extensive studies of the convergence theory of ideals in fuzzy topological spaces by several authors [6– 9 , 30] from different standpoints.

In posets, the concept of ideal is very useful when studying problems concerning ordered structures. Since an (L, M)-DFTML is, in fact, an ordered structure, it is natural to use (L, M)-double fuzzy ideal to study its properties.

Our aim of this paper is to establish the convergence theory of (L, M)-double fuzzy ideals in (L, M)-DFTML. The structure of this paper is organized as follows: In Section 2, we give some preliminary concepts. In Section 3, we introduce the concept of fuzzy DR-convergence structure and we present the relationship between (L, M)-topological double fuzzy remote neighborhood system and topological fuzzy DR-convergence structure. In Section 4, a concept of (L, M)-strong double fuzzy remote neighborhood system is proposed and its connections with the pretopological fuzzy DR-convergence structure are also investigated. In section 5, we discuss the relationship between topological fuzzy DR-convergence structure and pretopological fuzzy DR-convergence structure.

2 Preliminaries

Let a and b be elements in a complete lattice L. An element a ∈ L is said to be coprime if a ≤ b ∨ c implies that a ≤ b or a ≤ c. The set of all coprimes of L is denoted by M (L). Let e ∈ M (L) and e|a denote the set {b ∈ L : e≰b, b ≥ a}. We say a is a wedge below b, in symbol, a) b or b ( a, if for every subset D ⊆ L, ⋁D ≥ b implies a ≤ d for some d ∈ D. We denote β (a) = {b : b) a} and β^* (a) = β (a) ∩ M (L) for each a ∈ L. Thus a = ⋁ β^* (a) = ⋁ β (a) holds for each a ∈ L. 0_L (1_L) and 0_M (1_M) denotes the smallest (largest) elements of lattice L and M respectively.

Definition 2.1. [27] Let L and M be complete lattices and f : L → M. f will be called a generalized order-homomorphism (briefly, GOH), if

f (a) =0_M if and only if a = 0_L,

f is union-preserving,

f^⊢ is union-preserving, where f^⊢ (b) = ⋁ {a ∈ L : f (a) ≤ b} for all b ∈ M.

Theorem 2.2. [27] Let f : L → M be a GOH. Then

f and f^⊢ are order-preserving.

f^⊢ (f (a)) ≥ a, for each a ∈ L.

f (f^⊢ (b)) ≤ b, for each b ∈ M.

f (a) ≤ b if and only if a ≤ f^⊢ (b).

f (a) = ⋀ {b ∈ M : f^⊢ (a) ≥ a}, for each a ∈ L.

f^⊢ : M → L is intersection-preserving.

Definition 2.3. [3] (1) Category A is said to be a subcategory of category B provided that the following conditions are satisfied:

Ob (A) ⊆ Ob (B),

for each A, A^′ ∈ Ob(A), hom_A^{(A, A′)} ⊆ hom_B^{(A, A′)},

for each A-object A, the B-identity on A is the A-identity on A,

the composition law in A is the restriction of the composition law in B to the morphisms of A.

(2) A is called a full subcategory of B if, in addition to the above, for each A, A^′ ∈ Ob(A), hom_A^(A,A′) = hom_B^(A,A′).

(3) Let B be a category and E be a class of B-bimorphisms. A full subcategory A of B is called E-reflective (or bireflective) in B provided that each B-object has an A-reflection arrow in E as a bimorphism. This means that, for any B-object B, there exists an A-reflection (or A-reflection bimorphism) r : B → A from B to an A-object A with the following universal property: for any morphism f : B → A′ from B into some A-object A′, there exists a unique A-morphism f′ : A → A′ such that f′ ∘ r = f.

Definition 2.4. [3] Category A is topological w.r.t. category X and functor V : A → X if and only if each V-structured source in X has a unique, initial V-lift in A.

Remark 2.5. In [20], Rodabaugh also gives a longer definition of topological category to make it absolutely clear what is the intention of [3] in this definition w.r.t. each of the terms “V-structured source”, “V-lift”, “initial”, and “unique”.

Definition 2.6. [2] Let L and M be completely distributive lattices. The pair (η, η^*) of maps η, η^* : L → M is called an (L, M)-double fuzzy co-topology on L if it satisfies the following conditions:

(DFCT1) η (u) ≤ (η^* (u)) ′, for each u ∈ L,

(DFCT2) η (0_L) = η (1_L) =1_M, η^* (0_L) = η^* (1_L) =0_M,

(DFCT3) η (u ∨ v) ≥ η (u) ∧ η (v), and η^* (u ∨ v) ≤ η^* (u) ∨ η^* (v), for any u, v ∈ L,

(DFCT4) η (⋀ _i∈Γu_i) ≥ ⋀ _i∈Γη (u_i), and η^* (⋀ _i∈Γu_i) ≤ ⋁ _i∈Γη^* (u_i), for any family {u_i : i ∈ Γ} ⊆ L.

If (η, η^*) is an (L, M)-double fuzzy co-topology, then we say that (L, η, η^*) is an (L, M)-double fuzzy topological molecular lattice (briefly, (L, M)-DFTML). The value of η (u) can be interpreted as the degree of closeness of u ∈ L, and the value of η^* (u) can be interpreted as the degree of non-closeness of u ∈ L.

A continuous map between two (L, M)-DFTMLs (L₁, η, η^*) and (L₂, δ, δ^*) is a GOH f : L₁ → L₂ such that: $η (f^{⊢} (u)) \geq δ (u) and η^{*} (f^{⊢} (u)) \leq δ^{*} (u), \forall u \in L_{2} .$

The category of (L, M)-DFTMLs with their continuous maps is denoted by (L, M)-DFTML.

Definition 2.7. [2] Let L and M be completely distributive lattices. An (L, M)-double fuzzy remote neighborhood system is defined to be a set $(R, R^{*}) = {(R_{e}, R_{e}^{*}) : e \in M (L)}$ of maps $R_{e}, R_{e}^{*} : L \to M$ such that ∀u, v ∈ L,

(DFR1) $R_{e} (u) \leq (R_{e}^{*} (u))^{'}$ ,

(DFR2) R_e (0_L) =1_M, R_e (1_L) =0_M, and $R_{e}^{*} (0_{L}) = 0_{M}$ , $R_{e}^{*} (1_{L}) = 1_{M}$ ,

(DFR3) R_e (u) ≠0_M, $R_{e}^{*} (u) \neq 1_{M}$ implies e≰u,

(DFR4) R_e (u ∨ v) = R_e (u) ∧ R_e (v) and $R_{e}^{*} (u \lor v) = R_{e}^{*} (u) \lor R_{e}^{*} (v)$ .

The triplet (L, R, R^*) is called an (L, M)-double fuzzy remote neighborhood space (briefly, (L, M)-DFRNS), and it will be called (L, M)-topological double fuzzy remote neighborhood space (briefly, (L, M)-TDFRNS) if it satisfies moreover, for all e ∈ M (L), u ∈ L,

(DFR5) R_e (u) = ⋁ _v∈e|u ⋀ _s≰vR_s (v), and $R_{e}^{*} (u) = ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*} (v)$ .

If (R, R^*) and (S, S^*) are two (L, M)-double fuzzy remote neighborhood systems, we say that (R, R^*) is finer than (S, S^*) (or (S, S^*) is coarser than (R, R^*)), denoted by (S, S^*) ≤ (R, R^*) if and only if S_e (u) ≤ R_e (u) and $S_{e}^{*} (u) \geq R_{e}^{*} (u)$ , for each u ∈ L.

A continuous map between two (L, M)-DFRNSs (L₁, R, R^*) and (L₂, S, S^*) is a GOH f : L₁ → L₂ such that for each e ∈ c (L₁) and u ∈ L₂, $S_{f (e)} (u) \leq R_{e} (f^{⊢} (u)), S_{f (e)}^{*} (u) \geq R_{e}^{*} (f^{⊢} (u)) .$

The category of (L, M)-DFRNSs with their continuous maps is denoted by (L, M)-DFRN, and (L, M)-TDFRN the full subcategory of (L, M)-DFRN consisting of (L, M)-TDFRNSs.

Theorem 2.8. [2] (L, M)-TDFRN is isomorphic to (L, M)-DFTML.

Definition 2.9. [2] The pair $(I, I^{*})$ of maps $I, I^{*} : L \to M$ is called an (L, M)-double fuzzy ideal on L if it satisfies the following conditions:

(DFI1) $I (u) \leq (I^{*} (u))^{'}$ , ∀ u ∈ L,

(DFI2) $I (0_{L}) = 1_{M}$ , $I (1_{L}) = 0_{M}$ and $I^{*} (0_{L}) = 0_{M}$ , $I^{*} (1_{L}) = 1_{M}$ ,

(DFI3) $I (u \lor v) \geq I (u) \land I (v)$ and $I^{*} (u \lor v) \leq I^{*} (u) \lor I^{*} (v)$ , for each u, v ∈ L,

(DFI4) If u ≤ v, then $I (u) \geq I (v)$ and $I^{*} (u) \leq I^{*} (v)$ .

If $(I_{1}, I_{1}^{*})$ and $(I_{2}, I_{2}^{*})$ are two (L, M)-double fuzzy ideals on L, we say that $(I_{1}, I_{1}^{*})$ is finer than $(I_{2}, I_{2}^{*})$ (or $(I_{2}, I_{2}^{*})$ is coarser than $(I_{1}, I_{1}^{*})$ ), denoted by $(I_{2}, I_{2}^{*}) \leq (I_{1}, I_{1}^{*})$ if and only if $I_{2} (u) \leq I_{1} (u)$ and $I_{2}^{*} (u) \geq I_{1}^{*} (u)$ , for each u ∈ L.

The set of all (L, M)-double fuzzy ideals on L is denoted by $D I (L^{M})$ .

Example 2.10. [2] For each e ∈ M (L), we define r (e), r^* (e) : L → M as follows: $\begin{matrix} \forall u \in L, r (e) (u) & = & {\begin{matrix} 1_{M}, & if e ≰ u \\ 0_{M}, & if e \leq u . \end{matrix} \\ r^{*} (e) (u) & = & {\begin{matrix} 0_{M}, & if e ≰ u \\ 1_{M}, & if e \leq u . \end{matrix} \end{matrix}$ Then, (r (e), r^* (e)) is an (L, M)-double fuzzy ideal.

Proposition 2.11. [2] Let $(I, I^{*})$ be an (L, M)-double fuzzy ideal and f : L → L₁ be a GOH. Define the maps $f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}) : L_{1} \to M$ by: $\begin{matrix} f^{\Rightarrow} (I) (u) & = & ⋁_{f^{⊢} (u) \leq v} I (v) = I (f^{⊢} (u)), \\ f^{\Rightarrow} (I^{*}) (u) & = & ⋀_{f^{⊢} (u) \leq v} I (v) = I^{*} (f^{⊢} (u)), \end{matrix}$ for each u ∈ L₁. Then $(f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))$ is an (L, M)-double fuzzy ideal on L₁ and is called the image of $(I, I^{*})$ under f.

Lemma 2.12. [2] Suppose that ${(I_{i}, I_{i}^{*})}_{i \in Γ}$ is the set of (L, M)-double fuzzy ideals. Then, $⋀_{i \in Γ} (I_{i}, I_{i}^{*}) = (⋀_{i \in Γ} I_{i}, ⋁_{i \in Γ} I_{i}^{*})$ is an (L, M)-double fuzzy ideal where, $⋀_{i \in Γ} I_{i}, ⋁_{i \in Γ} I_{i}^{*} : L \to M$ and $(⋀_{i \in Γ} I_{i}) (u) = ⋀_{i \in Γ} I_{i} (u)$ , $(⋁_{i \in Γ} I_{i}^{*}) (u) = ⋁_{i \in Γ} I_{i}^{*} (u), \forall u \in L$ .

Lemma 2.13. Let $(I, I^{*}) \in D I (L^{M})$ , f : L₁ → L₂ and g : L₂ → L₃ be a GOHs. Then we have: $(g \circ f)^{\Rightarrow} (I) = g^{\Rightarrow} \circ f^{\Rightarrow} (I)$ and $(g \circ f)^{\Rightarrow} (I^{*}) = g^{\Rightarrow} \circ f^{\Rightarrow} (I^{*})$ .

Proof. It is obvious.

Definition 2.14. [2] The pair $(B, B^{*})$ of maps $B, B^{*} : L \to M$ is called an (L, M)-double fuzzy ideal base if it satisfies the following conditions:

(DFIB1) $B (u) \leq (B^{*} (u))^{'}$ , ∀ u ∈ L,

(DFIB2) $B (1_{L}) = 0_{M}$ , $⋁_{u \in L} B (u) = 1_{M}$ and $B^{*} (1_{L}) = 1_{M}$ , $⋀_{u \in L} B^{*} (u) = 0_{M}$ ,

(DFIB3) $〈 B 〉 (u \lor v) \geq B (u) \land B (v)$ and $〈 B^{*} 〉 (u \lor v) \leq B^{*} (u) \lor B^{*} (v)$ , ∀ u, v ∈ L.

Proposition 2.15. [2] If $(B, B^{*})$ is an (L, M)-double fuzzy ideal base, then $(〈 B 〉, 〈 B^{*} 〉)$ is an (L, M)-double fuzzy ideal, and $(B, B^{*})$ is called an (L, M)-double fuzzy ideal base of $(〈 B 〉, 〈 B^{*} 〉)$ .

Proposition 2.16. [2] Let (L, R, R^*) be an (L, M)-topological double fuzzy remote neighborhood space. Then, for each e ∈ M (L), the pair $(B_{e}, B_{e}^{*})$ of maps $B_{e}, B_{e}^{*} : L \to M$ defined by $B_{e} (u) = R_{e} (u) \land (⋀_{s #x2270; u} R_{s} (u))$ and $B_{e}^{*} (u) = R_{e}^{*} (u) \lor (⋁_{s ≰ u} R_{s}^{*} (u))$ , for each u ∈ L, is an (L, M)-double fuzzy ideal base of $(R_{e}, R_{e}^{*})$ .

For more details about double fuzzy concepts see [4 , 21].

3 Fuzzy DR-convergence structures

Definition 3.1. The map $c : D I (M^{L}) \to L$ is called a fuzzy DR-convergence structure (briefly, FDRCS) on L if it satisfies the following conditions:

(FDRC1) e ≤ c (r (e), r^* (e)), for each e ∈ M (L),

(FDRC2) $c (I, I^{*}) \leq c (G, G^{*})$ , for each $(I, I^{*}), (G, G^{*}) \in D I (M^{L})$ with $(I, I^{*}) \leq (G, G^{*})$ .

The triplet $(D I (M^{L}), L, c)$ is called a fuzzy DR-convergence space (briefly, FDRCS space), and it will be called pretopological if it satisfies

(FDRC3) $e \leq c (I_{e}^{c}, {I^{*}}_{e}^{c})$ , where $I_{e}^{c} = ⋀_{e \leq c (I, I^{*})} I$ and ${I^{*}}_{e}^{c} = ⋁_{e \leq c (I, I^{*})} I^{*}$ .

The triplet $(D I (M^{L}), L, c)$ is called a topological fuzzy DR-convergence space if it satisfies moreover,

(FDRC4) For all e ∈ M (L), the (L, M)-double fuzzy ideal $(I_{e}^{c}, {I^{*}}_{e}^{c})$ has an (L, M)-double fuzzy ideal base $(B_{e}^{c}, {B^{*}}_{e}^{c})$ such that $B_{e}^{c} (u) \leq B_{s}^{c} (u)$ and ${B^{*}}_{e}^{c} (u) \geq {B^{*}}_{s}^{c} (u)$ , ∀u ∈ L with s≰u.

A continuous map between two FDRCS spaces $(D I (M^{L_{1}}), L_{1}, c)$ and $(D I (M^{L_{2}}), L_{2}, d)$ is a GOH f : L₁ → L₂ such that for each $(I, I^{*}) \in I (M^{L_{1}})$ , $c (I, I^{*}) \leq f^{⊢} (d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))),$ or equivalently $f (c (I, I^{*})) \leq d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*})) .$

The category of pretopological FDRCS spaces and their continuous maps is denoted by PrFDRCS and TFDRCS denotes the full subcategory of PrFDRCS consisting of topological FDRCS spaces.

Lemma 3.2. Suppose that $(D I (M^{L}), L, c)$ is a pretopological FDRCS space. Then, we have

$e \leq c (I, I^{*})$ if and only if $(I_{e}^{c}, {I^{*}}_{e}^{c}) \leq (I, I^{*})$ for each e ∈ M (L), $(I, I^{*}) \in D I (M^{L})$ .

For e ∈ M (L) and u ∈ L, $I_{e}^{c} (u) = 0_{M}$ and ${I^{*}}_{e}^{c} (u) = 1_{M}$ whenever e ≤ u.

Proof. (i) Suppose that, $e \leq c (I, I^{*})$ . Then $I_{e}^{c} = ⋀_{e \leq c (I, I^{*})} I \leq I$ and ${I^{*}}_{e}^{c} = ⋁_{e \leq c (I, I^{*})} I^{*} \geq I^{*}$ .

Conversely, suppose that $(I_{e}^{c}, {I^{*}}_{e}^{c}) \leq (I, I^{*})$ . By (DFRC2), $c (I_{e}^{c}, {I^{*}}_{e}^{c}) \leq c (I, I^{*})$ and by (DFRC3), we have $e \leq c (I, I^{*})$ .

(ii) By (FRC1), e ≤ c (r (e), r^* (e)). Then, $I_{e}^{c} (u) = ⋀_{e \leq c (I, I^{*})} I (u) \leq r (e) (u) = 0_{M}$ and ${I^{*}}_{e}^{c} (u) = ⋁_{e \leq c (I, I^{*})} I^{*} (u) \geq r^{*} (e) (u) = 1_{M}$ .

Definition 3.3. Suppose that e ∈ M (L), $(I, I^{*}) \in D I (M^{L})$ and $e \leq c (I, I^{*})$ . Then, we say that $(I, I^{*})$ convergence to e.

Let $(R, R^{*}) = {(R_{e}, R_{e}^{*}) : e \in M (L)}$ be an (L, M)-TDFRNS. Define a map $c^{R, R^{*}} : D I (M^{L}) \to L$ such that for each $(I, I^{*}) \in D I (M^{L})$ , $c^{R, R^{*}} (I, I^{*}) = ⋁_{(R_{a}, R_{a}^{*}) \leq (I, I^{*})} a .$ We will prove that the map $c^{R, R^{*}} : D I (M^{L}) \to L$ is a topological FDRCS on L, which means that there is a way to obtain topological fuzzy DR-convergence structure from (L, M)-topological double fuzzy remote neighborhood system. In order to do this, the following lemma is necessary.

Lemma 3.4. For each e ∈ M (L), $(I, I^{*}) \in D I (M^{L})$ , $(R_{e}, R_{e}^{*}) \leq (I, I^{*})$ if and only if $e \leq c^{R, R^{*}} (I, I^{*})$ .

Proof. Suppose that $(R_{e}, R_{e}^{*}) \leq (I, I^{*})$ . Then, $c^{R, R^{*}} (I, I^{*}) = ⋁_{(R_{a}, R_{a}^{*}) \leq (I, I^{*})} a \geq e .$

Conversely, suppose that $e \leq c^{R, R^{*}} (I, I^{*})$ and we will prove that $R_{e} (u) \leq I (u)$ and $R_{e}^{*} (u) \geq I^{*} (u)$ , for each u ∈ L. Let α) R_e (u), since $R_{e} (u) = ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s} (v),$ there exists some v_α ∈ L such that e≰v_α, v_α ≥ u and ∀s≰v_α, α ≤ R_s (v_α). Therefore, since e≰v_α and $e \leq c^{R, R^{*}} (I, I^{*})$ , we have $c^{R, R^{*}} (I, I^{*}) ≰ v_{α}$ . Then, there exists z_m ∈ L such that $z_{m} \leq c^{R, R^{*}} (I, I^{*})$ and z_m≰v_α. Then, by the definition of $c^{R, R^{*}} (I, I^{*})$ , $(R_{z_{m}}, R_{z_{m}}^{*}) \leq (I, I^{*})$ . Since z_m≰v_α, $α \leq R_{z_{m}} (v_{α}) \leq I (v_{α}) \leq I (u) .$

Thus, $I (u) \geq ⋁ {α \in L : α) R_{e} (u)} = R_{e} (u) .$

The last equality holds for the reason that ∀a ∈ L, a = ⋁ {b ∈ L : b) a}, which holds if and only if L is complete distributive lattice.

It remains to prove that $R_{e}^{*} (u) \geq I^{*} (u)$ , for each u ∈ L. So, suppose that there exist u ∈ L and k ∈ M (L) such that $R_{e}^{*} (u) \leq k < I^{*} (u)$ . Take α ∈ M (L) such that $R_{e}^{*} (u)) α \leq k < I^{*} (u)$ . Since $R_{e}^{*} (u) = ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*} (v),$ for each v ∈ L such that e≰v, v ≥ u we have $⋁_{s ≰ v} R_{s}^{*} (v) \leq α$ . Then, $R_{s}^{*} (v) \leq α$ , for each s≰v. Therefore, since e≰v and $e \leq c^{R, R^{*}} (I, I^{*})$ , we have $c^{R, R^{*}} (I, I^{*}) ≰ v$ . Then, there exists z_m ∈ L such that $z_{m} \leq c^{R, R^{*}} (I, I^{*})$ and z_m≰v. Then, by the definition of $c^{R, R^{*}} (I, I^{*})$ , $(R_{z_{m}}, R_{z_{m}}^{*}) \leq (I, I^{*})$ . Since z_m≰v, $I^{*} (u) \leq I^{*} (v) \leq R_{z_{m}}^{*} (v) \leq α .$ A contradiction. Thus $R_{e}^{*} (u) \geq I^{*} (u)$ , for each u ∈ L.

Proposition 3.5. (i) $c^{R, R^{*}} : D I (M^{L}) \to L$ defined above is a topological FDRCS on L, called induced topological FDRCS from $(R, R^{*}) = {(R_{e}, R_{e}^{*}) : e \in M (L)}$ . Moreover, if (P, P^*) and (R, R^*) are (L, M)-topological double fuzzy remote neighborhood systems on L which induce the same topological FDRCS (i.e., c^{P,P^*} = c^{R,R^*}), then (P, P^*) = (R, R^*).

(ii) If a map f : (L₁, P, P^*) → (L₂, R, R^*) is continuous between (L, M)-TDFRNSs, then f is continuous with respect to the induced topological FDRCS spaces.

Proof. (i) (FDRC1) Since $(R_{e}, R_{e}^{*}) \leq (r (e), r^{*} (e))$ , we have $c^{R, R^{*}} (r (e), r^{*} (e)) = ⋁_{(R_{a}, R_{a}^{*}) \leq (r (e), r^{*} (e))} a \geq e .$

(FDRC2) For each $(I, I^{*}), (G, G^{*}) \in D I (M^{L})$ such that $(I, I^{*}) \leq (G, G^{*})$ , we have $\begin{matrix} c^{R, R^{*}} (I, I^{*}) & = & ⋁_{(R_{a}, R_{a}^{*}) \leq (I, I^{*})} a \\ \leq & ⋁_{(R_{a}, R_{a}^{*}) \leq (G, G^{*})} a \\ = & c^{R, R^{*}} (G, G^{*}) . \end{matrix}$

(FDRC3) $\begin{matrix} (R_{e}, R_{e}^{*}) & = & ⋀_{(R_{e}, R_{e}^{*}) \leq (I, I^{*})} (I, I^{*}) \\ = & ⋀_{e \leq c^{R, R^{*}} (I, I^{*})} (I, I^{*}) (by lemma 3.4) \\ = & (⋀_{e \leq c^{R, R^{*}} (I, I^{*})} I, ⋁_{e \leq c^{R, R^{*}} (I, I^{*})} I^{*}) \\ = & (I_{e}^{c^{R, R^{*}}}, {I^{*}}_{e}^{c^{R, R^{*}}}), \end{matrix}$ then $e \leq c^{R, R^{*}} (I_{e}^{c^{R, R^{*}}}, {I^{*}}_{e}^{c^{R, R^{*}}})$ .

(FDRC4) From the proof of (FDRC3), $(R_{e}, R_{e}^{*}) = (I_{e}^{c^{R, R^{*}}}, {I^{*}}_{e}^{c^{R, R^{*}}})$ holds. We define $(B_{e}^{c^{R, R^{*}}}, {B^{*}}_{e}^{c^{R, R^{*}}})$ as follows: for each u ∈ L, $\begin{matrix} B_{e}^{c^{R, R^{*}}} (u) = R_{e} (u) \land (⋀_{m ≰ u} R_{m} (u)), \\ {B^{*}}_{e}^{c^{R, R^{*}}} (u) = R_{e}^{*} (u) \lor (⋁_{m ≰ u} R_{m}^{*} (u)) . \end{matrix}$

By Proposition 2.16, $(B_{e}^{c^{R, R^{*}}}, {B^{*}}_{e}^{c^{R, R^{*}}})$ is an (L, M)-double fuzzy ideal base of $(I_{e}^{c^{R, R^{*}}}, {I^{*}}_{e}^{c^{R, R^{*}}})$ . Moreover, for each u ∈ L with s≰u, we have: $\begin{matrix} B_{e}^{c^{R, R^{*}}} (u) & = & R_{e} (u) \land (⋀_{m ≰ u} R_{m} (u)) \\ \leq & ⋀_{m ≰ u} R_{m} (u) \\ = & R_{s} (u) \land (⋀_{m ≰ u} R_{m} (u)) \\ = & B_{s}^{c^{R, R^{*}}} (u), \\ {B^{*}}_{e}^{c^{R, R^{*}}} (u) & = & R_{e}^{*} (u) \lor (⋁_{m ≰ u} R_{m}^{*} (u)) \\ \geq & ⋁_{m ≰ u} R_{m}^{*} (u) \\ = & R_{s}^{*} (u) \lor (⋁_{m ≰ u} R_{m}^{*} (u)) \\ = & {B^{*}}_{s}^{c^{R, R^{*}}} (u) . \end{matrix}$

Hence, c^{R,R^*} is a topological FDRCS on L.

In addition, suppose that c is the same topological FDRCS. By Lemma 3.4, for each e ∈ M (L), $(I, I^{*}) \in D I (M^{L})$ , it holds that $\begin{matrix} (R_{e}, R_{e}^{*}) \leq (I, I^{*}) & \Leftrightarrow & e \leq c (I, I^{*}) \\ \Leftrightarrow & (P_{e}, P_{e}^{*}) \leq (I, I^{*}), \end{matrix}$ which means $(R_{e}, R_{e}^{*}) = (P_{e}, P_{e}^{*})$ for each e ∈ M (L), or equivalently (R, R^*) = (P, P^*).

(ii) Let $(D I (M^{L_{1}}), L_{1}, c^{P, P^{*}})$ and $(D I (M^{L_{2}}), L_{2}, c^{R, R^{*}})$ be two topological FDRCS spaces induced by (L₁, P, P^*) and (L₂, R, R^*) respectively.

For the continuity of $f : (D I (M^{L_{1}}), L_{1}, c^{P, P^{*}}) \to (D I (M^{L_{2}}), L_{2}, c^{R, R^{*}})$ we only need to prove, for each $(I, I^{*}) \in D I (M^{L_{1}})$ , $c^{P, P^{*}} (I, I^{*}) \leq f^{⊢} (c^{R, R^{*}} (f^{\Rightarrow} (I)), f^{\Rightarrow} (I^{*})) .$

For any α ∈ M (L₁) with the property $α \leq c^{P, P^{*}} (I, I^{*})$ , we have: $\begin{matrix} α & \leq & c^{P, P^{*}} (I, I^{*}) \\ \Leftrightarrow & (P_{α}, P_{α}^{*}) \leq (I, I^{*}) (by Lemma 3.4) \\ \Rightarrow & R_{f (α)} (u) \leq P_{α} (f^{⊢} (u)) \leq I (f^{⊢} (u)) and \\ R_{f (α)}^{*} (u) \geq P_{α}^{*} (f^{⊢} (u)) \geq I^{*} (f^{⊢} (u)), \forall u \in L_{2}, \\ (since f : (L_{1}, P, P^{*}) \to (L_{2}, R, R^{*}) \\ is continuous) \\ \Leftrightarrow & (R_{f (α)}, R_{f (α)}^{*}) \leq (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*})) \\ \Leftrightarrow & f (α) \leq c^{R, R^{*}} (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*})) \\ (by Lemma 3.4) \\ \Rightarrow & α \leq f^{⊢} (c^{R, R^{*}} (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))) . \end{matrix}$

Then for each $(I, I^{*}) \in D I (M^{L_{1}})$ , $c^{P, P^{*}} (I, I^{*}) \leq f^{⊢} (c^{R, R^{*}} (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))) .$

Hence, $f : (D I (M^{L_{1}}), L_{1}, c^{P, P^{*}}) \to (D I (M^{L_{2}}), L_{2}, c^{R, R^{*}})$ is continuous.

Conversely, suppose that $c : D I (M^{L}) \to L$ is a topological FDRCS. Define a set $(R^{c}, R^{*^{c}}) = {(R_{e}^{c}, R_{e}^{*^{c}}) : e \in M (L)}$ , where $R_{e}^{c} = I_{e}^{c}$ and $R_{e}^{*^{c}} = I_{e}^{*^{c}}$ for any e ∈ M (L). Then, we have

Proposition 3.6. (i) The set $(R^{c}, R^{*^{c}}) = {(R_{e}^{c}, R_{e}^{*^{c}}) : e \in M (L)}$ defined above is an (L, M)-topological fuzzy remote neighborhood system on L, called induced (L, M)-topological fuzzy remote neighborhood system from c. Moreover, if c and d are topological FDRCSs on L which induce the same (L, M)-topological fuzzy remote neighborhood system (i.e., (R^c, R^{*^c}) = (R^d, R^{*^d})), then c = d.

(ii) If a map $f : (D I (M^{L_{1}}), L_{1}, c) \to (D I (M^{L_{2}}), L_{2}, d)$ is continuous between topological FDRCS spaces, then f : (L₁, R^c, R^{*^c}) → (L₂, R^d, R^{*^d}) is continuous with respect to the induced TFRNSs.

Proof. (i)(DFRN1), (DFRN2) and (DFRN4) are straightforward.

(DFRN3) If $R_{e}^{c} (u) \neq 0_{M}$ , $R_{e}^{*^{c}} (u) \neq 1_{M}$ , then $I (u) \neq 0_{M}$ , $I^{*} (u) \neq 1_{M}$ for each $(I, I^{*}) \in D I (M^{L})$ with $e \leq c (I, I^{*})$ . Since e ≤ c (r (e), r^* (e)), then r (e) (u) ≠0_M and r^* (e) (u) ≠1_M, which implies that e≰u.

(DFRN5) It can be easily shown that:

$⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c} (v) \leq I_{e}^{c} (u) and ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v) \geq I_{e}^{*^{c}} (u), \forall u \in L, e \in M (L) .$

What remains to prove is that:

$I_{e}^{c} (u) \leq ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c} (v) and I_{e}^{*^{c}} (u) \geq ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v), \forall u \in L, e \in M (L) .$

According to (FDRC4), for each e ∈ M (L), the (L, M)-double fuzzy ideal $(I_{e}^{c}, I_{e}^{*^{c}})$ has an (L, M)-double fuzzy ideal base $(B_{e}^{c}, B_{e}^{*^{c}})$ such that $B_{e}^{c} (v) \leq B_{s}^{c} (v)$ and $B_{e}^{*^{c}} (v) \geq B_{s}^{*^{c}} (v)$ for each v ∈ L with s≰v. First let α ∈ M (L) with the property $α) I_{e}^{c} (u)$ . It follows from $I_{e}^{c} (u) = 〈 B_{e}^{c} 〉 (u) = ⋁_{v \geq u} B_{e}^{c} (v)$ that there exists some v_α ∈ L with v_α ≥ u such that $α \leq B_{e}^{c} (v_{α})$ . Then $α \leq B_{e}^{c} (v_{α}) \leq I_{e}^{c} (v_{α}) .$

Considering the property of $(B_{e}^{c}, B_{e}^{*^{c}})$ , $B_{e}^{c} (v_{α}) \leq B_{s}^{c} (v_{α}) \leq I_{s}^{c} (v_{α}), for each s ≰ v_{α} .$

Then, we have $B_{e}^{c} (v_{α}) \leq ⋀_{s ≰ v_{α}} I_{s}^{c} (v_{α}) .$ Then: $\begin{matrix} α \leq B_{e}^{c} (v_{α}) & \leq & I_{e}^{c} (v_{α}) \land ⋀_{s ≰ v_{α}} I_{s}^{c} (v_{α}) \\ \leq & ⋁_{v \geq u} (I_{e}^{c} (v) \land ⋀_{s ≰ v} I_{s}^{c} (v)) \\ = & ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c} (v) . (by Lemma 3.2 (ii)) \end{matrix}$

Therefore, $I_{e}^{c} (u) \leq ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c} (v)$ .

Second, suppose that there exist u ∈ L and k ∈ M (L) such that $I_{e}^{*^{c}} (u) \leq k < ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v)$ . Take α ∈ M (L) such that $I_{e}^{*^{c}} (u)) α \leq k < ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v) .$

Since $I_{e}^{*^{c}} (u) = 〈 B_{e}^{*^{c}} 〉 (u) = ⋀_{v \geq u} B_{e}^{*^{c}} (v)$ , for each v ∈ L with v ≥ u, $B_{e}^{*^{c}} (v) \leq α$ . Then, $I_{e}^{*^{c}} (v_{α}) \leq B_{e}^{*^{c}} (v_{α}) \leq α .$

Considering the property of $(B_{e}^{c}, B_{e}^{*^{c}})$ , $B_{e}^{*^{c}} (v) \geq B_{s}^{*^{c}} (v) \geq I_{s}^{*^{c}} (v), for each s ≰ v_{α} .$

Then, we have $B_{e}^{*^{c}} (v) \geq ⋁_{s ≰ v} I_{s}^{*^{c}} (v) .$ Then: $\begin{matrix} α \geq B_{e}^{*^{c}} (v) & \geq & I_{e}^{*^{c}} (v) \lor ⋁_{s ≰ v} I_{s}^{*^{c}} (v) \\ \geq & ⋀_{v \geq u} (I_{e}^{*^{c}} (v) \lor ⋁_{s ≰ v} I_{s}^{*^{c}} (v)) \\ = & ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v) . (by Lemma 3.2 (ii)) \end{matrix}$

A contradiction, then $I_{e}^{*^{c}} (u) \geq ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v)$ , for each u ∈ L, e ∈ M (L) .

Additionally, if $(R, R^{*}) = {(R_{e}, R_{e}^{*}) : e \in M (L)}$ is the same induced (L, M)-topological double fuzzy remote neighborhood system, then we have $(I_{e}^{c}, I_{e}^{*^{c}}) = (R_{e}, R_{e}^{*}) = (I_{e}^{d}, I_{e}^{*^{d}}), \forall e \in M (L) .$

By Lemma 3.2(i), it follows immediately that for each $(I, I^{*}) \in D I (M^{L})$ , $\begin{matrix} e \leq c (I, I^{*}) & \Leftrightarrow & (I_{e}^{c}, I_{e}^{*^{c}}) \leq (I, I^{*}) \\ \Leftrightarrow & (I_{e}^{d}, I_{e}^{*^{d}}) \leq (I, I^{*}) \\ \Leftrightarrow & e \leq d (I, I^{*}), \end{matrix}$ hence c = d.

(ii) For each e ∈ M (L₁), u ∈ L₂, we have: $\begin{matrix} R_{e}^{c} (f^{⊢} (u)) & = & ⋀_{e \leq c (I, I^{*})} I (f^{⊢} (u)) \\ = & ⋀_{e \leq c (I, I^{*})} f^{\Rightarrow} (I) (u) \\ \geq & ⋀_{f (e) \leq f (c (I, I^{*}))} f^{\Rightarrow} (I) (u) \\ \geq & ⋀_{f (e) \leq d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))} f^{\Rightarrow} (I) (u) \\ \geq & ⋀_{f (e) \leq d (G, G^{*})} G (u) \\ = & R_{f (e)}^{d} (u), \\ R_{e}^{*^{c}} (f^{⊢} (u)) & = & ⋁_{e \leq c (I, I^{*})} I^{*} (f^{⊢} (u)) \\ = & ⋁_{e \leq c (I, I^{*})} f^{\Rightarrow} (I^{*}) (u) \\ \leq & ⋁_{f (e) \leq f (c (I, I^{*}))} f^{\Rightarrow} (I^{*}) (u) \\ \leq & ⋁_{f (e) \leq d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))} f^{\Rightarrow} (I^{*}) (u) \\ \leq & ⋁_{f (e) \leq d (G, G^{*})} G^{*} (u) \\ = & R_{f (e)}^{*^{d}} (u), \end{matrix}$

Hence, f : (L₁, R^c, R^{*^c}) → (L₂, R^d, R^{*^d}) is continuous.

Remark 3.7. From the process of checking (DFRN5) in the above proposition, we can easily find that if $(D I (M^{L}), L, c) \in ∣ TFDRCS ∣$ , then $I_{e}^{c} (u) = ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c} (v)$ and $I_{e}^{*^{c}} (u) = ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v)$ hold.

From Propositions 3.5 and 3.6, we obtain the main result in this section.

Theorem 3.8. (L, M)-TDFRN is isomorphic to TFDRCS.

Since (L, M)-DFTML is isomorphic to (L, M)-TDFRN, we obtain

Corollary 3.9. (L, M)-DFTML is isomorphic to TFDRCS.

4 The category (L, M)-SDFRN is isomorphic to PrFDRCS

In order to obtain an (L, M)-double fuzzy remote neighborhood space in [2] that can be identified with a pretopological fuzzy DR-convergence space, an additional condition is needed.

We call (L, R, R^*)∈ ∣ (L, M) - DFRN ∣ an (L, M)-strong double fuzzy remote neighborhood space if it satisfies

(DFR) $(R_{e}, R_{e}^{*}) = (⋁_{s \in β^{*} (e)}^{I} R_{s}, ⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*})$ .

The category of (L, M)-strong double fuzzy remote neighborhood spaces as a full subcategory of (L, M)-DFRN is denoted by (L, M)-SDFRN.

Remark 4.1. In general, $⋁_{i \in Γ} (I_{i}, I_{i}^{*}) = (⋁_{i \in Γ} I_{i}, ⋀_{i \in Γ} I_{i}^{*})$ , where $⋁_{i \in Γ} I_{i}, ⋀_{i \in Γ} I_{i}^{*} : L \to M$ in the sense of $(⋁_{i \in Γ} I_{i}) (u) = ⋁_{i \in Γ} I_{i} (u)$ , $(⋀_{i \in Γ} I_{i}^{*}) (u) = ⋀_{i \in Γ} I_{i}^{*} (u)$ , for all u ∈ L is not (L, M)-double fuzzy ideal when $(I_{i}, I_{i}^{*})$ is an (L, M)-double fuzzy ideal for each i ∈ Γ. But the set of ${(R_{s}, R_{s}^{*}) : s \in β^{*} (e)}$ has the property that R_s (resp. $R_{s}^{*}$ ) has a least upper bound and greatest lower bound which are denoted here by $⋁_{s \in β^{*} (e)}^{I} R_{s}$ , $⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*}$ .

Next, we establish the relationship between PrFDRCS and (L, M)-SDFRN.

Let $R = {(R_{e}, R_{e}^{*}) : e \in M (L)}$ be an (L, M)-strong double fuzzy remote neighborhood system on L. We also define $c^{R, R^{*}} : D I (M^{L}) \to L$ by: $c^{R, R^{*}} (I, I^{*}) = ⋁_{(R_{s}, R_{s}^{*}) \leq (I, I^{*})} s, \forall (I, I^{*}) \in D I (M^{L}) .$

For c^{R,R^*}, we have the following lemma.

Lemma 4.2. For (L, R, R^*)∈ ∣ (L, M) - SDFRN ∣, the following statements hold.

For t, s ∈ M (L), t ≤ s implies $(R_{t}, R_{t}^{*}) \leq (R_{s}, R_{s}^{*})$ .

For e ∈ M (L), $(I, I^{*}) \in D I (M^{L})$ , $(R_{e}, R_{e}^{*}) \leq (I, I^{*})$ if and only if $e \leq c^{R, R^{*}} (I, I^{*})$ .

Proof. (i) It is easily obtained from (DFR).

(ii) Suppose that $(R_{e}, R_{e}^{*}) \leq (I, I^{*})$ , then $c^{R, R^{*}} (I, I^{*}) = ⋁_{(R_{s}, R_{s}^{*}) \leq (I, I^{*})} s \geq e .$

Conversely, suppose that $e \leq c^{R, R^{*}} (I, I^{*})$ . It suffices to prove that $(⋁_{s \in β^{*} (e)}^{I} R_{s}, ⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*}) \leq (I, I^{*})$ . For each s ∈ β^* (e), $s ◃ e \leq c^{R, R^{*}} (I, I^{*}) = ⋁_{(R_{a}, R_{a}^{*}) \leq (I, I^{*})} a .$

Thus, there exists some a ∈ M (L) such that $(R_{a}, R_{a}^{*}) \leq (I, I^{*})$ and s ≤ a. From conclusion (i), we have $(R_{s}, R_{s}^{*}) \leq (R_{a}, R_{a}^{*}) \leq (I, I^{*})$ . Thus, $(R_{e}, R_{e}^{*}) = (⋁_{s \in β^{*} (e)}^{I} R_{s}, ⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*}) \leq (I, I^{*}) .$

Proposition 4.3. (i) $c^{R, R^{*}} : D I (M^{L}) \to L$ defined above is a pretopological FDRCS on L, called induced pretopological FDRCS from $(R, R^{*}) = {(R_{e}, R_{e}^{*}) : e \in M (L)}$ . Moreover, if (P, P^*) and (R, R^*) are (L, M)-strong topological double fuzzy remote neighborhood systems on L which induce the same pretopological FDRCS (i.e., c^{P,P^*} = c^{R,R^*}), then (P, P^*) = (R, R^*).

(ii) If a map f : (L₁, P, P^*) → (L₂, R, R^*) is continuous between (L, M)-strong double fuzzy remote neighborhood spaces, then f is continuous with respect to the induced pretopological FDRCS spaces.

Proof. By Lemma 4.2, it can be verified in the similar way of Proposition 3.5.

Conversely, suppose that $c : D I (M^{L}) \to L$ is a pretopological FDRCS on L. Let $(R_{e}^{c}, R_{e}^{*^{c}}) = (I_{e}^{c}, I_{e}^{*^{c}})$ , for all e ∈ M (L). Then, we have

Proposition 4.4. (i) The set $(R^{c}, R^{*^{c}}) = {(R_{e}^{c}, R_{e}^{*^{c}}) : e \in M (L)}$ defined above is an (L, M)-strong double fuzzy remote neighborhood system on L, called induced (L, M)-strong double fuzzy remote neighborhood system from c. Moreover, if c and d are two pretopological FDRCSs on L which induce the same (L, M)-strong double fuzzy remote neighborhood system (i.e., (R^c, R^{*^c}) = (R^d, R^{*^d})),then c = d.

(ii) If a map $f : (D I (M^{L_{1}}), L_{1}, c) \to (D I (M^{L_{2}}), L_{2}, d)$ is continuous between pretopological FDRCS spaces. Then f : (L₁, R^c, R^{*^c}) → (L₂, R^d, R^{*^d}) is continuous with respect to the induced (L, M)-strong double fuzzy remote neighborhood spaces.

Proof. At first, we check if (DFR) holds. On one hand, for each t, s ∈ M (L) with s ≤ t, by the definition of (R^c, R^{*^c}), $\begin{matrix} (R_{s}^{c}, R_{s}^{*^{c}}) & = & (⋀_{s \leq c (I, I^{*})} I, ⋁_{s \leq c (I, I^{*})} I^{*}) \\ \leq & (⋀_{t \leq c (I, I^{*})} I, ⋁_{t \leq c (I, I^{*})} I^{*}) \\ = & (R_{t}^{c}, R_{t}^{*^{c}}) . \end{matrix}$

Then, $(R_{s}^{c}, R_{s}^{*^{c}}) \leq (R_{t}^{c}, R_{t}^{*^{c}})$ , for each s ∈ β^* (t). Therefore, for each s ∈ β^* (e), we have $(⋁_{s \in β^{*} (e)}^{I} R_{s}^{c}, ⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*^{c}}) \leq (R_{e}^{c}, R_{e}^{*^{c}}) .$

On the other hand, $\begin{matrix} c (⋁_{s \in β^{*} (e)}^{I} R_{s}^{c}, ⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*^{c}}) & \geq & ⋁_{s \in β^{*} (e)} c (R_{s}^{c}, R_{s}^{*^{c}}) \\ = & ⋁_{s \in β^{*} (e)} c (I_{s}^{c}, I_{s}^{*^{c}}) \\ \geq & ⋁_{s \in β^{*} (e)} s = e . \end{matrix}$

Since $(R_{e}^{c}, R_{e}^{*^{c}}) = (⋀_{e \leq c (I, I^{*})} I, ⋁_{e \leq c (I, I^{*})} I^{*})$ and $e \leq c (⋁_{s \in β^{*} (e)}^{I} R_{s}^{c}, ⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*^{c}})$ , we have $(R_{e}^{c}, R_{e}^{*^{c}}) \leq (⋁_{s \in β^{*} (e)}^{I} R_{s}^{c}, ⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*^{c}}) .$

Hence, $(R_{e}^{c}, R_{e}^{*^{c}}) = (⋁_{s \in β^{*} (e)}^{I} R_{s}^{c}, ⋀_{s \in β^{*} (e)}^{I^{*}} R_{s}^{*^{c}}) .$

The remains follow in the same way as in the proof of Proposition 3.6.

From Proposition 4.3, and 4.4, we obtain

Theorem 4.5. (L, M)-SDFRN is isomorphic toPrFDRCS.

Let CD denote the category of completely distributive lattices with complete lattice morphisms as morphisms. We know that the category of completely distributive lattices with GOHs as its morphisms is the dual category of CD [30]. At the end of this section, we will give the conclusion that PrFDRCS is topological over CD^OP. For this the following lemma is necessary.

Lemma 4.6. For each e ∈ M (L), GOH f : L₁ → L₂, we have $(f^{\Rightarrow} (r (e)), f^{\Rightarrow} (r^{*} (e))) = (r (f (e)), r^{*} (f (e))) .$

Proof. We need to prove that (r (e) (f^⊢ (u)), r^* (e) (f^⊢ (u))) = (r (f (e)) (u), r^* (f (e)) (u)), ∀ u ∈ L₂ . In fact, f (e) ≰ u ⇔ e≰f^⊢ (u) . Thus from the definition of (r (e), r^* (e)), we have (r (e) (f^⊢ (u)), r^* (e) (f^⊢ (u))) = (r (f (e)) (u), r^* (f (e)) (u)).

Proposition 4.7. PrFDRCS is topological over CD^OP.

Proof. The fiber-smallness condition holds trivially. We need to prove that PrFDRCS fulfill the initial lift property. Let $(D I (M^{L_{i}}), L_{i}, c_{i}))_{i \in Γ}$ be a family of pretopological FDRCS spaces. Let further $(f_{i} : N \to (D I (M^{L_{i}}), L_{i}, c_{i}))_{i \in Γ}$ (f_i is GOH for all i ∈ Γ) be a source in PrFDRCS. Define $c^{N} : D I (M^{N}) \to N$ as for each $(I, I^{*}) \in D I (M^{N})$ , $c^{N} (I, I^{*}) = ⋀_{i \in Γ} f_{i}^{⊢} (c_{i} (f_{i}^{\Rightarrow} (I), f_{i}^{\Rightarrow} (I^{*}))) .$ We will show that c^N is a pretopological FDRCS on N which is initial with respect to the source $(f_{i} : N \to (D I (M^{L_{i}}), L_{i}, c_{i}))_{i \in Γ}$ .

Step 1 (V-lift): $(D I (M^{N}), N, c^{N}) \in ∣ PrFDRCS ∣$ .

(FDRC1) $\begin{matrix} c^{N} (r (e), r^{*} (e)) & = & ⋀_{i \in Γ} f_{i}^{⊢} (c_{i} (f_{i}^{\Rightarrow} (r (e)), f_{i}^{\Rightarrow} (r^{*} (e)))) \\ = & ⋀_{i \in Γ} f_{i}^{⊢} (c_{i} (r (f_{i} (e)), r^{*} (f_{i} (e)))) \\ (by Lemma 4.6) \\ \geq & ⋀_{i \in Γ} f_{i}^{⊢} (f_{i} (e)) \\ (by (FDRC 1) of Definition 3.1) \\ \geq & e . (by Theorem 2.2 (ii)) \end{matrix}$

(FRC2) It is straightforward.

(FRC3) $\begin{matrix} c^{N} (I_{e}^{c^{N}}, I_{e}^{*^{c^{N}}}) \\ = ⋀_{i \in Γ} f_{i}^{⊢} (c_{i} (f_{i}^{\Rightarrow} (I_{e}^{c^{N}}), f_{i}^{\Rightarrow} (I_{e}^{*^{c^{N}}}))) \\ = ⋀_{i \in Γ} f_{i}^{⊢} (c_{i} (f_{i}^{\Rightarrow} (⋀_{e \leq c^{N} (I, I^{*})} I), \\ f_{i}^{\Rightarrow} (⋁_{e \leq c^{N} (I, I^{*})} I^{*}))) \\ = ⋀_{i \in Γ} f_{i}^{⊢} (c_{i} (⋀_{e \leq ⋀_{j \in Γ} f_{j}^{⊢} (c_{j} (f_{j}^{\Rightarrow} (I), f_{j}^{\Rightarrow} (I^{*})))} f_{i}^{\Rightarrow} (I), \\ ⋁_{e \leq ⋀_{j \in Γ} f_{j}^{⊢} (c_{j} (f_{j}^{\Rightarrow} (I)), f_{j}^{\Rightarrow} (I^{*})} f_{i}^{\Rightarrow} (I^{*}))) \\ = ⋀_{i \in Γ} ⋁ {a \in N : f_{i} (a) \leq \\ c_{i} (⋀_{e \leq ⋀_{j \in Γ} f_{j}^{⊢} (c_{j} (f_{j}^{\Rightarrow} (I)), f_{j}^{\Rightarrow} (I^{*})))} f_{i}^{\Rightarrow} (I), \\ ⋁_{e \leq ⋀_{j \in Γ} f_{j}^{⊢} (c_{j} (f_{j}^{\Rightarrow} (I)), f_{j}^{\Rightarrow} (I^{*})))} f_{i}^{\Rightarrow} (I^{*}))} \\ \geq ⋀_{i \in Γ} ⋁ {a \in N : f_{i} (a) \leq \\ c_{i} (⋀_{e \leq f_{i}^{⊢} (c_{i} (f_{i}^{\Rightarrow} (I), f_{i}^{\Rightarrow} (I^{*})))} f_{i}^{\Rightarrow} (I), \\ ⋁_{e \leq f_{i}^{⊢} (c_{i} (f_{i}^{\Rightarrow} (I), f_{i}^{\Rightarrow} (I^{*})))} f_{i}^{\Rightarrow} (I^{*}))} \\ = ⋀_{i \in Γ} ⋁ {a \in N : f_{i} (a) \leq \\ c_{i} (⋀_{f_{i} (e) \leq c_{i} (f_{i}^{\Rightarrow} (I), f_{i}^{\Rightarrow} (I^{*}))} f_{i}^{\Rightarrow} (I), \\ ⋁_{f_{i} (e) \leq c_{i} (f_{i}^{\Rightarrow} (I), f_{i}^{\Rightarrow} (I^{*}))} f_{i}^{\Rightarrow} (I^{*}))} \\ \geq ⋀_{i \in Γ} ⋁ {a \in N : f_{i} (a) \leq \\ c_{i} (⋀_{f_{i} (e) \leq c_{i} (G, G^{*})} G, ⋁_{f_{i} (e) \leq c_{i} (G, G^{*})} G^{*})} \\ \geq ⋀_{i \in Γ} ⋁ {a \in N : f_{i} (a) \leq c_{i} (I_{f_{i} (e)}^{c_{i}}, I_{f_{i} (e)}^{*^{c_{i}}})} \\ \geq ⋀_{i \in Γ} ⋁ {a \in N : f_{i} (a) \leq f_{i} (e)} \\ = ⋀_{i \in Γ} f_{i}^{⊢} (f_{i} (e)) \\ \geq e . \end{matrix}$

Step 2 (initial V-lift): Now, we show that c^N is initial V-lift, i.e., for each $(D I (M^{K}), K, c^{K}) \in ∣ PrFDRCS ∣$ , a GOH f : K → N is continuous if and only if f_i ∘ f is continuous for every i ∈ Γ.

For each $(I, I^{*}) \in D I (M^{K})$ we have, $\begin{matrix} c^{N} (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*})) \\ = ⋀_{i \in Γ} f_{i}^{⊢} (c_{i} (f_{i}^{\Rightarrow} \circ f^{\Rightarrow} (I), f_{i}^{\Rightarrow} \circ f^{\Rightarrow} (I^{*}))) \\ = ⋀_{i \in Γ} f_{i}^{⊢} (c_{i} ((f_{i} \circ f)^{\Rightarrow} (I), (f_{i} \circ f)^{\Rightarrow} (I^{*}))) \\ (be Lemma 2.13) \\ \geq ⋀_{i \in Γ} f_{i}^{⊢} (f_{i} \circ f) (c^{K} (I, I^{*})) \\ (since f_{i} \circ f is continuous) \\ \geq ⋀_{i \in Γ} f (c^{K} (I, I^{*})) \\ = f (c^{K} (I, I^{*})) . \end{matrix}$

On the other hand, we want to show $(f_{i} \circ f) (c^{K} (I, I^{*})) \leq c_{i} ((f_{i} \circ f)^{\Rightarrow} (I), (f_{i} \circ f)^{\Rightarrow} (I^{*})),$ $\forall i \in Γ, (I, I^{*}) \in D I (M^{K})$ .

For each $i \in Γ, (I, I^{*}) \in D I (M^{K})$ , we have $\begin{matrix} (f_{i} \circ f) (c^{K} (I, I^{*})) \\ \leq (f_{i} \circ f) (f^{⊢} (c^{N} (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*})))) \\ (since f is continuous) \\ \leq f_{i} (c^{N} (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))) \\ = f_{i} (⋀_{i \in Γ} f_{i}^{⊢} (c_{i} (f_{i}^{\Rightarrow} \circ f^{\Rightarrow} (I), f_{i}^{\Rightarrow} \circ f^{\Rightarrow} (I^{*})))) \\ = f_{i} (⋀_{i \in Γ} f_{i}^{⊢} (c_{i} ((f_{i} \circ f)^{\Rightarrow} (I), (f_{i} \circ f)^{\Rightarrow} (I^{*})))) \\ (by Lemma 2.13) \\ = f_{i} (f_{i}^{⊢} (⋀_{i \in Γ} c_{i} ((f_{i} \circ f)^{\Rightarrow} (I), (f_{i} \circ f)^{\Rightarrow} (I^{*}))) \\ \leq ⋀_{i \in Γ} c_{i} ((f_{i} \circ f)^{\Rightarrow} (I), (f_{i} \circ f)^{\Rightarrow} (I^{*})) \\ \leq c_{i} (f_{i} \circ f)^{\Rightarrow} (I, I^{*}) . \end{matrix}$

Step 3 (unique initial V-lift): Suppose that $(D I (M^{N}), N, c^{*^{N}})$ is another initial V-lift with respect to the source $(f_{i} : N \to (D I (M^{L_{i}}), L_{i}, c_{i}))_{i \in Γ}$ . Let ${id}_{N} : (D I (M^{N}), N, c^{*^{N}}) \to (D I (M^{N}), N, c^{N})$ . Since $(D I (M^{N}), N, c^{N})$ is initial V-lift and f_i ∘ id_N = f_i is continuous for each i ∈ Γ, then id_N is continuous. Hence $c^{*^{N}} (I, I^{*}) \geq c^{N} (I, I^{*})$ , for each $(I, I^{*}) \in D I (M^{N})$ , i.e., c^{*^N} ≥ c^N. Using the similar argument, we have c^{*^N} ≤ c^N. Therefore, c^{*^N} = c^N.

Hence PrFDRCS is topological over CD^OP.

Corollary 4.8. (L, M)-SDFRN is topological over CD^OP.

5 The relationship between PrFDRCS and TFDRCS

In this section, we will discuss the relationship between topological fuzzy DR-convergence structure and pretopological fuzzy DR-convergence structure. It is proved that TFDRCS is concretely bireflective in PrFDRCS in the sense of Definition 2.3(3).

Suppose $(D I (M^{L}), L, c) \in ∣ PrFDRCS ∣$ . Define a map $c^{Q} : D I (M^{L}) \to L$ by $c^{Q} (I, I^{*}) = ⋁_{(I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}}) \leq (I, I^{*})} e, \forall (I, I^{*}) \in D I (M^{L}),$

where $I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}} : L \to M$ are defined by $\begin{matrix} I_{e}^{c^{*}} (u) = ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c} (v), \forall u \in L . \\ I_{e}^{*^{c^{*}}} (u) = ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v), \forall u \in L . \end{matrix}$

For $(I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}})$ , we have the following lemma.

Lemma 5.1. For all u ∈ L, e ∈ M (L), $(I, I) \in D I (M^{L})$ , the following statements hold.

$I_{e}^{c^{*}} (u) = ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c^{*}} (v)$ and $I_{e}^{*^{c^{*}}} (u) = ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{* c^{*}} (v),$ i.e., $(I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}})$ satisfies the axiom (DFRN5),

$(I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}}) \leq (I, I^{*})$ if and only if $e \leq c^{Q} (I, I^{*})$ ,

$(I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}}) = (I_{e}^{c^{Q}}, I_{e}^{*^{c^{Q}}})$ .

Proof. (i) The inequalities $I_{e}^{c^{*}} (u) \geq ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c^{*}} (v)$ , $I_{e}^{*^{c^{*}}} (u) \leq ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c^{*}}} (v)$ aretrivial.

Conversely, Firstly we will prove that, $I_{e}^{c^{*}} (u) \leq ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c^{*}} (v)$ . suppose that, $α ◃ I_{e}^{c^{*}} (u)$ , where α ∈ M (L) is arbitrary. Since, $I_{e}^{c^{*}} (u) = ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c} (v),$ there exists some v_α ∈ L such that v_α ∈ e|u and $⋀_{s ≰ v_{α}} I_{s}^{c} (v_{α}) \geq α$ , therefore, $\begin{matrix} ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c^{*}} (v) & = & ⋁_{v \in e | u} ⋀_{s ≰ v} ⋁_{a \in s | v} ⋀_{t ≰ a} I_{t}^{c} (a) \\ \geq & ⋀_{s ≰ v_{α}} ⋀_{t ≰ v_{α}} I_{t}^{c} (v_{α}) \\ \geq & α . \end{matrix}$

Then, $I_{e}^{c^{*}} (u) \leq ⋁_{v \in e | u} ⋀_{s ≰ v} I_{s}^{c^{*}} (v)$ . Secondly, we will prove that $I_{e}^{*^{c^{*}}} (u) \geq ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c^{*}}} (v)$ . Suppose that there exist u ∈ L and k ∈ M (L) such that $I_{e}^{*^{c^{*}}} (u) \leq k < ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c^{*}}} (v)$ . Take α ∈ M (L) such that $I_{e}^{*^{c^{*}}} (u)) α \leq k < ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c^{*}}} (v)$ . Since, $I_{e}^{* c^{*}} (u) = ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c}} (v),$ for each v ∈ L such that v ∈ e|u, $⋁_{s ≰ v} I_{s}^{*^{c}} (v) \leq α$ , therefore, $\begin{matrix} ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c^{*}}} (v) & = & ⋀_{v \in e | u} ⋁_{s ≰ v} ⋀_{a \in s | v} ⋁_{t ≰ a} I_{t}^{*^{c}} (a) \\ \leq & ⋁_{s ≰ v} ⋁_{t ≰ v} I_{t}^{*^{c}} (v) \\ \leq & α . \end{matrix}$

It is a contradiction. Thus, $I_{e}^{*^{c^{*}}} (u) \geq ⋀_{v \in e | u} ⋁_{s ≰ v} I_{s}^{*^{c^{*}}} (v)$ .

(ii) It can be obtained in the same manner of Lemma 3.4, and using presupposition of (i).

(iii) considering (ii) and Lemma 3,2(i), the following result holds, $\begin{matrix} (I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}}) \leq (I, I^{*}) & \Leftrightarrow & e \leq c^{Q} (I, I^{*}) \\ \Leftrightarrow & (I_{e}^{c^{Q}}, I_{e}^{*^{c^{Q}}}) \leq (I, I^{*}) . \end{matrix}$

For the main result in this section, we give the following lemma.

Lemma 5.2. Suppose that $(D I (M^{L_{1}}), L_{1}, c), (D I (M^{L_{2}}), L_{2}, d) \in ∣ PrFDRCS ∣$ , f : L₁ → L₂ is a GOH. Then $f : (D I (M^{L_{1}}), L_{1}, c) \to (D I (M^{L_{2}}), L_{2}, d)$ is continuous if and only if $(f^{\Rightarrow} (I_{e}^{c}), f^{\Rightarrow} (I_{e}^{*^{c}})) \geq (I_{f (e)}^{d}, I_{f (e)}^{*^{d}})$ , for each e ∈ M(L₁).

Proof. Suppose that $f : (D I (M^{L_{1}}), L_{1}, c) \to (D I (M^{L_{2}}), L_{2}, d)$ is continuous. Then, for each e ∈ M (L₁), $(I, I^{*}) \in D I (M^{L_{1}})$ , we have $c (I, I^{*}) \leq f^{⊢} (d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))) .$

For any u ∈ L₂, $\begin{matrix} f^{\Rightarrow} (I_{e}^{c}) (u) & = & I_{e}^{c} (f^{⊢} (u)) \\ = & ⋀_{e \leq c (I, I^{*})} I (f^{⊢} (u)) \\ = & ⋀_{e \leq c (I, I^{*})} f^{\Rightarrow} (I) (u) \\ \geq & ⋀_{f (e) \leq f (c (I, I^{*}))} f^{\Rightarrow} (I) (u) \\ \geq & ⋀_{f (e) \leq d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))} f^{\Rightarrow} (I) (u) \\ \geq & ⋀_{f (e) \leq d (G, G^{*})} G (u) \\ = & I_{f (e)}^{d} (u), \\ f^{\Rightarrow} (I_{e}^{*^{c}}) (u) & = & I_{e}^{*^{c}} (f^{⊢} (u)) \\ = & ⋁_{e \leq c (I, I^{*})} I^{*} (f^{⊢} (u)) \\ = & ⋁_{e \leq c (I, I^{*})} f^{\Rightarrow} (I^{*}) (u) \\ \leq & ⋁_{f (e) \leq f (c (I, I^{*}))} f^{\Rightarrow} (I^{*}) (u) \\ \leq & ⋁_{f (e) \leq d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))} f^{\Rightarrow} (I^{*}) (u) \\ \leq & ⋁_{f (e) \leq d (G, G^{*})} G^{*} (u) \\ = & I_{f (e)}^{*^{d}} (u) . \end{matrix}$

Then, $(f^{\Rightarrow} (I_{e}^{c}), f^{\Rightarrow} (I_{e}^{*^{c}})) \geq (I_{f (e)}^{d}, I_{f (e)}^{*^{d}})$ .

Conversely, for each e ∈ M (L₁), $(I, I^{*}) \in D I (M^{L_{1}})$ , $\begin{matrix} e \leq c (I, I^{*}) \\ \Rightarrow (I_{e}^{c}, I_{e}^{*^{c}}) \leq (I, I^{*}) (by Lemma 3.2 (i)) \\ \Rightarrow (I_{f (e)}^{d}, I_{f (e)}^{*^{d}}) \leq (f^{\Rightarrow} (I_{e}^{c}), f^{\Rightarrow} (I_{e}^{*^{c}})) \\ \leq (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*})) \\ \Rightarrow f (e) \leq d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*})) \\ (by Lemma 3.2 (i)) \\ \Rightarrow e \leq f^{⊢} (d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))), \end{matrix}$

we can conclude that: $\begin{matrix} c (I, I^{*}) \\ = ⋁ {e \in M (L_{1}) : e \leq c (I, I^{*})} \\ \leq ⋁ {e \in M (L_{1}) : e \leq f^{⊢} (d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*})))} \\ = f^{⊢} (d (f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))) . \end{matrix}$

Thus $f : (D I (M^{L_{1}}), L_{1}, c) \to (D I (M^{L_{2}}), L_{2}, d)$ is continuous.

c^Q defined above plays an important role in displaying the relationship between topological fuzzy DR-convergence structure and pretopological fuzzy DR-convergence structure. The following theorem gives the relationship between them.

Theorem 5.3. TFDRCS is concretely bireflective in PrFDRCS.

Proof. Assume that $(D I (M^{L}), L, c)$ is a pretopological FDRCS space. We assert that its TFDRCS-reflection is given by ${id}_{L} : (D I (M^{L}), L, c) \to (D I (M^{L}), L, c^{Q})$ , where c^Q is defined above. The conclusion can be proved by the followingsteps:

Step1 : $(D I (M^{L}), L, c^{Q}) \in ∣ TFDRCS ∣$ : It is obvious that $(D I (M^{L}), L, c^{Q})$ fulfills (FDRC1) and (FDRC2), and the verification of (FDRC3) is routine. What remains it to prove (FDRC4).

Let $B_{e}^{c^{Q}}, B_{e}^{*^{c^{Q}}} : L \to M$ be defined by: $\begin{matrix} B_{e}^{c^{Q}} (u) = I_{e}^{c^{Q}} (u) \land ⋀_{s ≰ u} I_{s}^{c^{Q}} (u), \\ B_{e}^{*^{c^{Q}}} (u) = I_{e}^{*^{c^{Q}}} (u) \lor ⋁_{s ≰ u} I_{s}^{*^{c^{Q}}} (u), \forall u \in L . \end{matrix}$

We can see that $(B_{e}^{c^{Q}}, B_{e}^{*^{c^{Q}}})$ is an (L, M)-double fuzzy ideal base that is just needed in (FDRC4). The process of checking is the same as that in Proposition 3.5.

Step2 : The map ${id}_{L} : (D I (M^{L}), L, c) \to (D I (M^{L}), L, c^{Q})$ is continuous and hence is a bimorphism in PrFDRCS. We only need to prove that:

${id}_{L} (c (I, I^{*})) \leq c^{Q} ({id}_{L}^{\Rightarrow} (I), {id}_{L}^{\Rightarrow} (I^{*})), \forall (I, I^{*}) \in D I (M^{L}),$ equivalently, $c (I, I^{*}) \leq c^{Q} (I, I^{*}) .$

For each e ∈ M (L) with the property of $e \leq c (I, I^{*})$ , according to the definition of $(I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}})$ , $\begin{matrix} (I_{e}^{c^{*}}, I_{e}^{*^{c^{*}}}) & \leq & (I_{e}^{c}, I_{e}^{*^{c}}) = (⋀_{e \leq c (G, G)} G, ⋁_{e \leq c (G, G)} G^{*}) \\ \leq & (I, I^{*}), \end{matrix}$ by Lemma 5.1(ii), we have $e \leq c^{Q} (I, I^{*})$ . Then $c (I, I^{*}) \leq c^{Q} (I, I^{*})$ holds.

Step3 : For each topological FDRCS space $(D I (M^{L_{1}}), L_{1}, d)$ the continuity of each map $f : (D I (M^{L}), L, c) \to (D I (M^{L_{1}}), L_{1}, d)$ implies the continuity of $f : (D I (M^{L}), L, c^{Q}) \to (D I (M^{L_{1}}), L_{1}, d)$ . By Lemma 5.2, it suffices to prove that: $(f^{\Rightarrow} (I_{e}^{c^{Q}}), f^{\Rightarrow} (I_{e}^{*^{c^{Q}}})) \geq (I_{f (e)}^{d}, I_{f (e)}^{*^{d}}), \forall e \in M (L) .$

Since $f : (D I (M^{L}), L, c) \to (D I (M^{L_{1}}), L_{1}, d)$ is continuous, then by Lemma 5.2, we have $(f^{\Rightarrow} (I_{e}^{c}), f^{\Rightarrow} (I_{e}^{*^{c}})) \geq (I_{f (e)}^{d}, I_{f (e)}^{*^{d}}) .$

Now, we conclude that for all u ∈ L, $\begin{matrix} I_{f (e)}^{d} (u) & = & ⋁_{v \in f (e) | u} ⋀_{s ≰ v} I_{s}^{d} (v) (see Remark 3.7) \\ = & ⋁_{f^{⊢} (v) \in e | f^{⊢} (u)} ⋀_{s ≰ v} I_{s}^{d} (v) \\ (since v \in f (e) | u \Leftrightarrow f^{⊢} (v) \in e | f^{⊢} (u)) \\ \leq & ⋁_{f^{⊢} (v) \in e | f^{⊢} (u)} ⋀_{f (t) ≰ v} I_{f (t)}^{d} (v) \\ = & ⋁_{f^{⊢} (v) \in e | f^{⊢} (u)} ⋀_{t ≰ f^{⊢} (v)} I_{f (t)}^{d} (v) \\ \leq & ⋁_{f^{⊢} (v) \in e | f^{⊢} (u)} ⋀_{t ≰ f^{⊢} (v)} I_{t}^{c} (f^{⊢} (v)) \\ (since I_{f (t)}^{d} (v) \leq f^{\Rightarrow} (I_{t}^{c}) (v) \\ = I_{t}^{c} (f^{⊢} (v))) \\ \leq & ⋁_{f^{⊢} (v) \in e | f^{⊢} (u)} ⋀_{t ≰ D} I_{t}^{c} (D) \\ = & I_{e}^{c^{*}} (f^{⊢} (u)) \\ = & I_{e}^{c^{Q}} (f^{⊢} (u)) (by Lemma 5.1 (3)) \\ = & f^{\Rightarrow} (I_{e}^{c^{Q}}) (u) . \end{matrix}$

Similarly, $I_{f (e)}^{*^{d}} (u) \geq f^{\Rightarrow} (I_{e}^{*^{c^{Q}}}) (u)$ . Thus, $(f^{\Rightarrow} (I_{e}^{c^{Q}}), f^{\Rightarrow} (I_{e}^{*^{c^{Q}}})) \geq (I_{f (e)}^{d}, I_{f (e)}^{*^{d}}), \forall e \in M (L) .$

Therefore, $f : (D I (M^{L}), L, c^{Q}) \to (D I (M^{L_{1}}), L_{1}, d)$ is continuous.

The following conclusion is obvious by Proposition 4.7, and Theorem 5.3.

Corollary 5.4. TFDRCS is topological over CD^OP.

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