SP -compactness and SP -connectedness degree in L -fuzzy pretopological spaces

Abstract

The notions of SP-compactness degree, SP-separatedness degree and SP-connectedness degree of an L-subset are discussed in L-fuzzy pretopological spaces via L-fuzzy semi-preopen operator and implication operator. The new notions do not based on the basis lattice structure, so the distributivity can be neglected. The notions SP-compactness, SP-separatedness and SP-connectedness degrees of L-fuzzy topological spaces can be viewed as a special cases of compactness, separatedness and connectedness degrees in L-fuzzy pretopological spaces. Furthermore, some properties of the new notions are obtained and investigated.

1 Introduction

Compactness and connectedness are two of the most significant concepts in topology and pure mathematics. Consequently it is natural to pay significant interest to these notions in fuzzy topology. To the best of our knowledge, an I-topological space’s compactness has been discussed for the first time in [2] by using the notion of open covers. Later on, many authors gave several forms of compactness [1 , 31].

In the previous approaches of compactness, L-subset does not have a degree of compactness except ⊥ (i.e., non-compact L-subset) and ⊤ (i.e., compact L-subset). In 1988, E. Lowen and R. Lowen [12] considered compactness of I-topological spaces as a matter of degree. Furthermore, Šostak [26, 27] defined the compactness degree $c (A)$ of an L-subset A in the case L = I. Recently, Shi et al. [9] studied the fuzzy compactness degree in L-fuzzy topological spaces.

In I-fuzzy topological spaces, Šostak [25, 26] defined the degree of connectedness via the level I-topological spaces. Yue and Fang [30] introduced the connectivity for the whole L-fuzzy topological space. In 2009, Shi [19] presented separatedness degree and connectedness degree of L-subset based on L-fuzzy closure operator.

Recently, Shi [23] used L-fuzzy semiopen andL-fuzzy preopen operators in L-fuzzy pretopological spaces as a tool to measure the semiopenness and preopenness degree of an L-subset, respectively. Based on Shi’s operator, the notion of semicompactness is introduced and characterized in [22, 29]. Although a definition of preconnectedness degree was also presented by Ghareeb [5], it was defined by using L-fuzzy preopen operator. In 2012, Ghareeb [6] used L-fuzzy preopen operator to present a new operator to measure the degree of semi-preopenness of an L-subset.

In this paper, we aim to define fuzzy SP-compactness degree, fuzzy SP-separatedness degree and fuzzy SP-connectedness degree in L-fuzzy pretopological spaces by using L-fuzzy semi-preopen operator. Also, some of their properties will be briefly discussed.

2 Preliminaries

By (L, ⋁ , ⋀ , ′), we denote a complete DeMorgan algebra. The collection of non-unit prime elements, the collection of non-zero coprime elements, the smallest and the largest elements in L are denoted by P (L), M (L), ⊥ and ⊤, respectively. For each α, β ∈ L, α is wedge below β and is expressed as α ⪡ β, if for each E ⊆ L, ⋁E ≥ β, then γ ≥ α for some γ ∈ E. For any β ∈ L, Θ (β) and Ω (β) are the greatest minimal family and the greatest maximal family of β, respectively. Θ^* (β) = Θ (β) ∩ M (L) and Ω^* (β) = Ω (β) ∩ P (L). The completely distributive lattice is a complete lattice L such that β = ⋁ {α ∈ L : α ⪡ β} for all β ∈ L. The binary operation “↦” in the complete DeMorgan algebra L is given by $α \mapsto β = ⋁ {γ \in L : α \land γ \leq β} .$ For all α, β, γ, δ ∈ L and {α_i}, {β_i} ⊆ L, we have:

(α ↦ β) ≥ γ iff α ∧ γ ≤ β,

α↦ β = ⊤ iff α ≤ β,

α ↦ (⋀ _iβ_i) = ⋀ _i (α ↦ β_i),

(⋁ _iα_i) ↦ β = ⋀ _i (α_i ↦ β),

(α ↦ γ) ∧ (γ ↦ β) ≤ α ↦ β,

α ≤ β ⇒ γ ↦ α ≤ γ ↦ β,

α ≤ β ⇒ β ↦ γ ≤ α ↦ γ,

(α ↦ β) ∧ (γ ↦ δ) ≤ α ∧ γ ↦ β ∧ δ.

By L^X, we denote the family of all L-subsets of X (X≠ ∅). The smallest and the largest L-subsets in L^X are

\underline{⊥}

and

\underline{⊤}

, respectively. For the sub-collection Φ ⊆ L^X, 2^(Φ) and 2^[Φ] symbolize to the collection of each finite sub-collection and the collection of each countable sub-collection of Φ, respectively.

An L-fuzzy function $f_{L}^{\to} : L^{X} ⟶ L^{Y}$ is defined by the crisp function f : X ⟶ Y as usual, i.e., $f_{L}^{\to} (G) (y) = ⋁_{f (x) = y} G (x)$ and $f_{L}^{\leftarrow} (H) = H \circ f$ for all G ∈ L^X, y ∈ Y and H ∈ L^Y. The mapping $[\tilde{\subset}] : L^{X} \times L^{X} \to L$ refers to the L-fuzzy inclusion [8, 24] on X, where $[G \tilde{\subset} H] = ⋀_{x \in X} (G^{'} (x) \lor H (x)) .$

The pair $(X, T)$ is called an L-topological space (briefly, L-ts) [2], where $T$ is a subcollection of L^X which includes the L-subsets $\underline{⊥}$ , $\underline{⊤}$ , and $T$ is closed for finite infima and any suprema. The elements of $T$ are open L-subsets. For any L-ts $(X, T)$ and H ∈ L^X is called fuzzy compact [18] if for each collection $V$ of open L-subsets, wehave $\begin{matrix} ⋀_{y \in X} (H^{'} (y) \lor ⋁_{A \in V} G (y)) \\ \leq ⋁_{W \in 2^{(V)}} ⋀_{y \in X} (H^{'} (y) \lor ⋁_{G \in W} G (y)) \end{matrix}$ i.e., $[H \tilde{\subset} ⋁ V] \leq ⋁_{W \in 2^{(V)}} [H \tilde{\subset} ⋁ W] .$

Definition 2.1. Let X≠ ∅. A mapping $T : L^{X} ⟶ L$ which satisfies:

$T (\underline{⊤}) = T (\underline{⊥}) = ⊤$ ,

$T (⋁_{i \in Γ} G_{j}) \geq ⋀_{i \in Γ} T (G_{i})$ for each G_i ∈ L^X, i ∈ Γ,

is called an L-fuzzy pretopology [4]. The pair

(X, T)

is called L-fuzzy pretopological space (briefly, L-fpts). An L-fuzzy pretopology

T

is called L-fuzzy topology (L-ft, for short) on X [8, 24] if it satisfies the following additional condition:

for each G, H ∈ L^X, $T (G \land H) \geq T (G) \land T (H)$ .

In this case,

(X, T)

is said to be an L-fuzzy topological space (briefly, L-fts). The gradation of closedness of G is given by

T^{*} (G) = T (G^{'})

. A function

f : (X, T_{1}) ⟶ (Y, T_{2})

is said to be L-fuzzy continuous function iff

T_{1} (f_{L}^{\leftarrow} (C)) \geq T_{2} (C)

for each C ∈ L^Y.

Definition 2.2. [23] For an L-fpts $(X, T)$ and E ∈ L^X. The degree of fuzzy compactness ${com}_{T} (E)$ of E is defined by: $\begin{matrix} {com}_{T} (E) \\ = ⋀_{V \subseteq L^{X}} {(⋀_{A \in V} T (A) \land ⋀_{x \in X} (E^{'} \lor ⋁_{A \in V} A) (x)) \\ \mapsto ⋁_{W \in 2^{(V)}} ⋀_{x \in X} (E^{'} \lor ⋁_{A \in W} A) (x)} \\ = ⋀_{V \subseteq L^{X}} {(⋀_{A \in V} T (A) \land [E \tilde{\subseteq} ⋁ V]) \\ \mapsto ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W]} . \end{matrix}$ E is called fuzzy compact iff ${com}_{T} (G) = ⊤$ .

Definition 2.3. For an L-fpts $(X, T)$ and G, H ∈ L^X. Define $\begin{matrix} S_{T} (G, H) & = & (⋀_{x_{α} \leq G} (Cl (H) (x_{α}))^{'}) \\ \land & (⋀_{y_{β} \leq H} (Cl (G) (y_{β}))^{'}), \end{matrix}$ where Cl is the closure operator defined by $T$ [20]. Then $S_{T} (G, H)$ is said to be the fuzzy separatedness degree between G and H. For any A ∈ L^X, we have ${Con}_{T} (A) = ⋀_{E, F \in L^{X} ∖ {\underline{⊥}}, A = E \lor F} {S_{T} (E, F)^{'}} .$ Then ${Con}_{T} (A)$ is called the degree of fuzzy connectedness of A.

Definition 2.4. [15, 16] For an L-ts $(X, T)$ , α ∈ L ∖ {⊥} and H ∈ L^X, A collection $V \subseteq L^{X}$ is called a Θ_α-cover of H if for each x ∈ X, we have $α \in Θ (H^{'} (x) \lor ⋁_{A \in V} A (x))$ . The family $V$ is said to be a strong Θ_α-cover of H if $α \in Θ (⋀_{x \in X} (H^{'} (x) \lor ⋁_{A \in V} A (x)))$ .

Definition 2.5. [15, 16] For an L-ts $(X, T)$ , α ∈ L ∖ {⊥} and H ∈ L^X, A collection $V \subseteq L^{X}$ is called a Q_α-cover of H if for each x ∈ X, we have $H^{'} (x) \lor ⋁_{A \in V} A (x) \geq α$ .

Definition 2.6. [15, 16] For an L-ts $(X, T)$ , α ∈ L ∖ {⊥} and H ∈ L^X, A collection $V \subseteq L^{X}$ is called:

an α-shading of H if for each x ∈ X, $(H^{'} (x) \lor ⋁_{A \in V} A (x)) ≰ α$ .

a strong α-shading of H if $⋀_{x \in X} (H^{'} (x) \lor ⋁_{A \in V} A (x)) ≰ α$ .

an α-remote collection of H if for each x ∈ X, $(H (x) \land ⋀_{B \in V} B (x)) ≱ α$ .

a strong α-remote collection of H if $⋁_{x \in X} (H (x) \land ⋀_{B \in V} B (x)) ≱ α$ .

Lemma 2.7. [18] Let $f_{L}^{\to}$ be the extension of the crisp functionf : X → Y, then for each collection $V \subset L^{X}$ , it follows that $\begin{matrix} ⋀_{y \in Y} (f_{L}^{\to} (A)^{'} (y) \lor ⋁_{B \in V} B (y)) \\ = & ⋀_{x \in X} (A (x)^{'} \lor ⋁_{B \in V} f_{L}^{\leftarrow} (B) (x)) . \end{matrix}$

Definition 2.8. [23] Let $(X, T)$ be an L-fts and G ∈ L^X. An L-fuzzy preopen operator $T_{p} : L^{X} \to L$ is given by $\begin{matrix} T_{p} (G) \\ = ⋀_{x_{α} ⪡ G} ⋁_{x_{α} ⪡ H} {T (H) \land ⋀_{y_{β} ⪡ H} ⋀_{y_{β} ≰ F \geq G} (T (F^{'}))^{'}} . \end{matrix}$

Definition 2.9. [6] Let $(X, T)$ be an L-fts and G ∈ L^X. An L-fuzzy semi-preopen operator $T_{sp} : L^{X} \to L$ is given by $T_{sp} (G) = ⋁_{H \leq G} {T_{p} (H) \land ⋀_{x_{α} ⪡ G} ⋀_{x_{α} ≰ F \geq H} (T (F^{'}))^{'}},$ where $T_{sp} (G)$ and $T_{sp} (G^{'})$ can be regarded as the semi-preopenness degree and the semi-preclosedness degree of G, respectively.

Lemma 2.10.If $(X, T)$ is anL-fpts, then $T_{sp}$ verifies the next statements:

$T_{sp} (\underline{⊥}) = T_{sp} (\underline{⊤}) = ⊤$ ,

for all G_i ∈ L^X and i ∈ Γ, $T_{sp} (⋁_{i \in Γ} G_{i}) \geq ⋀_{i \in Γ} T_{sp} (G_{i})$ .

Proof. Trivial. □

Definition 2.11. The function spCl : L^X → L^{M(L^X)} is called an L-fuzzy semi-preclosure operator on X if it satisfying the following conditions:

spCl (G) (x_α) = ⋀ _β⪡αspCl (G) (x_β), for each x_α ∈ M (L^X),

$spCl (\underline{⊥}) (x_{α}) = ⊥$ for each x_α ∈ M (L^X),

spCl (G) (x_α) =⊤ for each x_α ≤ G,

for each β ∈ L_⊥, (spCl (⋁ (spCl (G)) _[β])) _[β] ⊂ (spCl (G)) _[β].

spCl (G) (x_α) is the degree of belonging of x_α to the semi-preclosure of spCl (G).

Theorem 2.12.If $T_{sp}$ is anL-fuzzy semi-preopen operator onXandspClbe the inducedL-fuzzy semi-preclosure operator. Then, we have $spCl (G) (x_{α}) = ⋀_{x_{α} ≰ F \geq G} (T_{sp} (F^{'}))^{'},$ for eachx_α ∈ M (L^X) and G ∈ L^X.

Proof. Straightforward. □

3 Degree of fuzzy SP-compactness in L-fuzzy pretopological space

Definition 3.1. For an L-fpts $(X, T)$ and G ∈ L^X, the fuzzy SP-compactness degree spCom (G) of G is given by: $\begin{matrix} spCom (G) \\ = ⋀_{V \subseteq L^{X}} {(⋀_{A \in V} T_{sp} (A) \land ⋀_{x \in X} (G^{'} \lor ⋁_{A \in V} A) (x)) \\ \mapsto ⋁_{W \in 2^{(V)}} ⋀_{x \in X} (G^{'} \lor ⋁_{A \in W} A) (x)} \\ = ⋀_{V \subseteq L^{X}} {(⋀_{A \in V} T_{sp} (A) \land [G \tilde{\subseteq} ⋁ V]) \\ \mapsto ⋁_{W \in 2^{(V)}} [G \tilde{\subseteq} ⋁ W]} . \end{matrix}$

Remark 3.2. If $(X, T)$ is an L-fts. By using Lemma 2.10, we have ${Com}_{T_{sp}} (G) = spCom (G)$ , for each G ∈ L^X.

Example 3.3. Let $X = ℝ$ be the real numbers and let τ be ordinary topology. Now for each $t \in ℝ$ , let $A_{t} = (- \infty, t) \cap ℚ$ , where $ℚ$ is the rational numbers. Moreover for every $t \in ℝ$ , define a fuzzy set B_t ∈ [0, 1] ^X by $B_{t} = {\begin{matrix} 1, & when x \in A_{t}; \\ 0.5, & when x \notin A_{t} . \end{matrix}$ Now, we define a [0, 1]-fuzzy topology by $T (G) = {\begin{matrix} 1, & when G \in {\emptyset, X}; \\ 0.5, & when G = B_{t}, t \in ℝ; \\ 0, & otherwise . \end{matrix}$ Then $T_{sp} (G) = {\begin{matrix} 1, & when G \in {\emptyset, X}; \\ 0.5, & when G = B_{t}, t \in ℝ; \\ 0, & otherwise . \end{matrix}$ Moreover, we can check that spCom (X) =1 and $spCom (ℚ) = 0.5$ .

Theorem 3.4. For any L-pts $(X, T)$ and H ∈ L^X. H is fuzzy SP-compact iff ${spCom}_{χ_{T}} (H) = ⊤$ , where the mapping $χ_{T} : L^{X} ⟶ L$ defined by $χ_{T} (H) = {\begin{matrix} ⊤, & H \in T; \\ ⊥, & H \notin T . \end{matrix}$

Proof. Let $T$ be an L-pt on X. It is clear that $χ_{T}$ is L-fpt. An L-subset A ∈ L^X is semi-preopen set with respect to $T$ if and only if ${(χ_{T})}_{sp} (A) = ⊤$ . By using the definition of fuzzy SP-compactness, an L-subset H ∈ L^X is fuzzy SP-compact provided that for each collection $V \subseteq L^{X}$ , we have ${(χ_{T})}_{sp} (V) \leq [[H \tilde{\subseteq} ⋁ V] \leq ⋁_{W \in 2^{(V)}} [H \tilde{\subseteq} ⋁ W]] .$ By using the operation ↦ properties, H is fuzzy SP-compact if and only if for each family $V \subseteq L^{X}$ , we have $\begin{matrix} {(χ_{T})}_{sp} (V) & \mapsto & ([H \tilde{\subseteq} ⋁ V] \mapsto ⋁_{W \in 2^{(V)}} [H \tilde{\subseteq} ⋁ W]) \\ = & ⊤ . \end{matrix}$ By substitution in the definition of ${spCom}_{χ_{T}} (H)$ , we get ${spCom}_{χ_{T}} (H) = ⊤$ . □

Theorem 3.5. For an L-fpts $(X, T)$ and H ∈ L^X. An L-subset H is L-fuzzy SP-compact if and only if spCom (H) =⊤.

Proof. By using Definition 2.2 and “↦” properties, the conclusion is hold. □

Theorem 3.6. For any L-fpts $(X, T)$ and G ∈ L^X, we have $spCom (G) \leq {Com}_{T} (G)$ .

Proof. Straightforward. □

Lemma 3.7.For anyL-fpts $(X, T)$ andG ∈ L^X, we havespCom (G) ≥ αis and only if $T_{sp} (V) \land [G \tilde{\subseteq} ⋁ V] \land α \leq ⋁_{W \in 2^{(V)}} [G \tilde{\subseteq} ⋁ W],$ for each $V \subseteq L^{X}$ .

Proof. For each α ∈ L, G ∈ L^X and $V \subseteq L^{X}$ , we have $\begin{matrix} spCom (G) \geq α & \Leftrightarrow & ⋀_{V \subseteq L^{X}} (T_{sp} (V) \mapsto ([G \tilde{\subseteq} ⋁ V] \\ \mapsto & ⋁_{W \in 2^{(V)}} [G \tilde{\subseteq} ⋁ W])) \geq α \\ \Leftrightarrow & T_{sp} (V) \mapsto ([G \tilde{\subseteq} ⋁ V] \\ \mapsto & ⋁_{W \in 2^{(V)}} [G \tilde{\subseteq} ⋁ W]) \geq α \\ \Leftrightarrow & (T_{sp} (V) \land [G \tilde{\subseteq} ⋁ V]) \\ \mapsto & ⋁_{W \in 2^{(V)}} [G \tilde{\subseteq} ⋁ W] \geq α \\ \Leftrightarrow & T_{sp} (V) \land [G \tilde{\subseteq} ⋁ V] \land α \\ \leq & ⋁_{W \in 2^{(V)}} [G \tilde{\subseteq} ⋁ W] . □ \end{matrix}$

Theorem 3.8.For anL-fpts $(X, T)$ andE ∈ L^X, we havespCom (E) ≥ αif and only if $\begin{matrix} ⋁_{H \in M} T_{sp}^{*} (H)^{'} & \lor & (⋁_{x \in X} (E (x) \land ⋀_{H \in M} H (x))) \lor α^{'} \\ \geq & ⋀_{N \in 2^{(M)}} ⋁_{x \in X} (E (x) \land ⋀_{H \in N} H (x)), \end{matrix}$ for each $M \subseteq L^{X}$ .

Proof. The proof is clear from the definition of $T_{sp}^{*}$ and Lemma 3.7. □

Theorem 3.9.For anL-fpts $(X, T)$ andE ∈ L^X, we have $\begin{matrix} spCom (E) & = & ⋁ {α \in L | ⋀_{A \in V} T_{sp} (A) \land [E \tilde{\subseteq} ⋁ V] \\ \land α \leq ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{X}} . \end{matrix}$

Proof. From Lemma 3.7, we could know that spCom (E) is the upper bound of $\begin{matrix} {α \in L | ⋀_{A \in V} T_{sp} (A) \land [E \tilde{\subseteq} ⋁ V] \land α \\ \leq ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{X}} . \end{matrix}$ By using the Definition 3.1, we have $\begin{matrix} spCom (E) \\ \leq ⋀_{A \in V} T_{sp} (A) \mapsto ([E \tilde{\subseteq} ⋁ V] \mapsto ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W]) \\ = (⋀_{A \in V} T_{sp} (A) \land [E \tilde{\subseteq} ⋁ V]) \mapsto ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W], \end{matrix}$ for each $V \subseteq L^{X}$ . By applying the properties of the operation “↦”, we have $\begin{matrix} ⋀_{A \in V} T_{sp} (A) \land [E \tilde{\subseteq} ⋁ V] \land spCom (E) \\ \leq ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W], \end{matrix}$ and hence $\begin{matrix} spCom (E) \in ⋁ {α \in L | ⋀_{A \in V} T_{sp} (A) \land [E \tilde{\subseteq} ⋁ V] \\ \land α \leq ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{X}} . \end{matrix}$ Therefore, we complete the proof. □

Theorem 3.10.For anyL-fpts $(X, T)$ andE, F ∈ L^X, we have $spCom (E \lor F) \geq spCom (E) \land spCom (F) .$

Proof. We can prove the theorem by using the next inequality: $\begin{matrix} spCom (E \lor F) \\ = ⋁ {α \in L | ⋀_{A \in V} T_{sp} (A) \land [E \lor F \tilde{\subseteq} ⋁ V] \land α \\ \leq ⋁_{W \in 2^{(V)}} [E \lor F \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{X}} \\ = ⋁ {α \in L | ⋀_{A \in V} T_{sp} (A) \land [E \tilde{\subseteq} ⋁ V] \land [F \tilde{\subseteq} ⋁ V] \\ \land α \leq ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W] \land [F \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{X}} \\ \geq ⋁ {α \in L | ⋀_{A \in V} T_{sp} (A) \land [E \tilde{\subseteq} ⋁ V] \\ \land α \leq ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{X}} \\ \land ⋁ {α \in L | ⋀_{A \in V} T_{sp} (A) \land [F \tilde{\subseteq} ⋁ V] \\ \land & α \leq ⋁_{W \in 2^{(V)}} [F \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{X}} \\ = spCom (E) \land spCom (F) . □ \end{matrix}$

Theorem 3.11.For anyL-fpts $(X, T)$ andE, F ∈ L^X, we have $spCom (E \land F) \geq spCom (E) \land T_{sp} (F^{'}) .$

Proof. We can prove the theorem by using the next inequality: $\begin{matrix} spCom (E \land F) \\ = & ⋁ {α \in L | ⋀_{A \in V} T_{sp} (A) \land [E \land F \tilde{\subseteq} ⋁ V] \\ \land α \leq ⋁_{W \in 2^{(V)}} [E \land F \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{X}} \\ = ⋁ {α \in L | ⋀_{A \in V} T_{sp} (A) \land [E \tilde{\subseteq} F^{'} \lor ⋁ V] \\ \land α \leq ⋁_{W \in 2^{(V)}} [E \tilde{\subseteq} F^{'} \lor ⋁ W], \forall V \subseteq L^{X}} \\ \geq spCom (E) \land T_{sp} (F^{'}) . □ \end{matrix}$

Corollary 3.12.For anL-fpts $(X, T)$ andH ∈ L^X, we have $spCom (H) \geq spCom (\underline{⊤}) \land T_{sp} (G^{'}) .$

Theorem 3.13.For any twoL-fts’s $(X, T_{1})$ and $(X, T_{2})$ such that $T_{1} \leq T_{2}$ and for anyG ∈ L^X, we have ${spCom}_{T_{2}} (G) \leq {spCom}_{T_{1}} (G)$ .

Corollary 3.14.For anyL-fpts $(X, T)$ with the base or the subbase $B$ , we have $spCom (G) \leq {spCom}_{B} (G)$ , for anyG ∈ L^X.

Theorem 3.15.For any twoL-fpts’s $(X, T_{1})$ and $(Y, T_{2})$ . If $f : (X, T_{1}) ⟶ (Y, T_{2})$ isL-fuzzy semi-preirresolute function, then ${spCom}_{T_{2}} (f_{L}^{\to} (H)) \geq {spCom}_{T_{1}} (H),$ for everyH ∈ L^X.

Proof. For every H ∈ L^X, we have $\begin{matrix} {spCom}_{T_{2}} (f_{L}^{\to} (H)) \\ = ⋁ {α \in L | {T_{sp}}_{2} (V) \land [f_{L}^{\to} (H) \tilde{\subseteq} ⋁ V] \\ \land & α \leq ⋁_{W \in 2^{(V)}} [f_{L}^{\to} (H) \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{Y}} \\ \geq ⋁ {α \in L | {T_{sp}}_{1} (f_{L}^{\leftarrow} (V)) \land [H \tilde{\subseteq} ⋁ f_{L}^{\leftarrow} (V)] \\ \land α \leq ⋁_{W \in 2^{(V)}} [H \tilde{\subseteq} ⋁ f_{L}^{\leftarrow} (W)], \forall V \subseteq L^{Y}} \\ \geq ⋁ {α \in L | {T_{sp}}_{1} (P) \land [H \tilde{\subseteq} ⋁ P] \\ \land α \leq ⋁_{R \in 2^{(P)}} [H \tilde{\subseteq} ⋁ R], \forall R \subseteq L^{X}} \\ = {spCom}_{T}_{1} (H) . □ \end{matrix}$

Theorem 3.16.For any twoL-fpts’s $(X, T_{1})$ and $(Y, T_{2})$ . If $f : (X, T_{1}) ⟶ (Y, T_{2})$ isL-fuzzy semi-precontinuous function, then ${Com}_{T_{2}} (f_{L}^{\to} (H)) \geq {spCom}_{T_{1}} (H),$ for everyH ∈ L^X.

Proof. For each H ∈ L^X, we have $\begin{matrix} {Com}_{T_{2}} (f_{L}^{\to} (H)) \\ = & ⋁ {α \in L | T_{2} (V) \land [f_{L}^{\to} (H) \tilde{\subseteq} ⋁ V] \\ \land α \leq ⋁_{W \in 2^{(V)}} [f_{L}^{\to} (H) \tilde{\subseteq} ⋁ W], \forall V \subseteq L^{Y}} \\ \geq ⋁ {α \in L | {T_{sp}}_{1} (f_{L}^{\leftarrow} (V)) \land [H \tilde{\subseteq} ⋁ f_{L}^{\leftarrow} (V)] \\ \land α \leq ⋁_{W \in 2^{(V)}} [H \tilde{\subseteq} ⋁ f_{L}^{\leftarrow} (W)], \forall V \subseteq L^{Y}} \\ \geq ⋁ {α \in L | {T_{sp}}_{1} (P) \land [H \tilde{\subseteq} ⋁ P] \\ \land α \leq ⋁_{R \in 2^{(P)}} [H \tilde{\subseteq} ⋁ R], \forall R \subseteq L^{X}} \\ = {spCom}_{T}_{1} (H) . □ \end{matrix}$

Theorem 3.17.For anyL-fpts $(X, T)$ , G ∈ L^Xandα ∈ L \ {⊥}. The next statements are equivalent:

spCom (H) ≥ α.

For each β ∈ P (L), β≱α, every strongβ-shading $V$ ofHwith $T_{sp} (U) ≰ β$ has a finite sub-collection $W$ which is a strongβ-shading ofH.

For eachβ ∈ P (L), β≱α, every strongβ-shading $V$ ofHwith $T_{sp} (U) ≰ β$ , there exists a finite sub-collection $W$ of $V$ andγ ∈ Ω^* (β) where $W$ is aγ-shading ofH.

For eachβ ∈ P (L), β≱α, every strongβ-shading $V$ ofHwith $T_{sp} (V) ≰ β$ , there is a finite sub-collection $W$ of $V$ andγ ∈ Ω^* (β) where $W$ is a strongγ-shading ofH.

For eachβ ∈ M (L), β ≰ α′, every strongβ-remote collection $R$ ofHwith $T_{sp}^{*} (R) ≰ β^{'}$ has a finite sub-collection $H$ which is a strongβ-remote collection ofH.

For eachβ ∈ M (L), β ≰ α′, every strongβ-remote collection $R$ ofHwith $T_{sp}^{*} (R) ≰ β^{'}$ , there is a finite sub-collection $H$ of $R$ andγ ∈ Θ^* (β) where $H$ is anγ-remote collection ofH.

For eachβ ∈ M (L), β ≰ α′, every strongβ-remote collection $R$ ofHwith $T_{sp}^{*} (R) ≰ β^{'}$ , there is a finite sub-collection $H$ of $R$ andγ ∈ Θ^* (β) where $H$ is a strongγ-remote collection ofH.

For eachβ ≤ α, α ∈ Θ (β), β, γ≠ ⊥, everyQ_β-cover $V \subseteq (T_{sp})_{β}$ ofHhas a finite sub-collection $W$ which is aQ_γ-cover ofH.

For eachβ ≤ α, γ ∈ Θ (β), β, γ≠ ⊥, every Q_β-cover $V \subseteq (T_{sp})_{β}$ ofHhas a finite sub-collection $W$ which is a strongΘ_γ-coverofH.

For eachβ ≤ α, γ ∈ Θ (β), β, γ≠ ⊥, every Q_β-cover $V \subseteq (T_{sp})_{β}$ ofHhas a finite sub-collection $W$ which is aΘ_γ-cover ofH.

For eachβ ≤ α, γ ∈ Θ (β), β, α≠ ⊥, every strongΘ_β-cover $V \subseteq (T_{sp})_{β}$ ofHhas a finite sub-collection $W$ which is aQ_γ-cover ofH.

For eachβ ≤ α, γ ∈ Θ (β), β, γ≠ ⊥, every strongΘ_β-cover $V \subseteq (T_{sp})_{β}$ ofHhas a finite sub-collection $W$ which is a strongΘ_γ-cover ofH.

For eachβ ≤ α, γ ∈ Θ (β), β, γ≠ ⊥, every strongΘ_β-cover $V \subseteq (T_{sp})_{β}$ of Hhas a finite sub-collection $W$ which is aΘ_γ-cover ofH.

Theorem 3.18.For anyL-fpts $(X, T)$ , H ∈ L^X, and α ∈ L \ {⊥}. If Θ (γ ∧ δ) = Θ (γ) ∧ Θ (δ) for eachγ, δ ∈ L, then the next statements will be equivalent:

spCom (H) ≥ α.

For eachβ ∈ Θ (α), β≠ ⊥, every strongΘ_β-cover $V$ ofHwith $β \in Θ (T_{sp} (V))$ has a finite sub-collection $W$ which is aQ_β-cover ofH.

For eachβ ∈ Θ (α), β≠ ⊥, every strongΘ_β-cover $V$ ofHwith $β \in Θ (T_{sp} (V))$ has a finite sub-collection $W$ which is a strongΘ_β-cover ofH.

For eachβ ∈ Θ (α), β≠ ⊥, every strongΘ_β-cover $V$ ofHwith $β \in Θ (T_{sp} (V))$ has a finite sub-collection $W$ which is aΘ_β-cover ofH.

4 Degree of fuzzy SP-separatedness in L-fuzzy pretopological space

Definition 4.1. For an L-fpts $(X, T)$ and A, B ∈ L^X. Let $\begin{matrix} spS (A, B) & = & (⋀_{x_{α} \leq A} (spCl (B) (x_{α}))^{'}) \\ \land & (⋀_{y_{β} \leq B} (spCl (A) (y_{β}))^{'}) . \end{matrix}$ Then spS (A, B) is called the degree of fuzzy SP-separatedness between A and B.

Proposition 4.2.For anyL-pts $(X, T)$ andG, H ∈ L^X, ${spS}_{χ_{T}} (G, H) = ⊤$ iffG, HareSP-separated.

Lemma 4.3.For anyL-fpts $(X, T)$ andG, H ∈ L^X. If $G \land H \neg = \underline{⊥}$ , thenspS (G, H) =⊥.

Proof. Suppose that z_β ∈ M (L^X) with z_β ≤ G ∧ H, then we have $\begin{matrix} spS (G, H) \\ = (⋀_{x_{α} \leq G} (spCl (H) (x_{α}))^{'}) \land (⋀_{x_{α} \leq H} (spCl (G) (x_{α}))^{'}) \\ \leq {(spCl (H) (z_{β}))}^{'} \land {(spCl (G) (z_{β}))}^{'} = ⊤^{'} \land ⊤^{'} = ⊥ . □ \end{matrix}$

Lemma 4.4.For anyL-fpts $(X, T)$ andG, H, F, D ∈ L^X, spS (G, H) ≤ spS (F, D) if F ≤ GandD ≤ H.

Proof. If F ≤ G and D ≤ H, thenspCl (F) ≤ spCl (G) and spCl (D) ≤ spCl (H). Thus we have $\begin{matrix} spS (G, H) \\ = (⋀_{x_{α} \leq G} (spCl (H) (x_{α}))^{'}) \land (⋀_{y_{β} \leq H} (spCl (G) (y_{β}))^{'}) \\ \leq (⋀_{x_{α} \leq G} (spCl (D) (x_{α}))^{'}) \land (⋀_{y_{β} \leq H} (spCl (F) (y_{β}))^{'}) \\ \leq (⋀_{x_{α} \leq F} (spCl (D) (x_{α}))^{'}) \land (⋀_{y_{β} \leq D} (spCl (F) (y_{β}))^{'}) \\ = spS (F, D) . □ \end{matrix}$

Lemma 4.5. For any L-fpts $(X, T)$ , G, H ∈ L^X and α ∈ M (L), (spS (G, H)) ′≱α if and only if there exist D, E ∈ L^X with D ≥ G, E ≥ H, $D \land H = E \land G = \underline{⊥}$ and $(T_{sp} (D^{'}))^{'} \lor (T_{sp} (E^{'}))^{'} ≱ α$ .

Proof. Let (spS (G, H)) ′≱α, then (spS (G, H)) ′≱γ for some γ ∈ Θ^* (α). Then $⋁_{x_{α} \leq G} spCl (H) (x_{α}) \lor ⋁_{y_{β} \leq H} spCl (A) (y_{β}) ≱ γ .$ Moreover, $\begin{matrix} ⋁_{x_{α} \leq G} ⋀_{x_{α} ≰ E \geq H} (T_{sp} (E^{'}))^{'} \lor ⋁_{y_{β} \leq H} ⋀_{y_{β} ≰ D \geq G} (T_{sp} (D^{'}))^{'} ≱ γ . \end{matrix}$ Therefore, for each x_α ≤ G and for eachy_β ≤ H, there exist D_{y
_β}, E_{x
_α} ∈ L^X with x_α ≰ E_{x
_α} ≥ H, y_β ≰ D_{y
_β} ≥ G and $(T_{sp} (D_{y_{β}}^{'}))^{'} \lor (T_{sp} (E_{x_{α}}^{'}))^{'} ≱ γ$ . Let E = ⋀ _{x_α≤G}E_{x
_α} and D = ⋀ _{y_β≤H}D_{y
_β}. Then, we get D ≥ G, E ≥ H, $D \land H = E \land G = \underline{⊥}$ and $\begin{matrix} (T_{sp} (D^{'}))^{'} \lor (T_{sp} (E^{'}))^{'} \\ = {(T_{sp} (⋁_{y_{β} \leq H} D_{y_{β}}^{'}))}^{'} \lor {(T_{sp} (⋁_{x_{α} \leq H} E_{x_{α}}^{'}))}^{'} \\ \leq ⋁_{y_{β} \leq H} {(T_{sp} (D_{y_{β}}^{'}))}^{'} \lor ⋁_{x_{α} \leq G} {(T_{sp} (E_{x_{α}}^{'}))}^{'} \\ ≱ α . \end{matrix}$ Now, suppose there exist D, E ∈ L^X with D ≥ G, E ≥ H, $D \land H = E \land G = \underline{⊥}$ and $(T_{sp} (D^{'}))^{'} \lor (T_{sp} (E^{'}))^{'} ≱ α$ . But $\begin{matrix} (spS (G, H))^{'} \\ = ⋁_{x_{α} \leq G} spCl (H) (x_{α}) \lor ⋁_{y_{β} \leq H} spCl (G) (y_{β}) \\ = ⋁_{x_{α} \leq G} ⋀_{x_{α} ≰ G \geq H} (T_{sp} (G^{'}))^{'} \\ \lor ⋁_{y_{β} \leq H} ⋀_{y_{β} ≰ F \geq G} (T_{sp} (F^{'}))^{'} \\ \leq (T_{sp} (D^{'}))^{'} \lor (T_{sp} (E^{'}))^{'} . \end{matrix}$ Then (spS (G, H)) ′≱α. □

5 Degree of fuzzy SP-connectedness in L-fuzzy pretopological space

Definition 5.1. For an L-fpts $(X, T)$ and H ∈ L^X. Let $spCon (H) = ⋀_{\begin{matrix} A, B \in L^{X} ∖ {\underline{⊥}}, \\ H = A \lor B \end{matrix}} {spS (A, B)^{'}} .$ Then spCon (H) is called the degree of fuzzy SP-connectedness of H.

By using Definition 4.1, we get $\begin{matrix} spCon (H) & = & ⋀_{\begin{matrix} A, B \in L^{X} ∖ {\underline{⊥}}, \\ H = A \lor B \end{matrix}} {⋁_{x_{α} \leq A} spCl (B) (x_{α}) \\ \lor & ⋁_{y_{β} \leq B} spCl (A) (y_{β})} . \end{matrix}$

Remark 5.2. If $(X, T)$ is L-fts and by using Lemma 2.10, we have $S_{T_{sp}} (A, B) = spS (A, B)$ and ${Con}_{T_{sp}} (H) = spCon (H)$ for each A, B, G ∈ L^X.

Proposition 5.3. If $T$ is L-pretopology on X and H ∈ L^X. Then spCon (H) =⊤ iff H is SP-connected.

Theorem 5.4.For anyL-fpts $(X, T)$ andH ∈ L^X, we have $spCon (H) = ⋀_{\begin{matrix} H \land A \neg = \underline{⊥}, \\ H \land B \neg = \underline{⊥}, \\ H \land A \land B \neg = \underline{⊥}, \\ H \leq A \lor B \end{matrix}} {(T_{sp} (A^{'}))^{'} \lor (T_{sp} (B^{'}))^{'}} .$

Proof. $\begin{matrix} spCon (H) & = & ⋀_{\begin{matrix} A, B \in L^{X} ∖ {\underline{⊥}}, \\ H = A \lor B \end{matrix}} {⋁_{x_{α} \leq A} spCl (B) (x_{α}) \\ \lor ⋁_{y_{β} \leq B} spCl (A) (y_{β})} \\ = & ⋀_{\begin{matrix} A, B \in L^{X} ∖ {\underline{⊥}}, \\ H = A \lor B \end{matrix}} {⋁_{x_{α} \leq A} ⋀_{x_{α} ≰ D \geq B} (T_{sp} (D^{'}))^{'} \\ \lor ⋁_{y_{β} \leq B} ⋀_{y_{β} ≰ E \geq A} (T_{sp} (E^{'}))^{'}} \\ = & ⋀_{\begin{matrix} H \land A \neg = \underline{⊥}, \\ H \land B \neg = \underline{⊥}, \\ H \land A \land B = \underline{⊥}, \\ H = A \lor B \end{matrix}} {⋁_{x_{α} \leq H \land A} ⋀_{\begin{matrix} x_{α} ≰ D \\ D \geq H \land B \end{matrix}} (T_{sp} (D^{'}))^{'} \\ \lor ⋁_{y_{β} \leq G \land B} ⋀_{\begin{matrix} y_{β} ≰ E \\ E \geq H \land A \end{matrix}} (T_{sp} (E^{'}))^{'}} \\ \leq & ⋀_{\begin{matrix} H \land A \neg = \underline{⊥}, \\ H \land B \neg = \underline{⊥}, \\ H \land A \land B = \underline{⊥}, \\ H = A \lor B \end{matrix}} {⋁_{x_{α} \leq G \land A} (T_{sp} (D^{'}))^{'} \\ \lor ⋁_{y_{β} \leq H \land B} (T_{sp} (E^{'}))^{'}} \\ = & ⋀_{\begin{matrix} H \land A \neg = \underline{⊥}, \\ H \land B \neg = \underline{⊥}, \\ H \land A \land B = \underline{⊥}, \\ H = A \lor B \end{matrix}} {(T_{sp} (D^{'}))^{'} \lor (T_{sp} (E^{'}))^{'}} . \end{matrix}$ Now, let spCon (H) ≱α such that α ∈ M (L). Then there exist A, $B \in L^{X} ∖ {\underline{⊥}}$ with H = A ∨ B and (spS (A, B)) ≱α. By using Lemma 4.5, there are E, F ∈ L^X with E ≥ A, F ≥ B, $E \land B = F \land A = \underline{⊥}$ and $(T_{sp} (E^{'}))^{'} \lor (T_{sp} (F^{'}))^{'} ≱ α$ . Therefore, we get $⋀_{\begin{matrix} H \land A \neg = \underline{⊥}, \\ H \land B \neg = \underline{⊥}, \\ H \land A \land B = \underline{⊥}, H \leq A \lor B \end{matrix}} {(T_{sp} (B^{'}))^{'} \lor (T_{sp} (A^{'}))^{'}} ≱ α .$ Thus $\begin{matrix} spCon (H) \geq ⋀_{\begin{matrix} H \land A \neg = \underline{⊥}, \\ H \land B \neg = \underline{⊥}, \\ H \land A \land B = \underline{⊥}, H \leq A \lor B \end{matrix}} {(T_{sp} (B^{'}))^{'} \lor (T_{sp} (A^{'}))^{'}} . \end{matrix}$ □

Corollary 5.5. For any L-fpts $(X, T)$ , we have $spCon (\underline{⊤}) = ⋀_{\begin{matrix} A, B \neq \underline{⊥}, \\ A \land B = \underline{⊥}, \\ A \lor B = \underline{⊤} \end{matrix}} {(T_{sp} (A))^{'} \lor (T_{sp} (B))^{'}} .$

Theorem 5.6.For anyx_α ∈ M (L^X), it follows thatspCon (x_α) =⊤.

Proof. The proof is clear. □

Theorem 5.7.ForH ∈ L^X, we have $⋁_{β \in M (L)} spCon (⋁ (spCl (H))_{[β]}) \geq spCon (H) .$

Proof. Consider that α ≤ spCon (H) such that αinM (L) and we will show that spCon (⋁ (spCl (H)) _[β]) ≥ α. Suppose spCon (⋁ (spCl (H)) _[β]) ≱α. By Theorem 5.4, there exist A, B ∈ L^X with $(⋁ (spCl (H))_{[α]}) \land A \neg = \underline{⊥}$ , $(⋁ (spCl (H))_{[α]}) \land B \neg = \underline{⊥}$ , $(⋁ (spCl (H))_{[α]}) \land A \land B = \underline{⊥}$ , ⋁ (spCl (H)) _[α] ≤ A ∧ B and $(T_{sp} (B^{'}))^{'} \lor (T_{sp} (A^{'}))^{'} ≱ α$ . Since $(⋁ (spCl (H))_{[α]}) \land A \neg = \underline{⊥}$ , there exists x_γ ≤ A such that spCl (H) (x_γ) ≥ α. Furthermore, Since $(⋁ (spCl (H))_{[α]}) \land A \land B = \underline{⊥}$ , we have x_γ ≰ B. If $H \land A \neg = \underline{⊥}$ , then H ≤ ⋁ (spCl (H)) _[α] ≤ A ∨ B we have H ≤ B, hence $α \leq spCl (H) (x_{γ}) = ⋀_{x_{γ} ≰ E \geq H} (T_{sp} (E^{'}))^{'} \leq (T_{sp} (B^{'}))^{'},$ which is a contradiction. Similarly, we can show that $H \land B \neg = \underline{⊥}$ . Therefore by H ≤ A ∨ B, $H \land A \land B = \underline{⊥}$ , $H \land B \neg = \underline{⊥}$ , $H \land A \neg = \underline{⊥}$ , $(T_{sp} (B^{'}))^{'} \lor (T_{sp} (A^{'}))^{'} ≱ α$ and Theorem 5.4, we have spCon (H) ≱α, contradicting spCon (H) ≥ α. Thus $⋁_{β \in M (L^{X})} spCon (⋁ (spCl (H))_{[β]}) \geq spCon (H) . □$

Theorem 5.8. $\begin{matrix} spCon (E \lor F) \geq (spS (E, F))^{'} & \land & spCon (E) \\ \land & spCon (F) \end{matrix}$ for eachE, F ∈ L^X.

Proof. Suppose that α ≤ (spS (E, F)) ′ ∧ spCon (E) ∧ spCon (F) such that α ∈ M (L) and let spCon (E ∨ F) ≱α. Based on Theorem 5.4, there are A, B ∈ L^X such that E ∨ F ≤ A ∨ B, $(E \lor F) \land A \land B = \underline{⊥}$ , $(E \lor F) \land B \neg = \underline{⊥}$ , $(E \lor F) \land A \neg = \underline{⊥}$ , and $(T_{sp} (B^{'}))^{'} \lor (T_{sp} (A^{'}))^{'} ≱ α$ . But $(E \lor F) \land A \neg = \underline{⊥}$ , so we get $F \land A \neg = \underline{⊥}$ and $E \land A \neg = \underline{⊥}$ .

Let $E \land A \neg = \underline{⊥}$ (where $F \land A \neg = \underline{⊥}$ is analogous). Thus we get $E \land B = \underline{⊥}$ , else if $E \land B \neg = \underline{⊥}$ , by E ≤ A ∨ B, $E \land A \neg = \underline{⊥}$ , $E \land B \neg = \underline{⊥}$ , $E \land A \land B = \underline{⊥}$ , and $(T_{sp} (B^{'}))^{'} \lor (T_{sp} (A^{'}))^{'} ≱ α$ , we have spCon (E) ≱α, and this leads to a contradiction. For this case by $(E \lor F) \land B \neg = \underline{⊥}$ , we have $F \land B \neg = \underline{⊥}$ . Similarly, we can show that $F \land A = \underline{⊥}$ . Thus by E ∨ F ≤ A ∨ B, we get F ≤ B and E ≤ A. Then by $E \land B = F \land A = \underline{⊥}$ , F ≤ B, E ≤ A, $(T_{sp} (B^{'}))^{'} \lor (T_{sp} (A^{'}))^{'} ≱ α$ and Lemma 4.5, we get (spS (E, F)) ′≱α, and this leads to a contradiction. Therefore spCon (E ∨ F) ≥ α. □

Remark 5.9. For any L-fpts $(X, T)$ and G, H ∈ L^X. If A and B in the proof of the previous theorem can be chosen such that $A \land B \neg = \underline{⊥}$ , then $spCon (G \lor H) \geq spCon (G) \land spCon (H) .$

Theorem 5.10.For anyL-fpts $(X, T)$ andH ∈ L^X, $\begin{matrix} spCon (H) = ⋀_{x_{α}, y_{β} \leq H} ⋁_{x_{α}, y_{β} \leq D_{x_{α}, y_{β}} \leq H} {spCon (D_{x_{α}, y_{β}})} . \end{matrix}$

Proof. Suppose that ⋀_{x_α,y_β≤H} ⋁ {spCon (D_{x_α,y_β}) : x_α, y_β ≤ D_{x_α,y_β} ≤ H} ≥ γ where γ ∈ M (L). By fixing x_α ≤ H, then for every y_β ≤ H, there is D_{x_α,y_β} ∈ L^X such that x_α, y_β ≤ D_{x_α,y_β} ≤ H and spCon (D_{x_α,y_β}) ≥ a. Let D_{x
_α} = ⋁ _{y_β≤H}D_{x_α,y_β}. Then D_{x
_α} = H and $⋀_{y_{β} \leq H} D_{x_{α}, y_{β}} \neg = \underline{⊥}$ . By using Corollary 5.9, we have spCon (H) = spCon (D_{x
_α}) ≥ ⋀ _{y_β≤H}spCon (D_{x_α,y_β}) ≥ γ. This shows that $\begin{matrix} spCon (H) \geq ⋀_{x_{α}, y_{β} \leq H} ⋁_{x_{α}, y_{β} \leq D_{x_{α}, y_{β}} \leq H} {spCon (D_{x_{α}, y_{β}})} . \end{matrix}$ Since $\begin{matrix} spCon (H) \leq ⋀_{x_{α}, y_{β} \leq H} ⋁_{x_{α}, y_{β} \leq D_{x_{α}, y_{β}} \leq H} {spCon (D_{x_{α}, y_{β}})} \end{matrix}$ is clear, then we have $\begin{matrix} spCon (H) = ⋀_{x_{α}, y_{β} \leq H} ⋁_{x_{α}, y_{β} \leq D_{x_{α}, y_{β}} \leq H} {spCon (D_{x_{α}, y_{β}})} . \end{matrix}$ □

6 Conclusion

We applied L-fuzzy semi-preopen operator [6] on two essential concepts of L-fuzzy pretopology which are compactness and connectedness by using implication operator. Many of its characteristics are introduced and studied. Also, the relationship with the corresponding concepts in L-fuzzy topology have been clarified. We believe that the new concepts are applicable in many fields including the GIS (Geographic Information System).

Acknowledgments

We are really grateful to the referees for their precious comments and suggestions which have improved the quality of this work.

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