Further research of single valued neutrosophic rough sets

Abstract

Smarandache (1998) initiated neutrosophic sets (NSs) as a new mathematical tool for dealing with problems involving incomplete, indeterminant and inconsistent knowledge. By simplifying NSs, Smarandache (1998) and Wang et al. (2010) proposed the concept of single valued neutrosophic sets (SVNSs) and studied some properties of SVNSs. In this paper, we mainly investigate the topological structures of single valued neutrosophic rough sets which is constructed by combining SVNSs and rough sets. Firstly, we introduce the concept of single valued neutrosophic topological spaces. Then, we discuss the relationships between single valued neutrosophic approximation spaces and single valued neutrosophic topological spaces. Concretely, a reflexive and transitive single valued neutrosophic relation can induce a single valued neutrosophic topological space such that its single valued neutrosophic interior and closure operators are the lower and upper approximation operators induced by this single valued neutrosophic relation, respectively. Conversely, a single valued neutrosophic interior (closure, respectively) operator derived from a single valued neutrosophic topological space is just the single valued neutrosophic lower (upper, respectively) approximation operator derived from a single valued neutrosophic approximation space under some conditions. Finally, we show there exists a one-to-one correspondence between the set of all reflexive and transitive single valued neutrosophic relations and the set of all single valued neutrosophic rough topologies.

Keywords

Neutrosophic sets single valued neutrosophic sets single valued neutrosophic rough sets single valued neutrosophic topological spaces

1 Introduction

Rough set theory, initiated by Pawlak [28, 29], is an effective mathematical tool for the study of intelligent systems characterized by insufficient and incomplete information. Pawlak rough set model is established based on an equivalence relation. In the real application, the equivalence relation is a very stringent condition which limits the applications of rough sets in real world. For this reason, by replacing the equivalence relation with covering, similarity relation, tolerance relation, fuzzy relation, etc, different kinds of generalizations of Pawlak rough set model were proposed [4 , 43–46]. Rough set theory has been successfully applied to many fields, such as machine learning, knowledge acquisition, and decision analysis, etc.

Smarandache [34, 35] introduced the concept of NSs which consists of three membership functions (truth membership function, indeterminacy membership function and falsity membership function), where every function value is a real standard or non-standard subset of the nonstandard unit interval ] 0^-, 1⁺ [. The NS generalizes the concepts of the classical set, fuzzy set [50], interval-valued fuzzy set [37], intuitionistic fuzzy set [1] and interval-valued intuitionistic fuzzy set [2]. Broumi and Smarandache [7] studied rough neutrosophic sets by applying rough set idea to neutrosophic sets. Smarandache [34] and Wang et al. [38] proposed SVNSs by simplifying NSs. SVNSs can also be looked as an extension of intuitionistic fuzzy sets [1], in which three membership functions are unrelated and their function values belong to the unit closed interval. Many researchers have studied the applications of SVNSs as well as theory. Ye [47, 48] proposed decision making based on correlation coefficients and weighted correlation coefficient of SVNSs, and gave the application of proposed methods. Majumdar and Samant [26] studied distance, similarity and entropy of SVNSs from a theoretical aspect. Yang et al. [42] proposed SVNRs and studied some kinds of kernels and closures of SVNRs. Based on the notions of SVNRs, single valued (interval valued) neutrosophic graphs have been studied in depth [3 , 22]. Broumi and Smarandache [8] proposed single valued neutrosophic information systems based on rough set theory to exploit simultaneously the advantages of SVNSs and rough sets. They studied rough approximation of a SVNS in the single valued neutrosophic information systems and investigated the knowledge reduction and extension of the single valued neutrosophic information systems. Yang et al. [41] proposed single valued neutrosophic rough sets by combining SVNSs and rough sets, and explored a general framework of the study of single valued neutrosophic rough sets.

Topological structures and properties [23] of rough sets are important research issues for the study of rough sets. Many researchers have addressed the issues [5 , 40]. Wiweger [39] and Chuchro [19, 20] established the relationships between crisp rough sets and crisp topological spaces. Boixader et al. [5] investigated the connection between fuzzy rough sets and fuzzy topological spaces. Qin et al. [30, 31] discussed topological structures of fuzzy rough sets. Wu and Zhou [40] generalized the results to IF rough sets and established relationships between IF rough approximations and IF topologies. Ma and Hu [27] studied topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Li and Cui [25] studied similarity of fuzzy relations which is based on fuzzy topologies induced by fuzzy rough approximation operators. Zhang et al. [51] discussed the topological structures of interval-valued hesitant fuzzy rough set and its application. Along this line, in the present paper, we shall study topological structures of single valued neutrosophic rough sets and establish the relationships between the single valued neutrosophic approximation spaces and single valued neutrosophic topological spaces.

The rest of this paper is organized as follows. In the next section, we recall some basic notions on Pawlak rough sets, NSs, SVNSs and single valued neutrosophic rough sets. In Section 3, we give notions of single valued neutrosophic topology and its interior operation and closure operation. Some related properties are also studied. Section 4 investigates the relationships between single valued neutrosophic approximation spaces and single valued neutrosophic topology spaces. The last section summarizes the conclusion.

2 Preliminaries

In this section, we recall some basic notions and results which will be used in the paper.

2.1 Pawlak rough sets

Definition 2.1. [28, 29]. Let U be a nonempty finite universe and R be an equivalence relation in U. (U, R) is called a Pawlak approximation space. ∀X ⊆ U, the lower and upper approximations of X, denoted by $\underline{R} (X)$ and $\bar{R} (X)$ , are defined as follows,respectively: $\begin{matrix} \underline{R} (X) = {x \in U ∣ [x]_{R} \subseteq X}, \\ \bar{R} (X) = {x \in U ∣ [x]_{R} \cap X \neq \emptyset}, \end{matrix}$ where [x] _R = {y ∈ U ∣ (x, y) ∈ R}. $\underline{R}$ and $\bar{R}$ are called as lower and upper approximation operators, respectively. The pair $(\underline{R} (X), \bar{R} (X))$ is called a Pawlak rough set.

2.2 NSs and SVNSs

Definition 2.2. [34] Let U be a space of points (objects), with a generic element in U denoted by x. A NS $\tilde{A}$ in U is characterized by three membership functions, a truth-membership function $T_{\tilde{A}}$ , an indeterminacy membership function $I_{\tilde{A}}$ and a falsity-membership function $F_{\tilde{A}}$ , where ∀x ∈ U, $T_{\tilde{A}} (x)$ , $I_{\tilde{A}} (x)$ and $F_{\tilde{A}} (x)$ are real standard or non-standard subsets of ] 0^-, 1⁺ [.

There is no restriction on the sum of $T_{\tilde{A}} (x)$ , $I_{\tilde{A}} (x)$ and $F_{\tilde{A}} (x)$ , thus 0^-≤ sup $T_{\tilde{A}} (x) +$ sup $I_{\tilde{A}} (x) +$ sup $F_{\tilde{A}} (x) \leq 3^{+}$ .

Definition 2.3. [34, 38] Let U be a space of points (objects), with a generic element in U denoted by x. A SVNS $\tilde{A}$ in U is characterized by three membership functions, a truth-membership function $T_{\tilde{A}}$ , an indeterminacy membership function $I_{\tilde{A}}$ , and a falsity-membership function $F_{\tilde{A}}$ , where ∀x ∈ U, $T_{\tilde{A}} (x), I_{\tilde{A}} (x), F_{\tilde{A}} (x) \in [0, 1]$ .

The SVNS $\tilde{A}$ can be denoted by $\tilde{A}$ $= {〈 x, T_{\tilde{A}} (x),$ $I_{\tilde{A}} (x), F_{\tilde{A}} (x) 〉 ∣ x \in U}$ or $\tilde{A} = (T_{\tilde{A}}, I_{\tilde{A}}, F_{\tilde{A}})$ . ∀x ∈ U, $\tilde{A}$ $(x) = (T_{\tilde{A}} (x), I_{\tilde{A}} (x), F_{\tilde{A}} (x))$ , and $(T_{\tilde{A}} (x), I_{\tilde{A}} (x), F_{\tilde{A}} (x))$ is called a single valued neutrosophic number.

In this paper, SVNS(U) will denote the family of all SVNSs in U. Let $\tilde{A}$ be a SVNS in U. If ∀x ∈ U, $T_{\tilde{A}} (x) = 0$ and $I_{\tilde{A}} (x) = F_{\tilde{A}} (x) = 1$ , then $\tilde{A}$ is called an empty SVNS, denoted by $\tilde{\emptyset}$ . If ∀x ∈ U, $T_{\tilde{A}} (x) = 1$ , and $I_{\tilde{A}} (x) = F_{\tilde{A}} (x) = 0$ , then $\tilde{A}$ is called a full SVNS, denoted by $\tilde{U}$ . ∀α₁, α₂, α₃ ∈ [0, 1], $\hat{α_{1}, α_{2}, α_{3}}$ denotes a constant SVNS satisfying, $T_{\hat{α_{1}, α_{2}, α_{3}}} (x) = α_{1}, I_{\hat{α_{1}, α_{2}, α_{3}}} (x) = α_{2}, F_{\hat{α_{1}, α_{2}, α_{3}}} (x) = α_{3}$ .

For any y ∈ U, a single valued neutrosophic singleton set 1_y and its complement 1_U-{y} are defined as: ∀x ∈ U, $\begin{matrix} T_{1_{y}} (x) = {\begin{matrix} 1, x = y \\ 0, x \neq y \end{matrix}, \\ I_{1_{y}} (x) = F_{1_{y}} (x) = {\begin{matrix} 0, x = y \\ 1, x \neq y \end{matrix}; \\ T_{1_{U - {y}}} (x) = {\begin{matrix} 0, x = y \\ 1, x \neq y \end{matrix}, \\ I_{1_{U - {y}}} (x) = F_{1_{U - {y}}} (x) = {\begin{matrix} 1, x = y \\ 0, x \neq y \end{matrix} . \end{matrix}$

Definition 2.4. [49] Let $\tilde{A}$ and $\tilde{B}$ be two SVNSs in U. If for any x ∈ U, $T_{\tilde{A}} (x) \leq T_{\tilde{B}} (x)$ , $I_{\tilde{A}} (x) \geq I_{\tilde{B}} (x)$ and $F_{\tilde{A}} (x) \geq F_{\tilde{B}} (x)$ , then we called $\tilde{A}$ is contained in $\tilde{B}$ , i.e. $\tilde{A} ⋐ \tilde{B}$ .

If $\tilde{A} ⋐ \tilde{B}$ and $\tilde{B} ⋐ \tilde{A}$ , then we called $\tilde{A}$ is equal to $\tilde{B}$ , denoted by $\tilde{A} = \tilde{B}$ .

Definition 2.5. [38] Let $\tilde{A}$ be a SVNS in U. The complement of $\tilde{A}$ is denoted by $\tilde{A}$ ^c, where ∀x ∈ U, $T_{{\tilde{A}}^{c}} (x) = F_{\tilde{A}} (x)$ , $I_{{\tilde{A}}^{c}} (x) = 1 - I_{\tilde{A}} (x)$ , and $F_{{\tilde{A}}^{c}} (x) = T_{\tilde{A}} (x)$ .

Definition 2.6. [42] Let $\tilde{A}$ and $\tilde{B}$ be two SVNSs in U.

(1) The union of $\tilde{A}$ and $\tilde{B}$ is a SVNS $\tilde{C}$ , denoted by $\tilde{C} = \tilde{A} ⋓ \tilde{B}$ , where ∀x ∈ U,

$T_{\tilde{C}} (x) = T_{\tilde{A}} (x) \lor T_{\tilde{B}} (x)$ , $I_{\tilde{C}} (x) = I_{\tilde{A}} (x) \land I_{\tilde{B}} (x)$ and $F_{\tilde{C}} (x) = F_{\tilde{A}} (x) \land F_{\tilde{B}} (x)$ ;

(2) The intersection of $\tilde{A}$ and $\tilde{B}$ is a SVNS $\tilde{D}$ , denoted by $\tilde{D} = \tilde{A} ⋒ \tilde{B}$ , where ∀x ∈ U,

$T_{\tilde{D}} (x) = T_{\tilde{A}} (x) \land T_{\tilde{B}} (x)$ , $I_{\tilde{D}} (x) = I_{\tilde{A}} (x) \lor I_{\tilde{B}} (x)$ and $F_{\tilde{D}} (x) = F_{\tilde{A}} (x) \lor F_{\tilde{B}} (x)$ ,

where “∨” and “∧” denote maximum and minimum, respectively.

It is easy to verify that the union and intersection of SVNSs satisfy commutative law, associative law, and distributive law.

Proposition 2.7. [41] Let $\tilde{A}$ and $\tilde{B}$ be two SVNSs in U. The following results hold:

$\tilde{A} ⋐ \tilde{A} ⋓ \tilde{B}$ and $\tilde{B} ⋐ \tilde{A} ⋓ \tilde{B}$ ,

$\tilde{A} ⋒ \tilde{B} ⋐ \tilde{A}$ and $\tilde{A} ⋒ \tilde{B} ⋐ \tilde{B}$ ,

$({\tilde{A}}^{c})^{c} = \tilde{A}$ ,

$(\tilde{A} ⋓ \tilde{B})^{c} = {\tilde{A}}^{c} ⋒ {\tilde{B}}^{c}$ ,

$(\tilde{A} ⋒ \tilde{B})^{c} = {\tilde{A}}^{c} ⋓ {\tilde{B}}^{c}$ .

2.3 Single valued neutrosophic rough sets

Definition 2.8. [41] A SVNS $\tilde{R}$ in U × U is called a single valued neutrosophic relation (SVNR) in U, denoted by $\tilde{R} = {〈 (x, y), T_{\tilde{R}} (x, y), I_{\tilde{R}} (x, y), F_{\tilde{R}} (x, y) 〉 ∣ (x, y) \in U \times U}$ , where $T_{\tilde{R}} : U \times U ⟶ [0, 1]$ , $I_{\tilde{R}} : U \times U ⟶ [0, 1]$ , and $F_{\tilde{R}} : U \times U ⟶ [0, 1]$ denote the truth-membership function, indeterminacy membership function, and falsity-membership function of $\tilde{R}$ , respectively.

Let $\tilde{R}$ be a SVNR in U, the complement $\tilde{R}$ ^c of $\tilde{R}$ is defined as follows

${\tilde{R}}^{c}$ $= {〈 (x, y), T_{{\tilde{R}}^{c}} (x, y), I_{{\tilde{R}}^{c}} (x, y), F_{{\tilde{R}}^{c}} (x, y) 〉 ∣ (x, y) \in U \times U}$ ,

where ∀ (x, y) ∈ U × U, $T_{{\tilde{R}}^{c}} (x, y) = F_{\tilde{R}} (x, y)$ , $I_{{\tilde{R}}^{c}} (x, y) = 1 - I_{\tilde{R}} (x, y)$ and $F_{{\tilde{R}}^{c}} (x, y) = T_{\tilde{R}} (x, y)$ .

Definition 2.9. [41] Let $\tilde{R}$ be a SVNR in U. If ∀x ∈ U, $T_{\tilde{R}} (x, x) = 1$ and $I_{\tilde{R}} (x, x) = F_{\tilde{R}} (x, x) = 0$ , then $\tilde{R}$ is called a reflexive SVNR. If ∀x, y ∈ U, $T_{\tilde{R}} (x, y) = T_{\tilde{R}} (y, x)$ , $I_{\tilde{R}} (x, y) = I_{\tilde{R}} (y, x)$ and $F_{\tilde{R}} (x, y) = F_{\tilde{R}} (y, x)$ , then $\tilde{R}$ is called a symmetric SVNR. If ∀x ∈ U, $⋁_{y \in U} T_{\tilde{R}} (x, y) = 1$ and $⋀_{y \in U} I_{\tilde{R}} (x, y) = ⋀_{y \in U} F_{\tilde{R}} (x, y) = 0$ , then $\tilde{R}$ is called a serial SVNR. If ∀x, y, z ∈ U, $⋁_{y \in U} (T_{\tilde{R}} (x, y) \land T_{\tilde{R}} (y, z)) \leq T_{\tilde{R}} (x, z)$ , $⋀_{y \in U} (I_{\tilde{R}} (x, y) \lor I_{\tilde{R}} (y, z)) \geq I_{\tilde{R}} (x, z)$ and $⋀_{y \in U} (F_{\tilde{R}} (x, y) \lor F_{\tilde{R}} (y, z)) \geq F_{\tilde{R}} (x, z)$ , then $\tilde{R}$ is called a transitive SVNR, where “∨” and “∧” denote maximum and minimum, respectively.

Definition 2.10. [41] Let $\tilde{R}$ be a SVNR in U, the tuple $(U, \tilde{R})$ is called a single valued neutrosophic approximation space. $\forall \tilde{A} \in$ SVNS (U), the lower and upper approximations of $\tilde{A}$ w.r.t. $(U, \tilde{R})$ , denoted by $\underline{\tilde{R}} (\tilde{A})$ and $\bar{\tilde{R}} (\tilde{A})$ , are two SVNSs whose membership functions are defined as:∀x ∈ U, $\begin{matrix} T_{\underline{\tilde{R}} (\tilde{A})} (x) = ⋀_{y \in U} (F_{\tilde{R}} (x, y) \lor T_{\tilde{A}} (y)), \\ I_{\underline{\tilde{R}} (\tilde{A})} (x) = ⋁_{y \in U} ((1 - I_{\tilde{R}} (x, y)) \land I_{\tilde{A}} (y)), \\ F_{\underline{\tilde{R}} (\tilde{A})} (x) = ⋁_{y \in U} (T_{\tilde{R}} (x, y) \land F_{\tilde{A}} (y)); \\ T_{\bar{\tilde{R}} (\tilde{A})} (x) = ⋁_{y \in U} (T_{\tilde{R}} (x, y) \land T_{\tilde{A}} (y)), \\ I_{\bar{\tilde{R}} (\tilde{A})} (x) = ⋀_{y \in U} (I_{\tilde{R}} (x, y) \lor I_{\tilde{A}} (y)), \\ F_{\bar{\tilde{R}} (\tilde{A})} (x) = ⋀_{y \in U} (F_{\tilde{R}} (x, y) \lor F_{\tilde{A}} (y)) . \end{matrix}$

The pair $(\underline{\tilde{R}} (\tilde{A}), \bar{\tilde{R}} (\tilde{A}))$ is called a single valued neutrosophic rough set of $\tilde{A}$ w.r.t. $(U, \tilde{R})$ . $\underline{\tilde{R}}$ and $\bar{\tilde{R}}$ are referred to as the single valued neutrosophic lower and upper approximation operators, respectively.

Yang et al. [41] studied the properties of single valued neutrosophic lower and upper approximation operators as follows.

Proposition 2.11. [41] Let $(U, \tilde{R})$ be a single valued neutrosophic approximation space. The single valued neutrosophic lower and upper approximation operators have the following properties: $\forall \tilde{A}, \tilde{B} \in$ SVNS(U), ∀α₁, α₂, α₃ ∈ [0, 1],

$\underline{\tilde{R}} (\tilde{U}) = \tilde{U}$ , $\bar{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ ;

If $\tilde{A} ⋐ \tilde{B}$ , then $\underline{\tilde{R}} (\tilde{A}) ⋐ \underline{\tilde{R}} (\tilde{B})$ and $\bar{\tilde{R}} (\tilde{A}) ⋐ \bar{\tilde{R}} (\tilde{B})$ ;

$\underline{\tilde{R}} (\tilde{A} ⋒ \tilde{B}) = \underline{\tilde{R}} (\tilde{A}) ⋒ \underline{\tilde{R}} (\tilde{B})$ , $\bar{\tilde{R}} (\tilde{A} ⋓ \tilde{B}) = \bar{\tilde{R}} (\tilde{A}) ⋓ \bar{\tilde{R}} (\tilde{B})$ ;

$\underline{\tilde{R}} (\tilde{A} ⋓ \tilde{B}) ⋑ \underline{\tilde{R}} (\tilde{A}) ⋓ \underline{\tilde{R}} (\tilde{B})$ , $\bar{\tilde{R}} (\tilde{A} ⋒ \tilde{B}) ⋐ \bar{\tilde{R}} (\tilde{A}) ⋒ \bar{\tilde{R}} (\tilde{B})$ ;

$\underline{\tilde{R}} ({\tilde{A}}^{c}) = (\bar{\tilde{R}} (\tilde{A}))^{c}$ , $\bar{\tilde{R}} ({\tilde{A}}^{c}) = (\underline{\tilde{R}} (\tilde{A}))^{c};$

$\underline{\tilde{R}} (A ⋓ \hat{α_{1}, α_{2}, α_{3}}) = \underline{\tilde{R}} (\tilde{A}) ⋓ \hat{α_{1}, α_{2}, α_{3}}$ , $\bar{\tilde{R}} (A ⋒ \hat{α_{1}, α_{2}, α_{3}}) = \bar{\tilde{R}} (\tilde{A}) ⋒ \hat{α_{1}, α_{2}, α_{3}}$ ;

$\underline{\tilde{R}} (\hat{α_{1}, α_{2}, α_{3}}) = \hat{α_{1}, α_{2}, α_{3}} \Leftrightarrow \underline{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ , $\bar{\tilde{R}} (\hat{α_{1}, α_{2}, α_{3}}) = \hat{α_{1}, α_{2}, α_{3}} \Leftrightarrow \bar{\tilde{R}} (\tilde{U}) = \tilde{U}$ .

Proposition 2.12. [41] Let $(U, \tilde{R})$ be a single valued neutrosophic approximation space. $\underline{\tilde{R}}$ and $\bar{\tilde{R}}$ are the lower and upper approximation operators, then we have

$\tilde{R}$ is serial $\Leftrightarrow \underline{\tilde{R}} (\hat{α_{1}, α_{2}, α_{3}}) = \hat{α_{1}, α_{2}, α_{3}}$ , ∀α₁, α₂, α₃ ∈ [0, 1],

$\Leftrightarrow \underline{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ ,

$\Leftrightarrow \bar{\tilde{R}} (\hat{α_{1}, α_{2}, α_{3}}) = \hat{α_{1}, α_{2}, α_{3}}$ , ∀α₁, α₂, α₃ ∈ [0, 1],

$\Leftrightarrow \bar{\tilde{R}} (\tilde{U}) = \tilde{U}$ ;

$\tilde{R}$ is reflexive $\Leftrightarrow \underline{\tilde{R}} (\tilde{A}) ⋐ \tilde{A}$ , $\forall \tilde{A} \in$ SVNS(U),

$\Leftrightarrow \tilde{A} ⋐ \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in$ SVNS(U);

$\tilde{R}$ is symmetric $\Leftrightarrow \underline{\tilde{R}} (1_{U - {x}}) (y) = \underline{\tilde{R}} (1_{U - {y}}) (x)$ , ∀x, y ∈ U,

$\Leftrightarrow \bar{\tilde{R}} (1_{x}) (y) = \bar{\tilde{R}} (1_{y}) (x)$ , ∀x, y ∈ U;

$\tilde{R}$ is transitive $\Leftrightarrow \underline{\tilde{R}} (\tilde{A}) ⋐ \underline{\tilde{R}} (\underline{\tilde{R}} (\tilde{A}))$ , $\forall \tilde{A} \in$ SVNS(U),

$\Leftrightarrow \bar{\tilde{R}} (\bar{\tilde{R}} (\tilde{A})) ⋐ \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in$ SVNS(U).

3 Single valued neutrosophic topological spaces

In this section, we will introduce the concept of single valued neutrosophic topological spaces and basis concepts related to single valued neutrosophic topological spaces.

We first introduce the concept of a single valued neutrosophic topology as follows.

Definition 3.1. A single valued neutrosophic topology on a nonempty set U is a family τ of SVNSs in U that satisfies the following conditions:

$\tilde{\emptyset}, \tilde{U} \in τ,$

$\tilde{A} ⋒ \tilde{B} \in τ$ for any $\tilde{A}, \tilde{B} \in τ,$

$⋓_{i \in I} {\tilde{A}}_{i} \in τ$ for any ${\tilde{A}}_{i} \in τ$ , i ∈ I, where I is an index set.

The pair (U, τ) is called a single valued neutrosophic topological space and each SVNS $\tilde{A}$ in τ is referred to as a single valued neutrosophic open set in (U, τ). The complement of a single valued neutrosophic open set in (U, τ) is called a single valued neutrosophic closed set in (U, τ).

Example 3.2. Let U = {x₁, x₂}. $\tilde{A}$ , $\tilde{B}$ , $\tilde{C}$ , $\tilde{D}$ are four SVNSs in U defined as follows: $\begin{matrix} \tilde{A} & = & {〈 x_{1}, 0.2, 0.8, 0.1 〉, 〈 x_{2}, 1, 0.3, 0.1 〉}, \\ \tilde{B} & = & {〈 x_{1}, 0.2, 0.8, 0.6 〉, 〈 x_{2}, 0.5, 0.4, 1 〉}, \\ \tilde{C} & = & {〈 x_{1}, 0.3, 0.7, 0.1 〉, 〈 x_{2}, 1, 0.2, 0.1 〉}, \\ \tilde{D} & = & {〈 x_{1}, 0.1, 0.9, 0.8 〉, 〈 x_{2}, 0.4, 0.5, 1 〉} . \end{matrix}$

By Definitions 2.6 and 3.1, $τ = {\tilde{\emptyset}, \tilde{U}, \tilde{A}, \tilde{B}, \tilde{C}, \tilde{D}}$ is a single valued neutrosophic topology on U and (U, τ) is a single valued neutrosophic topological space.

Now, we define the single valued neutrosophic interior and closure operators in a single valued neutrosophic topological space.

Definition 3.3. Let (U, τ) be a single valued neutrosophic topological space. For any $\tilde{A} \in$ SVNS(U), the single valued neutrosophic interior and closure of $\tilde{A}$ are defined as follows: $\begin{matrix} int (\tilde{A}) & = & ⋓ {\tilde{M} | \tilde{M} \in τ and \tilde{M} ⋐ \tilde{A}}, \\ cl (\tilde{A}) & = & ⋒ {\tilde{N} | {\tilde{N}}^{c} \in τ and \tilde{A} ⋐ \tilde{N}}, \end{matrix}$ where int and cl: SVNS(U)⟶ SVNS(U) are called the single valued neutrosophic interior and closure operators of τ, respectively.

Next, we discuss the properties of the single valued neutrosophic interior and closure operators.

Theorem 3.4. Let (U, τ) be a single valued neutrosophic topological space. For any $\tilde{A} \in$ SVNS(U), we have

$\tilde{A}$ is a single valued neutrosophic open set in (U, τ) iff $int (\tilde{A}) = \tilde{A}$ ;

$\tilde{A}$ is a single valued neutrosophic closed set in (U, τ) iff $cl (\tilde{A}) = \tilde{A}$ .

Proof. It is straightforward from Definition 3.3.

Theorem 3.5. Let (U, τ) be a single valued neutrosophic topological space. For any $\tilde{A}, \tilde{B} \in$ SVNS(U), the following results hold:

$(int (\tilde{A}))^{c} = cl ({\tilde{A}}^{c});$

$(cl (\tilde{A}))^{c} = int ({\tilde{A}}^{c});$

$int (\tilde{U}) = \tilde{U}$ , $int (\tilde{\emptyset}) = \tilde{\emptyset};$

$cl (\tilde{U}) = \tilde{U}$ , $cl (\tilde{\emptyset}) = \tilde{\emptyset};$

$int (\tilde{A}) ⋐ \tilde{A};$

$\tilde{A} ⋐ cl (\tilde{A});$

$int (int (\tilde{A})) = int (\tilde{A});$

$cl (cl (\tilde{A})) = cl (\tilde{A});$

$int (\tilde{A} ⋒ \tilde{B}) = int (\tilde{A}) ⋒ int (\tilde{B});$

$cl (\tilde{A} ⋓ \tilde{B}) = cl (\tilde{A}) ⋓ cl (\tilde{B});$

If $\tilde{A} ⋐ \tilde{B}$ , then $int (\tilde{A}) ⋐ int (\tilde{B});$

If $\tilde{A} ⋐ \tilde{B}$ , then $cl (\tilde{A}) ⋐ cl (\tilde{B}) .$

Proof. We only prove (Int0) and (Int4). For any $\tilde{A}, \tilde{B} \in$ SVNS(U), by Definition 3.3 and Proposition 2.7, we have $\begin{matrix} (Int 0) (int (\tilde{A}))^{c} \\ = (⋓ {\tilde{M} ∣ \tilde{M} \in τ and \tilde{M} ⋐ \tilde{A}})^{c} \\ = ⋒ {{\tilde{M}}^{c} ∣ \tilde{M} \in τ and \tilde{M} ⋐ \tilde{A}} \\ = ⋒ {\tilde{N} ∣ {\tilde{N}}^{c} \in τ and {\tilde{N}}^{c} ⋐ \tilde{A}} \\ = ⋒ {\tilde{N} ∣ {\tilde{N}}^{c} \in τ and {\tilde{A}}^{c} ⋐ \tilde{N}} \\ = cl ({\tilde{A}}^{c}); \\ (Int 4) int (\tilde{A} ⋒ \tilde{B}) \\ = ⋓ {\tilde{M} ∣ \tilde{M} \in τ and \tilde{M} ⋐ (\tilde{A} ⋒ \tilde{B})} \\ = ⋓ {\tilde{M} ∣ \tilde{M} \in τ and \tilde{M} ⋐ \tilde{A} and \tilde{M} ⋐ \tilde{B}} \\ = (⋓ {\tilde{M} ∣ \tilde{M} \in τ and \tilde{M} ⋐ \tilde{A}}) \\ ⋒ (⋓ {\tilde{M} ∣ \tilde{M} \in τ and \tilde{M} ⋐ \tilde{B}}) \\ = int (\tilde{A}) ⋒ int (\tilde{B}) . \end{matrix}$

The following Theorem 3.6 shows that under some conditions, a single valued neutrosophic operator is the single valued neutrosophic interior operator (the single valued neutrosophic closure operator) of a certain topology.

Theorem 3.6. (1) If a single valued neutrosophic operator int: SVNS(U)⟶ SVNS(U) satisfies the properties (Int1)–(Int4), then there exists a single valued neutrosophic topology τ_int on U such that int_{τ
_int} = int.

(2) If a single valued neutrosophic operator cl: SVNS(U)⟶ SVNS(U) satisfies the properties (Cl1)–(Cl4), then there exists a single valued neutrosophic topology τ_cl on U such that cl_{τ
_cl} = cl.

Proof. (1) Define τ_int $= {\tilde{A} \in$ SVNS $(U) ∣ int (\tilde{A}) = \tilde{A}} .$ Next, we show τ_int satisfies (T₁)–(T₃).

(T₁) By (Int1), $\tilde{\emptyset}, \tilde{U} \in τ$ _int.

(T₂) For any $\tilde{A}, \tilde{B} \in τ$ _int, $int (\tilde{A}) = \tilde{A}$ and $int (\tilde{B}) = \tilde{B}$ . By (Int4), we have $int (\tilde{A} ⋒ \tilde{B}) = int (\tilde{A}) ⋒ int (\tilde{B}) = \tilde{A} ⋒ \tilde{B}$ . So $\tilde{A} ⋒ \tilde{B} \in τ$ _int.

(T₃) Suppose that ${\tilde{A}}_{i} \in τ$ _int, then $int ({\tilde{A}}_{i}) = {\tilde{A}}_{i}$ for any i ∈ I. By (Int2), we have $int (⋓_{i \in I} {\tilde{A}}_{i}) ⋐ ⋓_{i \in I} {\tilde{A}}_{i}$ .

Conversely, $int ({\tilde{A}}_{i}) ⋐ ⋓_{i \in I} int ({\tilde{A}}_{i})$ . By (Int3) and (Int5), we have $int (⋓_{i \in I} int ({\tilde{A}}_{i})) ⋑ int (int ({\tilde{A}}_{i})) = int ({\tilde{A}}_{i})$ for any i ∈ I. Thus $int (⋓_{i \in I} int ({\tilde{A}}_{i})) ⋑ ⋓_{i \in I} int ({\tilde{A}}_{i})$ . Since $int ({\tilde{A}}_{i}) = \tilde{A_{i}}$ , we have $int (⋓_{i \in I} {\tilde{A}}_{i}) ⋑ ⋓_{i \in I} {\tilde{A}}_{i} .$

Thus $int (⋓_{i \in I} ({\tilde{A}}_{i})) = ⋓_{i \in I} {\tilde{A}}_{i}$ . So $⋓_{i \in I} ({\tilde{A}}_{i}) \in τ$ _int.

Hence τ_int is a single valued neutrosophic topology on U.

It is obvious that int_{τ
_int} = int.

(2) Define τ_cl = ${\tilde{A} \in$ SVNS $(U) ∣ cl ({\tilde{A}}^{c}) = {\tilde{A}}^{c}}$ . Next, we show τ_cl satisfies (T₁)–(T₃).

(T₁) By Definition 2.5, we have ${\tilde{\emptyset}}^{c} = \tilde{U}$ and ${\tilde{U}}^{c} = \tilde{\emptyset}$ . Then by (Cl1), $cl ({\tilde{\emptyset}}^{c}) = cl (\tilde{U}) = \tilde{U} = {\tilde{\emptyset}}^{c}$ and $cl ({\tilde{U}}^{c}) = cl (\tilde{\emptyset}) = \tilde{\emptyset} = {\tilde{U}}^{c}$ , which means that $\tilde{\emptyset}, \tilde{U} \in τ_{cl}$ .

(T₂) For any $\tilde{A}, \tilde{B} \in τ$ _cl, $cl ({\tilde{A}}^{c}) = {\tilde{A}}^{c}$ and $cl ({\tilde{B}}^{c}) = {\tilde{B}}^{c}$ . By Proposition 2.7 and (Cl4), we have $cl ((\tilde{A} ⋒ \tilde{B})^{c}) = cl ({\tilde{A}}^{c} ⋓ {\tilde{B}}^{c}) = cl ({\tilde{A}}^{c}) ⋓ cl ({\tilde{B}}^{c}) = {\tilde{A}}^{c} ⋓ {\tilde{B}}^{c} = (\tilde{A} ⋒ \tilde{B})^{c}$ . So $\tilde{A} ⋒ \tilde{B} \in τ_{cl}$ .

(T₃) Suppose that ${\tilde{A}}_{i} \in τ$ _cl, then $cl ({\tilde{A}}_{i}^{c}$ $) = {\tilde{A}}_{i}$ ^c for any i ∈ I. By (Cl2), we have ${(⋓_{i \in I} {\tilde{A}}_{i})}^{c}$ $⋐ cl ((⋓_{i \in I} {\tilde{A}}_{i})$ ^c).

Conversely, by (Cl5), we have $cl ((⋓_{i \in I} {\tilde{A}}_{i})^{c}) = cl (⋒_{i \in I} {\tilde{A}}_{i}^{c}) ⋐ cl ({\tilde{A}}_{i}$ ^c) for any i ∈ I. Thus

$cl ((⋓_{i \in I} {\tilde{A}}_{i})^{c}) ⋐ ⋒_{i \in I} cl ({\tilde{A}}_{i}^{c}) = ⋒_{i \in I} {\tilde{A}}_{i}^{c} = (⋓_{i \in I} {\tilde{A}}_{i})^{c}$ .

Thus $cl ({(⋓_{i \in I} {\tilde{A}}_{i})}^{c})$ $= (⋓_{i \in I} {\tilde{A}}_{i})$ ^c. So $⋓_{i \in I} {\tilde{A}}_{i} \in τ$ _cl.

Hence τ_cl is a single valued neutrosophic topology on U.

It is obvious that cl_{τ
_cl} = cl.

Theorem 3.7. (1) Let int: SVNS(U)⟶ SVNS(U) be a single valued neutrosophic operator satisfying the properties (Int1)–(Int4). Define

τ_{int}^{'} = {int (\tilde{A}) ∣ \tilde{A} \in SVNS (U)},

then $τ_{int}^{'} = τ_{int}$ .

(2) Let cl: SVNS(U)⟶ SVNS(U) be a single valued neutrosophic operator satisfying the properties (Cl1)–(Cl4). Define

τ_{cl}^{'} = {(cl (\tilde{A}))^{c} ∣ \tilde{A} \in SVNS (U)},

then $τ_{cl}^{'} = τ_{cl}$ .

Proof. (1) Obviously, $τ_{int} = {\tilde{A} \in$ SVNS $(U) | int (\tilde{A}) = \tilde{A}} ⋐$ $τ_{int}^{'} .$ Conversely, for any $\tilde{A} \in$ SVNS(U), by (Int3), $int (int (\tilde{A})) = int (\tilde{A})$ , from which we have $int (\tilde{A}) \in τ$ _int. Hence $τ_{int}^{'} ⋐ τ$ _int. So $τ_{int}^{'} = τ$ _int.

(2) For any $\tilde{A} \in$ SVNS(U), $(cl (\tilde{A}))^{c} \in$ $τ_{cl}^{'}$ , we have $cl (((cl (\tilde{A}))^{c})^{c}) = cl (cl (\tilde{A})) = cl (\tilde{A}) = ((cl (\tilde{A}))^{c})^{c}$ , which means that $(cl (\tilde{A}))^{c} \in τ$ _cl. Then $τ_{cl}^{'} ⋐$ τ_cl. Conversely, for any $\tilde{A} \in τ$ _cl, $\tilde{A} = (cl (\tilde{A}$ ^c)) ^c. Since $\tilde{A}$ ^c∈ SVNS(U), we have $\tilde{A} = (cl (\tilde{A}$ ^c)) ^c ∈ $τ_{cl}^{'}$ , which means that τ_cl⋐ $τ_{cl}^{'}$ . So τ_cl = $τ_{cl}^{'}$ .

Theorem 3.8. Let int: SVNS(U)⟶ SVNS(U) be a single valued neutrosophic operator satisfying the properties (Int0)–(Int4) and cl: SVNS(U)⟶ SVNS(U) be a single valued neutrosophic operator satisfying the properties (Cl0)–(Cl4). Then $τ_{int}^{'} = τ_{int} = τ_{cl}^{'} = τ_{cl} .$

Proof. By Theorem 3.7, we have $τ_{int}^{'} = τ$ _int and $τ_{cl}^{'} = τ$ _cl. Thus, we only need to prove that $τ_{int}^{'} =$ $τ_{cl}^{'} .$

By (Int0) and (Cl0), we have $\begin{matrix} τ_{int}^{'} & = & {int (\tilde{A}) ∣ \tilde{A} \in SVNS (U)} \\ = & {(cl ({\tilde{A}}^{c}))^{c} ∣ \tilde{A} \in SVNS (U)} \\ = & {(cl (\tilde{A}))^{c} ∣ {\tilde{A}}^{c} \in SVNS (U)} \\ = & {(cl (\tilde{A}))^{c} ∣ \tilde{A} \in SVNS (U)} \\ = & τ_{cl}^{'} . \end{matrix}$

4 Relationships between single valued neutrosophic approximation spaces and single valued neutrosophic topological spaces

In this section, we will discuss the relationships between single valued neutrosophic approximation spaces and single valued neutrosophic topological spaces.

4.1 From single valued neutrosophic approximation spaces to single valued neutrosophic topological spaces

Let $(U, \tilde{R})$ be a single valued neutrosophic approximation space. Define $τ_{\tilde{R}} = {\tilde{A} \in SVNS (U) ∣ \underline{\tilde{R}} (\tilde{A}) = \tilde{A}} (i)$

The following Theorem 4.1 shows that, by Equation (i), a reflexive SVNR $\tilde{R}$ in U can induce a single valued neutrosophic topology $τ_{\tilde{R}}$ on U.

Theorem 4.1. If $\tilde{R}$ is a reflexive SVNR in U, then $τ_{\tilde{R}}$ is a single valued neutrosophic topology on U.

Proof. We verify $τ_{\tilde{R}}$ satisfies (T₁)–(T₃).

(T₁) Since $\tilde{R}$ is a reflexive relation, $\tilde{R}$ is serial. By Propositions 2.11 and 2.12, we have $\tilde{\underline{R}} (\tilde{\emptyset}) = (\tilde{\emptyset})$ and $\tilde{\underline{R}} (\tilde{U}) = (\tilde{U})$ . Hence $\tilde{\emptyset}, \tilde{U} \in$ $τ_{\tilde{R}}$ .

(T₂) According to the definition of $τ_{\tilde{R}}$ , for any $\tilde{A}, \tilde{B} \in$ $τ_{\tilde{R}}$ , we have $\tilde{\underline{R}} (\tilde{A}) = \tilde{A}$ and $\tilde{\underline{R}} (\tilde{B}) = \tilde{B}$ . By Proposition 2.11, we have $\tilde{\underline{R}} (\tilde{A} ⋒ \tilde{B}) = \tilde{\underline{R}} (\tilde{A}) ⋒ \tilde{\underline{R}} (\tilde{B}) = \tilde{A} ⋒ \tilde{B}$ , which implies that $\tilde{A} ⋒ \tilde{B} \in$ $τ_{\tilde{R}}$ .

(T₃) For any ${\tilde{A}}_{i} \in$ $τ_{\tilde{R}}$ , we have $\tilde{\underline{R}} ({\tilde{A}}_{i}) = {\tilde{A}}_{i}$ for any i ∈ I. By Proposition 2.12 and the reflexivity of $\tilde{R}$ , we have $\underline{\tilde{R}} (⋓_{i \in I} {\tilde{A}}_{i}) ⋐ ⋓_{i \in I} {\tilde{A}}_{i}$ . Conversely, by Proposition 2.11, ${\tilde{A}}_{i} = \underline{\tilde{R}} (\tilde{A_{i}}) ⋐ \underline{\tilde{R}} (⋓_{i \in I} {\tilde{A}}_{i})$ , which implies that $⋓_{i \in I} {\tilde{A}}_{i} ⋐ \underline{\tilde{R}} (⋓_{i \in I} {\tilde{A}}_{i}) .$ Thus $⋓_{i \in I} {\tilde{A}}_{i} = \underline{\tilde{R}} (⋓_{i \in I} {\tilde{A}}_{i})$ and $⋓_{i \in I} {\tilde{A}}_{i} \in$ $τ_{\tilde{R}}$ .

So $τ_{\tilde{R}}$ is a single valued neutrosophic topology on U.

Remark 4.2. If the SVNR $\tilde{R}$ is not reflexive, then $τ_{\tilde{R}}$ defined by Equations (i) may not be a single valued neutrosophic topology, as shown in the following example.

Example 4.3. Let $(U, \tilde{R})$ be a single valued neutrosophic approximation space, where U = {x₁, x₂, x₃} and $\tilde{R} \in$ SVNR(U × U) is given in Table 1.

Table 1
SVNR $\tilde{R}$

$\tilde{R}$ x ₁ x ₂ x ₃

x ₁ (0,0,1) (0.3,0.1,0.6) (1,0,0.4)

x ₂ (0,0.2,0.4) (0.6,0.5,1) (0.6,0,1)

x ₃ (1,0,1) (1,0.5,1) (1,0,0)

$\tilde{R}$	x ₁	x ₂	x ₃
x ₁	(0,0,1)	(0.3,0.1,0.6)	(1,0,0.4)
x ₂	(0,0.2,0.4)	(0.6,0.5,1)	(0.6,0,1)
x ₃	(1,0,1)	(1,0.5,1)	(1,0,0)

Obviously, $\tilde{R}$ is not reflexive. According to Definition 2.10, we have $\begin{matrix} T_{\underline{\tilde{R}} (\tilde{\emptyset})} (x_{1}) & = & ⋀_{y \in U} (F_{\tilde{R}} (x_{1}, y) \lor T_{\tilde{\emptyset}} (y)) = 0.4, \\ I_{\underline{\tilde{R}} (\tilde{\emptyset})} (x_{1}) & = & ⋁_{y \in U} ((1 - I_{\tilde{R}} (x_{1}, y)) \land I_{\tilde{\emptyset}} (y)) = 1, \\ F_{\underline{\tilde{R}} (\tilde{\emptyset})} (x_{1}) & = & ⋁_{y \in U} (T_{\tilde{R}} (x_{1}, y) \land F_{\tilde{\emptyset}} (y)) = 1, \\ T_{\bar{\tilde{R}} (\tilde{U})} (x_{1}) & = & ⋁_{y \in U} (T_{\tilde{R}} (x_{1}, y) \land T_{\tilde{U}} (y)) = 1, \\ I_{\bar{\tilde{R}} (\tilde{U})} (x_{1}) & = & ⋀_{y \in U} (I_{\tilde{R}} (x_{1}, y) \lor I_{\tilde{U}} (y)) = 0, \\ F_{\bar{\tilde{R}} (\tilde{U})} (x_{1}) & = & ⋀_{y \in U} (F_{\tilde{R}} (x_{1}, y) \lor F_{\tilde{U}} (y)) = 0.4 . \end{matrix}$

Hence $\underline{\tilde{R}} (\tilde{\emptyset}) (x_{1}) = (0.4, 1, 1)$ . Similarly, we can obtain $\underline{\tilde{R}} (\tilde{\emptyset}) (x_{2}) = (0.4, 1, 0.6)$ and $\underline{\tilde{R}} (\tilde{\emptyset}) (x_{3}) = (0, 1, 1)$ . Thus $\underline{\tilde{R}} (\tilde{\emptyset}) = {〈 x_{1}, 0.4, 1, 1 〉, 〈 x_{2}, 0.4, 1, 0.6 〉,$ $〈 x_{3}, 0, 1, 1 〉} \neq \tilde{\emptyset}$ , which means that $\tilde{\emptyset} \notin τ_{\tilde{R}}$ . So $τ_{\tilde{R}}$ do not form a single valued neutrosophic topology.

Lemma 4.4. If $\tilde{R}$ is a reflexive and transitive SVNR in U, for any $\tilde{A} \in$ SVNS(U), we have $\underline{\tilde{R}} (\underline{\tilde{R}} (\tilde{A})) = \underline{\tilde{R}} (\tilde{A}) and \bar{\tilde{R}} (\bar{\tilde{R}} (\tilde{A})) = \bar{\tilde{R}} (\tilde{A}) .$

Proof. It can be easily verified by Proposition 2.12.

Theorem 4.5. The following conclusions hold:

If $\tilde{R}$ is a reflexive and transitive SVNR in U, then $τ_{\tilde{R}} = {\underline{\tilde{R}} (\tilde{A}) ∣ \tilde{A} \in$ SVNS(U)};

$τ_{\tilde{R}} = {\tilde{A} \in$ SVNS $(U) ∣ \bar{\tilde{R}} ({\tilde{A}}^{c}) = {\tilde{A}}^{c}}$ .

Proof. (1) It is obvious that $τ_{\tilde{R}} ⋐ {\underline{\tilde{R}} (\tilde{A}) ∣ \tilde{A} \in$ SVNS(U)}. By Lemma 4.4, for any $\tilde{A} \in$ SVNS(U), we have $\underline{\tilde{R}} (\underline{\tilde{R}} (\tilde{A})) = \underline{\tilde{R}} (\tilde{A})$ , which implies that $\underline{\tilde{R}} (\tilde{A}) \in$ $τ_{\tilde{R}}$ and ${\underline{\tilde{R}} (\tilde{A}) ∣ \tilde{A} \in$ SVNS $(U)} ⋐ τ_{\tilde{R}}$ . So $τ_{\tilde{R}} = {\underline{\tilde{R}} (\tilde{A}) ∣ \tilde{A} \in$ SVNS(U)}.

(2) It follows immediately from the duality of $\tilde{\underline{R}}$ and $\bar{\tilde{R}}$ .

The above Theorem 4.5 (1) states a reflexive and transitive single valued neutrosophic relation can generate a single valued neutrosophic topology and this topology is just the family of all single valued neutrosophic lower approximations induced by the given single valued neutrosophic relation.

Proposition 4.6. Let I be an index set, and ${\tilde{A}}_{i} \in$ SVNS(U) for any i ∈ I. If $\tilde{R}$ is a reflexive and transitive SVNR in U, then $\tilde{\underline{R}} (⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})) = ⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})$ .

Proof. By the reflexivity of $\tilde{R}$ , we have $\tilde{\underline{R}} (⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i}))$ $⋐ ⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})$ .

On the other hand, since $⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i}) ⋑ \tilde{\underline{R}} ({\tilde{A}}_{i})$ , $\tilde{\underline{R}} (⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})) ⋑ \tilde{\underline{R}} (\tilde{\underline{R}} ({\tilde{A}}_{i})) .$ By Lemma 4.4, we have $\tilde{\underline{R}} (⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})) ⋑ \tilde{\underline{R}} ({\tilde{A}}_{i})$ , which means that $\tilde{\underline{R}} (⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})) ⋑ ⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})$ .

Thus $\tilde{\underline{R}} (⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})) = ⋓_{i \in I} \tilde{\underline{R}} ({\tilde{A}}_{i})$ .

The following Theorem 4.7 shows that the single valued neutrosophic lower and upper approximation operators are the interior and closure operators of a single valued topological space induced by a reflexive and transitive SVNR, respectively.

Theorem 4.7. Let (U, $τ_{\tilde{R}})$ be a single valued neutrosophic topological space induced by a reflexive and transitive SVNR w.r.t. $(U, \tilde{R})$ , i.e., $τ_{\tilde{R}} = {\underline{\tilde{R}} (\tilde{A}) ∣ \tilde{A} \in$ SVNS(U)}, then $\forall \tilde{A} \in$ SVNS(U), $\begin{matrix} (1) & \tilde{\underline{R}} (\tilde{A}) = {int}_{τ_{\tilde{R}}} (A) \\ = ⋓ {\tilde{\underline{R}} (\tilde{B}) ∣ \tilde{\underline{R}} (\tilde{B}) ⋐ \tilde{A}, \tilde{B} \in SVNS (U)}, \\ (2) & \bar{\tilde{R}} (\tilde{A}) = {cl}_{τ_{\tilde{R}}} (\tilde{A}) \\ = ⋒ {(\tilde{\underline{R}} (\tilde{B}))^{c} ∣ (\tilde{\underline{R}} (\tilde{B}))^{c} ⋑ \tilde{A}, \tilde{B} \in SVNS (U)} \\ = ⋒ {\bar{\tilde{R}} ({\tilde{B}}^{c}) ∣ \bar{\tilde{R}} ({\tilde{B}}^{c}) ⋑ \tilde{A}, \tilde{B} \in SVNS (U)} . \end{matrix}$

Proof. (1) By reflexivity of $\tilde{R}$ and Proposition 2.12, we have $\tilde{\underline{R}} (\tilde{A}) ⋐ \tilde{A}$ . So $\tilde{\underline{R}} (\tilde{A}) ⋐ ⋓ {\tilde{\underline{R}} (\tilde{B}) ∣ \tilde{\underline{R}} (\tilde{B}) ⋐ \tilde{A}, \tilde{B} \in$ SVNS(U)}. On the other hand, we have $⋓ {\tilde{\underline{R}} (\tilde{B}) ∣ \tilde{\underline{R}} (\tilde{B}) ⋐ \tilde{A}, \tilde{B} \in$ SVNS $(U)} ⋐ \tilde{A}$ , from which it follows that $\tilde{\underline{R}} (⋓ {\tilde{\underline{R}} (\tilde{B}) ∣ \tilde{\underline{R}} (\tilde{B}) ⋐ \tilde{A}, \tilde{B} \in$ SVNS $(U)}) ⋐ \tilde{\underline{R}} (\tilde{A})$ . By Proposition 4.6, $⋓ {\tilde{\underline{R}} (\tilde{B}) ∣ \tilde{\underline{R}} (\tilde{B}) ⋐ \tilde{A}, \tilde{B} \in$ SVNS $(U)} = \underline{\tilde{R}} (⋓ {\tilde{\underline{R}} (\tilde{B}) ∣ \tilde{\underline{R}} (\tilde{B})) ⋐ \tilde{\underline{R}} (\tilde{A}) .$ Hence we can conclude that

$\tilde{\underline{R}} (\tilde{A}) = ⋓ {\tilde{\underline{R}} (\tilde{B}) ∣ \tilde{\underline{R}} (\tilde{B}) ⋐ \tilde{A}, \tilde{B} \in$ SVNS(U)}.

(2) It follows immediately from the duality of $\tilde{\underline{R}}$ and $\bar{\tilde{R}}$ .

Proposition 4.8. Let $\tilde{R} \in$ SVNR(U × U), then ∀x, y ∈ U, we have ${\underline{\tilde{R}}}_{(1_{y})} (x) = (1, 0, 0) and {\bar{\tilde{R}}}_{(1_{y})} (x) = \tilde{R} (x, y) .$

Proof. It can easily be verified by Definition 2.10.

Theorem 4.9. Let $(U, \tilde{R})$ be a single valued neutrosophic approximation space and (U, $τ_{\tilde{R}})$ be a single valued neutrosophic topological space induced by $(U, \tilde{R})$ , where $\tilde{R}$ is a reflexive and transitive SVNR in U, then $\begin{matrix} \tilde{R} & = & {〈 (x, y), T_{\tilde{R}} (x, y), I_{\tilde{R}} (x, y), \\ F_{\tilde{R}} (x, y) 〉 ∣ (x, y) \in U \times U}, \end{matrix}$ where ∀x, y ∈ U, $\begin{matrix} T_{\tilde{R}} (x, y) & = & ⋀_{\tilde{B} \in (y)_{τ_{\tilde{R}}}} T_{\tilde{B}} (x), \\ I_{\tilde{R}} (x, y) & = & ⋁_{\tilde{B} \in (y)_{τ_{\tilde{R}}}} I_{\tilde{B}} (x), \\ F_{\tilde{R}} (x, y) & = & ⋁_{\tilde{B} \in (y)_{τ_{\tilde{R}}}} F_{\tilde{B}} (x), \\ (y)_{τ_{\tilde{R}}} & = & {\tilde{B} \in SVNS (U) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, \\ T_{\tilde{B}} (y) = 1, I_{\tilde{B}} (y) = 0, F_{\tilde{B}} (y) = 0} . \end{matrix}$

Proof. For any x, y ∈ U, by Theorem 4.7, we have $\bar{\tilde{R}} (1_{y}) = {cl}_{τ_{\tilde{R}}} (1_{y})$ . Moreover, by Proposition 4.8, we have $T_{\tilde{R}} (x, y) = T_{\bar{\tilde{R}} (1_{y})} (x) = T_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x)$ , $I_{\tilde{R}} (x, y) = I_{\bar{\tilde{R}} (1_{y})} (x) = I_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x)$ and $F_{\tilde{R}} (x, y) = F_{\bar{\tilde{R}} (1_{y})} (x) = F_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x) .$

Notice that ${cl}_{τ_{\tilde{R}}} (1_{y}) = ⋒ {\tilde{B} \in$ SVNS $(U) ∣ {\tilde{B}}^{c} \in$ $τ_{\tilde{R}}, 1_{y} ⋐ \tilde{B}} .$ Then for any x, y ∈ U, we have $\begin{matrix} T_{\tilde{R}} (x, y) & = & T_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x) \\ = & T_{⋒ {\tilde{B} \in SVNS (U) | {\tilde{B}}^{c} \in τ_{\tilde{R}}, 1_{y} ⋐ \tilde{B}}} (x) \\ = & ⋀ {T_{\tilde{B}} (x) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, T_{1_{y}} (t) \\ \leq & T_{\tilde{B}} (t) for any t \in U} \\ = & ⋀ {T_{\tilde{B}} (x) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, T_{\tilde{B}} (y) = 1} \\ = & ⋀_{\tilde{B} \in (y)_{τ_{\tilde{R}}}} T_{\tilde{B}} (x), \\ I_{\tilde{R}} (x, y) & = & I_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x) \\ = & I_{⋒ {\tilde{B} \in SVNS (U) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, 1_{y} ⋐ \tilde{B}}} (x) \\ = & ⋁ {I_{\tilde{B}} (x) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, I_{1_{y}} (t) \\ \geq & I_{\tilde{B}} (t) for any t \in U} \\ = & ⋁ {I_{\tilde{B}} (x) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, I_{\tilde{B}} (y) = 0} \\ = & ⋁_{\tilde{B} \in (y)_{τ_{\tilde{R}}}} I_{\tilde{B}} (x), \\ F_{\tilde{R}} (x, y) & = & F_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x) \\ = & F_{⋒ {\tilde{B} \in SVNS (U) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, 1_{y} ⋐ \tilde{B}}} (x) \\ = & ⋁ {F_{\tilde{B}} (x) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, F_{1_{y}} (t) \geq F_{\tilde{B}} (t) \\ for any t \in U} \\ = & ⋁ {F_{\tilde{B}} (x) ∣ {\tilde{B}}^{c} \in τ_{\tilde{R}}, F_{\tilde{B}} (y) = 0} \\ = & ⋁_{\tilde{B} \in (y)_{τ_{\tilde{R}}}} F_{\tilde{B}} (x) . \end{matrix}$

This completes the proof.

Theorem 4.9 above shows that a reflexive and transitive SVNR can be represented by its induced single valued neutrosophic topology.

4.2 From single valued neutrosophic topological spaces to single valued neutrosophic approximation spaces

In subsection 4.1, we obtain a reflexive single valued neutrosophic relation derived from a single valued neutrosophic approximation space can generate a single valued neutrosophic topologies. Furthermore, a reflexive and transitive single valued neutrosophic relation can induce a single valued neutrosophic topological space such that its single valued neutrosophic interior and closure operators are the lower and upper approximation operators induced by this single valued neutrosophic relation, respectively. In this subsection, we will discuss the reverse problem: If a single valued neutrosophic topological space can induce a single valued neutrosophic approximation space under some specifical conditions?

Theorem 4.10. Let (U, τ) be a single valued neutrosophic topological space and int, cl :SVNS(U)⟶ SVNS(U) be its single valued neutrosophic interior and closure operators, respectively. Then there exists a reflexive and transitive SVNR $\tilde{R_{τ}}$ in U such that $\underline{\tilde{R_{τ}}} (\tilde{A}) = int (\tilde{A})$ and $\bar{\tilde{R_{τ}}} (\tilde{A}) = cl (\tilde{A})$ for all $\tilde{A} \in$ SVNS(U) iff int satisfies the axioms (I1) and (I2), or equivalently, cl satisfies the axioms (C1) and (C2): $\forall \tilde{A}, \tilde{B} \in$ SVNS(U), ∀ α₁, α₂, α₃ ∈ [0, 1],

$int (\tilde{A} ⋓ \hat{α_{1}, α_{2}, α_{3}}) = int (\tilde{A}) ⋓ \hat{α_{1}, α_{2}, α_{3}}$ ,

$int (\tilde{A} ⋒ \tilde{B}) = int (\tilde{A}) ⋒ int (\tilde{B})$ ,

$cl (\tilde{A} ⋒ \hat{α_{1}, α_{2}, α_{3}}) = cl (\tilde{A}) ⋒ \hat{α_{1}, α_{2}, α_{3}}$ ,

$cl (\tilde{A} ⋓ \tilde{B}) = cl (\tilde{A}) ⋓ cl (\tilde{B})$ .

Proof. “⇒” Suppose that there exists a reflexive and transitive SVNR $\tilde{R_{τ}}$ in U such that $\underline{\tilde{R_{τ}}} (\tilde{A}) = int (\tilde{A})$ and $\bar{\tilde{R_{τ}}} (\tilde{A}) = cl (\tilde{A})$ for all $\tilde{A} \in$ SVNS(U), then it can be easily observed that the axioms (I1), (I2), (C1) and (C2) hold.

“⟸” Assume that the operator cl satisfies the axioms (C1) and (C2). By cl, we define a single valued neutrosophic relation as follows: $\begin{matrix} \tilde{R_{τ}} & = & {〈 (x, y), T_{cl (1_{y})} (x), I_{cl (1_{y})} (x), F_{cl (1_{y})} (x) 〉 \\ ∣ (x, y) \in U \times U} (ii) \end{matrix}$

Moreover, we can prove that for any $\tilde{A} \in$ SVNS(U), $\tilde{A} = ⋓_{y \in U} (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}) .$

In fact, for any x ∈ U, by Definition 2.6, we have $\begin{matrix} T_{⋓_{y \in U} (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x) \\ = ⋁_{y \in U} T_{1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)})} (x) \\ = ⋁_{y \in U} (T_{1_{y}} (x) \land T_{\hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = (1 \land T_{\tilde{A}} (x)) ⋁_{y \in U, y \neq x} (0 \land T_{\tilde{A}} (y)) \\ = T_{\tilde{A}} (x) . \\ I_{⋓_{y \in U} (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)})} (x) \\ = ⋀_{y \in U} I_{1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)})} (x) \\ = ⋀_{y \in U} (I_{1_{y}} (x) \lor I_{\hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = (0 \lor I_{\tilde{A}} (x)) ⋀_{y \in U, y \neq x} (1 \lor I_{\tilde{A}} (y)) \\ = I_{\tilde{A}} (x) . \\ F_{⋓_{y \in U} (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)})} (x) \\ = ⋀_{y \in U} F_{1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)})} (x) \\ = ⋀_{y \in U} (F_{1_{y}} (x) \lor F_{\hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = (0 \lor F_{\tilde{A}} (x)) ⋀_{y \in U, y \neq x} (1 \lor F_{\tilde{A}} (y)) \\ = F_{\tilde{A}} (x) . \end{matrix}$

So $\tilde{A} = ⋓_{y \in U} (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)})$ for any $\tilde{A} \in$ SVNS(U).

Next, we prove $cl (\tilde{A}) = \bar{\tilde{R_{τ}}} (\tilde{A})$ and $int (\tilde{A}) = \underline{\tilde{R_{τ}}} (\tilde{A})$ .

For any x ∈ U, by Definition 2.10 and the axioms (I1), (I2), (C1) and (C2), we have $\begin{matrix} T_{\bar{\tilde{R_{τ}}} (\tilde{A})} (x) & = & ⋁_{y \in U} (T_{\tilde{R_{τ}}} (x, y) \land T_{\tilde{A}} (y)) \\ = & ⋁_{y \in U} (T_{cl (1_{y})} (x) \land T_{\hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = & ⋁_{y \in U} (T_{cl (1_{y}) ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = & ⋁_{y \in U} (T_{cl (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y))}} (x)) \\ = & T_{⋓_{y \in U} cl (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y))}} (x) \\ = & T_{cl (⋓_{y \in U} (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}))} (x) \\ = & T_{cl (\tilde{A})} (x) . \\ I_{\bar{\tilde{R_{τ}}} (\tilde{A})} (x) & = & ⋀_{y \in U} (I_{\tilde{R_{τ}}} (x, y) \lor I_{\tilde{A}} (y)) \\ = & ⋀_{y \in U} (I_{cl (1_{y})} (x) \lor I_{\hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = & ⋀_{y \in U} (I_{cl (1_{y}) ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = & ⋀_{y \in U} (I_{cl (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y))}} (x)) \\ = & I_{⋓_{y \in U} cl (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y))}} (x) \\ = & I_{cl (⋓_{y \in U} (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}))} (x) \\ = & I_{cl (\tilde{A})} (x) . \\ F_{\bar{\tilde{R_{τ}}} (\tilde{A})} (x) & = & ⋀_{y \in U} (F_{\tilde{R_{τ}}} (x, y) \lor F_{\tilde{A}} (y)) \\ = & ⋀_{y \in U} (F_{cl (1_{y})} (x) \lor F_{\hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = & ⋀_{y \in U} (F_{cl (1_{y}) ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}} (x)) \\ = & ⋀_{y \in U} (F_{cl (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y))}} (x)) \\ = & F_{⋓_{y \in U} cl (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y))}} (x) \\ = & F_{cl (⋓_{y \in U} (1_{y} ⋒ \hat{T_{\tilde{A}} (y), I_{\tilde{A}} (y), F_{\tilde{A}} (y)}))} (x) \\ = & F_{cl (\tilde{A})} (x) . \end{matrix}$

Thus $cl (\tilde{A}) = \bar{\tilde{R_{τ}}} (\tilde{A})$ . Note that cl and int are dual to each other, we have $int (\tilde{A}) = \tilde{\underline{R_{τ}}} (\tilde{A}) .$ By Theorem 3.5, we have $\tilde{\underline{R_{τ}}} (\tilde{A}) ⋐ \tilde{A}$ . Then, by Proposition 2.12, $\tilde{R_{τ}}$ is reflexive. Moreover, by Theorem 3.5 again, we have $\underline{\tilde{R_{τ}}} (\underline{\tilde{R_{τ}}} (\tilde{A})) = \underline{\tilde{R_{τ}}} (\tilde{A})$ , which means that $\underline{\tilde{R_{τ}}} (\tilde{A}) ⋐ \underline{\tilde{R_{τ}}} (\underline{\tilde{R_{τ}}} (\tilde{A}))$ . By Proposition 2.12, $\tilde{R_{τ}}$ is transitive. Hence $\tilde{R_{τ}}$ is a reflexive and transitive SVNR.

The above Theorem 4.10 gives the sufficient and necessary conditions that a single valued neutrosophic interior (closure, respectively) operator derived from a single valued neutrosophic topological space is just the single valued neutrosophic lower (upper, respectively) approximation operator induced by a reflexive and transitive single valued neutrosophic relation. Based on Theorem 4.10, we give the following definition.

Definition 4.11. Let (U, τ) be a single valued neutrosophic topological space and int, cl: SVNS(U)⟶ SVNS(U) be the single valued neutrosophic interior and closure operators of τ, respectively. If int satisfies the axioms (I1) and (I2) (or equivalently, cl satisfies the axioms (C1) and (C2)), then we call (U, τ) a single valued neutrosophic rough topological space and τ a single valued neutrosophic roughtopology.

Let $\tilde{R}$ be the set of all reflexive and transitive SVNRs in U and $T$ be the set of all single valued neutrosophic rough topologies. We can obtain the following results.

Theorem 4.12. (1) If $\tilde{R} \in \tilde{R}$ , $τ_{\tilde{R}}$ is defined by Equation (i) and $τ_{{\tilde{R}}_{τ}}$ is defined by Equation (ii), then ${\tilde{R}}_{τ_{\tilde{R}}} = \tilde{R}$ .

(2) If τ $\in T$ , ${\tilde{R}}_{τ}$ is defined by Equation (ii) and $τ_{{\tilde{R}}_{τ}}$ is defined by Equation (i), then $τ_{{\tilde{R}}_{τ}} = τ$ .

Proof. (1) By Theorem 4.7, and the reflexivity and transitivity of $\tilde{R}$ , we have $\tilde{\underline{R}} = {int}_{τ_{\tilde{R}}}$ and $\bar{\tilde{R}} = {cl}_{τ_{\tilde{R}}}$ . According to Proposition 4.8, for any x, y ∈ U, we have $\begin{matrix} T_{{\tilde{R}}_{τ_{\tilde{R}}}} (x, y) & = & T_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x) = T_{\bar{\tilde{R}} (1_{y})} (x) = T_{\tilde{R}} (x, y), \\ I_{{\tilde{R}}_{τ_{\tilde{R}}}} (x, y) & = & I_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x) = I_{\bar{\tilde{R}} (1_{y})} (x) = I_{\tilde{R}} (x, y), \\ F_{{\tilde{R}}_{τ_{\tilde{R}}}} (x, y) & = & F_{{cl}_{τ_{\tilde{R}}} (1_{y})} (x) = F_{\bar{\tilde{R}} (1_{y})} (x) = F_{\tilde{R}} (x, y) . \end{matrix}$

Thus ${\tilde{R}}_{τ_{\tilde{R}}} = \tilde{R}$ .

(2) By Equation (i) and Theorem 4.10, we have $\begin{matrix} τ_{{\tilde{R}}_{τ}} & = & {\tilde{A} \in SVNS (U) ∣ \tilde{\underline{R_{τ}}} (\tilde{A}) = \tilde{A}} \\ = & {\tilde{A} \in SVNS (U) ∣ int (\tilde{A}) = \tilde{A}} \\ = & τ . \end{matrix}$

Theorem 4.13. There exists a one-to-one correspondence between $\tilde{R}$ and $T$ .

Proof. Define a mapping f: $\tilde{R} ⟶ T$ as $\forall \tilde{R} \in R, f (\tilde{R}) = τ_{\tilde{R}} .$

On the other hand, define a mapping g: $T ⟶ \tilde{R}$ as $\forall τ \in T$ , g (τ) = $\tilde{R_{τ}}$ . Then, by Theorem 4.12, it is easy to verify that both f and g are one-to-one correspondences between $R$ and $T$ .

Theorem 4.13 shows that there exists a one-to-one correspondence between the set of all reflexive and transitive single valued neutrosophic relations and the set of all single valued neutrosophic rough topologies such that the single valued neutrosophic lower and upper approximation operators induced by the reflexive and transitive single valued neutrosophic relations are the single valued neutrosophic interior andclosure operators of single valued neutrosophic rough topologies, respectively.

5 Conclusion

In this paper, we study the topological structures of single valued neutrosophic rough sets. Firstly, we prove that a reflexive and transition single valued neutrosophic relation can induce a single valued neutrosophic topological space such that its single valued neutrosophic interior and closure operators are the lower and upper approximation operators induced by this single valued neutrosophic relation, respectively. Then, we investigate the sufficient and necessary conditions that a single valued neutrosophic interior (closure, respectively) operator derived from a single valued neutrosophic topological space is just the single valued neutrosophic lower (upper, respectively) approximation operator derived from a single valued neutrosophic approximation space. Finally, we show there exists a one-to-one correspondence between the set of all reflexive and transitive single valued neutrosophic relations and the set of all single valued neutrosophic rough topologies.

In the forthcoming research, we will establish the more general single valued neutrosophic rough set model based on triangle norm.

Footnotes

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (No. 61473181) and the Fundamental Research Funds For the Central Universities (No. GK201702008).

References

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986), 87–96.

Atanassov

and Gargov

, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343–349.

Ashraf

, Naz

, Rashmanlou

and Malik

M.A.

, Regularity of graphs in single valued neutrosophic environment, Journal of Intelligent and Fuzzy Systems (2016). DOI: 10.3233/JIFS-161960

Bonikowski

, Bryniarski

and Skardowska

U.W.

, Extensions and intentions in the rough set theory, Information Sciences 107 (1998), 149–167.

Boixader

, Jacas

and Recasens

, Upper and lower approximation of fuzzy sets, International Journal of General Systems 29 (2000), 555–568.

Bryniaski

, A calculus of rough sets of the first order, Bulletin of the Polish Academy of Sciences Mathematics 36(16) (1989), 71–77.

Broumi

and Smarandache

, Rough neutrosophic sets, Italian Journal of Pure and Applied Mathematics 32 (2014), 493–502.

Broumi

and Smarandache

, Single valued neutrosophic information systems based on rough set theory, http://www.researchgate.net/publication/280742564.

Broumi

, Talea

, Bakali

and Smarandache

, Single valued neutrosophic graphs, Journal of New Theory 10 (2016), 86–101.

10.

Broumi

, Talea

, Smarandache

and Bakali

, Single valued neutrosophic graphs: Degree, order and size, IEEE International Conference on Fuzzy Systems (FUZZ) 2016, pp. 2444–2451.

11.

Broumi

, Bakali

, Talea

and Smarandache

, Isolated single valued neutrosophic graphs, Neutrosophic Sets and Systems 11 (2016), 74–78.

12.

Broumi

, Talea

, Bakali

and Smarandache

, Interval valued neutrosophic graphs, Critical Review 12 (2016), 5–33.

13.

Broumi

, Smarandache

, Talea

and Bakali

, Decisionmaking method based on the interval valued neutrosophic graph, Future Technologies Conference (2016), pp. 44–50.

14.

Broumi

, Bakali

, Talea

, Smarandache

and Ali

, Shortest path problem under bipolar neutrosphic setting, Applied Mechanics and Materials 859 (2016), 59–66.

15.

Broumi

, Bakali

, Talea

, Smarandache

and Vladareanu

, Computation of shortest path problem in a network with SV-trapezoidal neutrosophic numbers, Proceedings of the 2016 International Conference on Advanced Mechatronic Systems, Melbourne, Australia, 2016, pp. 417–422.

16.

Broumi

, Bakali

, Talea

, Smarandache

and Vladareanu

, Applying dijkstra algorithm for solving neutrosophic shortest path problem, Proceedings of the 2016 International Conference on Advanced Mechatronic Systems, Melbourne, Australia, 2016, pp. 412–416.

17.

Broumi

, Talea

, Bakali

and Smarandache

, On bipolar single valued neutrosophic graphs, Journal of New Theory 11 (2016), 84–102.

18.

Broumi

, Smarandache

, Talea

and Bakali

, An introduction to bipolar single valued neutrosophic graph theory, Applied Mechanics and Materials 841 (2016), 184–191.

19.

Chuchro

, A certain conception of rough sets in topological Boolean algebras, Bulletin of the Section of Logic 22 (1993), 9–12.

20.

Chuchro

, On rough sets in topological Boolean algebras, Ziarko

(ed), Rough sets, fuzzy sets and knowledge discovery, Sringer, Berlin, 1994, pp. 157–160.

21.

Dubois

and Prade

, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17 (1990), 191–209.

22.

Fathi

, Elchawalby

and Salama

A.A.

, A neutrosophic graph similarity measures, In Smarandache

and Pramanik

(editors), New Trends in Neutrosophic Theory and Applications, Brussels, Belgium, EU, 2016, pp. 223–230.

23.

Kelley

, General Topology, Van Nostrand Company, 1955.

24.

T.J.

and Zhang

W.X.

, Rough fuzzy approximation on two universes of discourse, Information Sciences 178 (2008), 892–906.

25.

and Cui

, Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators, Information Sciences 305 (2015), 219–233.

26.

Majudar

and Samant

S.K.

, On similarity and entropy of neutrosophic sets, Journal of Intelligent and Fuzzy Systems 26 (2014), 1245–1252.

27.

Z.M.

and Hu

B.Q.

, Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets, Information Sciences 218 (2013), 194–204.

28.

Pawlak

, Rough sets, International Journal of Computer and Information Sciences 11 (1982), 341–356.

29.

Pawlak

, Rough Sets: Theoretical Aspects of Reasoning about Data, Academic Kluwer Publishers, Dordrecht, Boston, 1991.

30.

Qin

K.Y.

and Pei

, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems 151 (2005), 601–613.

31.

Qin

K.Y.

, Yang

J.L.

and Pei

, Generalized rough sets based on reflexive and transitive relation, Information Sciences 178 (2008), 4138–4141.

32.

Radzikowska

A.M.

and Kerre

E.E.

, A comparative study of fuzzy rough sets, Fuzzy Sets and Systems 126 (2001), 137–155.

33.

Slowinski

and Vanderpooten

, A generalized definition of rough approximations based on similarity, IEEE Transaction on Knowledge and Data Engineering 12 (2000), 331–336.

34.

Smarandache

, Neutrosophy. Neutrosophic Probability, Set, and Logic, >American Research Press, Rehoboth, USA, 1998.

35.

Smarandache

, A unifying field in logics. Neutrosophy: neutrosophic probability, set and logic, >American Research Press, Rehoboth, USA, 1999.

36.

Thiele

, On aximatic characterisation of fuzzy approximation operators: The rough fuzzy sets case. Proceeding of the 31st IEEE Internation Symposium on Multiple-Valued Logic, 2001, pp. 330–335.

37.

Turksen

, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20 (1986), 191–210.

38.

Wang

, Smarandache

, Zhang

Y.Q.

and Sunderraman

, Single valued neutrosophic sets, Multispace and Multistruct 4 (2010), 410–413.

39.

Wiweger

, On topological rough sets, Bulletin of Polish Academy of Science: Mathematic 37 (1989), 89–93.

40.

W.Z.

and Zhou

, On intuitionistic fuzzy topologies based on intuitionistic fuzzy reflexive and transitive relation, Soft Computing 15 (2011), 1183–1194.

41.

Yang

H.L.

, Zhang

C.L.

, Guo

Z.L.

, Yiu

Y.L.

and Liao

X.W.

, A hybrid model of single valued neutrosophic sets and rough sets: single valued neutrosophic rough set model, Soft Computing, (2016). DOI: 10.1007/s00500-016-2356-y.

42.

Yang

H.L.

, Guo

Z.L.

, She

Y.H.

and Liao

X.W.

, On single valued neutrosophic relations, Journal of Intelligent and Fuzzy Systems 30 (2016), 1045–1056.

43.

Yang

and Li

, Reduction about approximation spaces of covering generalized rough sets, International Journal of Approximation Reasoning 51(3) (2010), 335–345.

44.

Yao

Y.Y.

, Constructive and algebraic methods of the theory of rough sets, Information Sciences 109 (1998), 21–47.

45.

Yao

Y.Y.

, On generalizing pawlak approximation operators, LNAI 1424 (1998), 298–307.

46.

Yao

Y.Y.

, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111 (1998), 239–259.

47.

, Multicriteria decision-making method using the correlation coefficient under single valued neutrosophic enviroment, International Journal of General Systems 42 (2013), 386–394.

48.

, Improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attributedecision making, Journal of Intelligent and Fuzzy Systems 27 (2014), 2453–2462.

49.

, A multicriteria decision-making method using aggregation operators for simplied neutrosophic sets, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2459–2466.

50.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

51.

Zhang

H.D.

, Shu

and Liao

S.L.

, Topological structures of interval-valued hesitant fuzzy rough set and its application, Journal of Intelligent and Fuzzy Systems 30 (2016), 1029–1043.

Further research of single valued neutrosophic rough sets

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Pawlak rough sets

2.2 NSs and SVNSs

2.3 Single valued neutrosophic rough sets

3 Single valued neutrosophic topological spaces

4 Relationships between single valued neutrosophic approximation spaces and single valued neutrosophic topological spaces

4.1 From single valued neutrosophic approximation spaces to single valued neutrosophic topological spaces

Table 1 SVNR R ˜ R ˜ x 1 x 2 x 3 x 1 (0,0,1) (0.3,0.1,0.6) (1,0,0.4) x 2 (0,0.2,0.4) (0.6,0.5,1) (0.6,0,1) x 3 (1,0,1) (1,0.5,1) (1,0,0)

5 Conclusion

Footnotes

Acknowledgements

References

Table 1
SVNR $\tilde{R}$

$\tilde{R}$ x ₁ x ₂ x ₃

x ₁ (0,0,1) (0.3,0.1,0.6) (1,0,0.4)

x ₂ (0,0.2,0.4) (0.6,0.5,1) (0.6,0,1)

x ₃ (1,0,1) (1,0.5,1) (1,0,0)