A New Single-Valued Neutrosophic Rough Sets and Related Topology

(Fuzzy) rough sets are closely related to (fuzzy) topologies. Neutrosophic rough sets and neutrosophic topologies are extensions of (fuzzy) rough sets and (fuzzy) topologies, respectively. In this paper, a new type of neutrosophic rough sets is presented, and the basic properties and the relationships to neutrosophic topology are discussed. .e main results include the following: (1) For a single-valued neutrosophic approximation space (U, R), a pair of approximation operators called the upper and lower ordinary single-valued neutrosophic approximation operators are defined and their properties are discussed..en the further properties of the proposed approximation operators corresponding to reflexive (transitive) single-valued neutrosophic approximation space are explored. (2) It is verified that the single-valued neutrosophic approximation spaces and the ordinary single-valued neutrosophic topological spaces can be interrelated to each other through our defined lower approximation operator. Particularly, there is a one-to-one correspondence between reflexive, transitive single-valued neutrosophic approximation spaces and quasidiscrete ordinary single-valued neutrosophic topological spaces.

(i) Broumi's rough neutrosophic sets [19]: let R be an equivalent relation (can be easily extended for an arbitrary binary relation) on U. en, for each neutrosophic set A on U, a pair of neutrosophic sets R(A) and R(A) on U are defined as the lower and upper approximations of A w.r.t. (U, R).
(ii) Sweety's neutrosophic rough sets [20]: let R be a neutrosophic relation on U. en, for each neutrosophic set A on U, a pair of neutrosophic sets R(A) and R(A) on U are defined as the lower and upper approximations of A w.r.t. (U, R). Yang [10] defined a similar model by considering the singlevalued neutrosophic relation and single-valued neutrosophic set on U.
In this paper, we shall introduce a new model of rough sets fusion with neutrosophic sets under the framework of single-valued neutrosophic approximation space (U, R) (i.e., a nonempty set U together with a single-valued neutrosophic relation R on U). For each ordinary subset A of U, we shall define a pair of single-valued neutrosophic sets R(A) and R(A) on U as the lower and upper approximations of A with respect to (U, R). Obviously, our model is different from Broumi-Sweety-Yang's models, since, in our model, the original sets are ordinary subsets of U and their approximations are single-valued neutrosophic sets, but, in Broumi-Sweety-Yang's models, the original sets and their approximations are all (single-valued) neutrosophic sets. Hence, our rough sets will be called ordinary single-valued neutrosophic rough sets.
(i) Yang's single-valued neutrosophic topological spaces [45]: for a nonempty set U, Yang defined the single-valued neutrosophic topology on U as a subset τ of Svns(U) (the set of all single-valued neutrosophic sets on U) with some conditions. Yang's space can be regarded as an extension of Lowen's fuzzy topological space [46]. Yang also proved that there is a one-to-one correspondence between reflexive and transitive single-valued neutrosophic approximation spaces and his single-valued neutrosophic rough topological spaces. (ii) Kim's ordinary single-valued neutrosophic topological spaces [47]: for a nonempty set U, Kim defined the ordinary single-valued neutrosophic topology on U as a neutrosophic set τ on P(U) (the power set of U ) with some conditions. Kim's space can be regarded as an extension of Sȏstak's fuzzy topology [48] (or Ying's fuzzifying topology [49]).
In this paper, we shall prove that there are close relationships between our ordinary single-valued neutrosophic rough sets and Kim's ordinary single-valued neutrosophic topological spaces. e close relationships exhibit that it is meaningful to investigate the new rough sets model. e method of this paper and the comparison with related literature can be summarized in Table 1. e remainder of this paper is organized as follows. In Section 2, we will recall some knowledge about neutrosophic sets and rough sets. In Section 3, we shall give the notion of ordinary single-valued neutrosophic upper and lower approximation operators and discuss their properties. en we will explore the further properties of the proposed approximations corresponding to reflexive (transitive) single-valued neutrosophic approximation space. In Section 4, we will prove that each single-valued neutrosophic approximation space induces an ordinary single valued neutrosophic topological space via our defined lower approximation. In Section 5, we shall verify that each ordinary single-valued neutrosophic topological space induces a single-valued neutrosophic approximation space. In Section 6, we will show that there is a one-to-one correspondence between reflexive and transitive singlevalued neutrosophic approximation spaces and quasidiscrete ordinary single-valued neutrosophic topological spaces.

Preliminaries
In this section, we recall some knowledge about neutrosophic rough sets and neutrosophic topologies used in this paper.
Unless otherwise stated, we always assume that U is a nonempty infinite set. We denote P(U) as the power set of U and define A c � U − A for A ∈ P(U).
Definition 1 (see [2]). An Svns A � (A T , A I , A F ) on U is defined as three membership functions A T , A I , A F : U ⟶ [0, 1], which are interpreted as truthmembership function, indeterminacy-membership function, and falsity-membership function, respectively. All Svnss are denoted by Svns(U).
Definition 3 (see [10]). An Svns R on U × U is referred to a single-valued neutrosophic relation (Svnr) on U. en the pair (U, R) is said to be a single-valued neutrosophic approximation space (Svnas). Furthermore, R is called 2 Journal of Mathematics (2) e pair (YR(A), YR (A)) is referred to the singlevalued neutrosophic rough sets of A. YR and YR are said to be the single-valued neutrosophic upper and lower approximation operators, respectively. Definition 5 (see Definition 8 in [47]). An Svns τ on P(U), 1], is referred to an ordinary single-valued neutrosophic topology (OSvnt) on U if τ fulfills the following conditions: For examples and more results about OSvnts, refer to [47]. e following lemma can be easily observed. We will use it without mentioning again.
. en the following conditions are equivalent:

Ordinary Single-Valued Neutrosophic Rough Sets for Svnas
In this section, we present the notions and properties of ordinary single-valued neutrosophic upper and lower approximation operators.  Notes: "+" represents that the set is a single-valued neutrosophic set, and "− " represents that it is not; "√" represents that there is a bijection between the considered rough sets and topologies, and "×" represents that there is no bijection. e pair (R(A), R(A)) is referred to the ordinary singlevalued neutrosophic rough sets of A. R and R are said to be the ordinary single-valued neutrosophic upper and lower approximation operators, respectively.

Remark 2
(1) e definition of R(A) T (x) is an interpretation of the fact that "the join of R T (x) and A is not empty," and the definition of ." (2) For a fuzzy relation r on U, it is easily observed that r induces an Svnr R r on U defined as follows: are the fuzzy approximations of ordinary subset w.r.t. fuzzy relation in the work of Yao [51]. erefore, the singlevalued neutrosophic approximations in this paper are a generalization of Yao's fuzzy approximations. (3) Obviously, the single-valued neutrosophic approximation operators in this paper are different from the single-valued neutrosophic approximation operators in the work of Yang [10], since our operators are defined from P(U) to Svns(U) and Yang's operators are defined from Svns(U) to Svns(U). 3 and let R be defined as in Table 2.
Hence, we obtain R(A) and R(A) as in Table 3.

Theorem 1. Let (U, R) be an Svnas.
en we have the following: Proof. For (1)-(3), we prove only the results for lower approximation.
e proofs for upper approximation are similar and hence are omitted.
(1) For any For any x ∈ U and A⊆B, we obtain 4 Journal of Mathematics
e following theorem gives a characterization on the approximation operators generated by reflexive Svnas. □ Theorem 2. Let (U, R) be an Svnas. en the following three are equivalent: Hence, R(A)⊑⊤ A .

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Hence, R is reflexive. (1) R is transitive.
(i) For any B⊆A, we have R (B) T (x) ≤ R (A) T (x) and so Note that, for any y ∈ B x , z ∉ B x , we have and then Hence, Note that, for any y ∈ B x , z ∉ B x , we have   (22) and then Hence, Note that, for any y ∈ B x , z ∉ B x , we have and then Hence, (2) ⇒ (1). Let x, y, z ∈ U.
(i) Note that Take any A⊆U − z { }; then y ∈ A or y ∉ A.
Case 2: if y ∉ A, then A⊆U − y and so By a combination of Cases 1 and 2, we obtain that is, R T (x, z) ≥ R T (x, y)∧R T (y, z), as desired. (ii) Note that Journal of Mathematics Similar to (i), we can prove that Take any A⊆U − z { }; then y ∈ A or y ∉ A.
Case 2: if y ∉ A, then A⊆U − y and so By a combination of Cases 1 and 2, we obtain that is, R F (x, z) ≤ R F (x, y)∨R F (y, z), as desired.
From (i)-(iii), we know that R is transitive.

Ordinary Single-Valued Neutrosophic Topological Space Induced by Single-Valued Neutrosophic Approximation Space
In this section, we shall consider the OSvnt induced by Svnas through the ordinary single-valued neutrosophic lower approximation operator. At first, we fix a subclass of ordinary single-valued neutrosophic topological spaces.
Definition 7. An OSvnts (U, τ) is said to be quasidiscrete if it fulfills the following: It is not difficult to see that quasidiscrete OSvnts is an extension of quasidiscrete topological space [10].
is a quasidiscrete OSvnt on U.
Proof. OSvnt1: it follows that Journal of Mathematics Similarly, we can prove that Similarly, we can prove that e definition of τ R (A) is an interpretation of the fact that "A is contained in its lower approximation."

Single-Valued Neutrosophic Approximation Space Induced by Ordinary Single-Valued Neutrosophic Topological Space
In this section, we shall consider the Svnas induced by OSvnt.
Theorem 5. Let (U, τ) be an OSvnts. en the Svnr R τ on U is defined as follows: for any (x, y) ∈ U × U, is reflexive and transitive.
Proof. Reflexivity: it follows that Transitivity: let x, y, z ∈ U.
(i) Note that Take any D ∈ P(U) with (x, z) ∈ D × D c ; then y ∈ D or y ∈ D c . Case 1: if y ∈ D, then (y, z) ∈ D × D c . So, By a combination of Cases 1 and 2, we obtain that (ii) Note that Take any D ∈ P(U) with (x, z) ∈ D × D c ; then y ∈ D or y ∈ D c . Case 1: if y ∈ D, then (y, z) ∈ D × D c . So, Case 2: if y ∈ D c , then (x, y) ∈ D × D c . So,

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By a combination of Cases 1 and 2, we obtain that (iii) Similar to (ii), one can prove that □ Remark 4. Note that neither of the topological conditions (OSvnt1)-(OSvnt3) is used in the above theorem. Hence, it can be extended to any single-valued neutrosophic relation on P(U).

One-to-One Correspondence between Reflexive and Transitive Single-Valued Neutrosophic Approximation Spaces and Quasidiscrete Ordinary Single-Valued Neutrosophic Topological Spaces
In this section, we prove that there is a one-to-one correspondence between reflexive and transitive Svnas and quasidiscrete OSvnts.
From eorems 6 and 7, we obtain the following corollary.

Corollary 1.
ere is a one-to-one correspondence between reflexive and transitive Svnas and quasidiscrete OSvnts with the same underlying set.
Remark 5. We can give a similar discussion on Svnas and ordinary single-valued neutrosophic cotopology in [47] via the ordinary single-valued neutrosophic upper approximation operator.

Conclusions
In this paper, we presented a new model of neutrosophic rough sets.
e difference between this model and the existing models is that, in our model, the original sets are ordinary subsets of U and their approximations are singlevalued neutrosophic sets; however, in the existing models, the original sets and their approximations are all (singlevalued) neutrosophic sets. We also discussed the basic properties of the proposed rough sets and gave their relationships with Kim's ordinary single-valued neutrosophic topology. Particularly, we proved by our lower approximation operator that there is a one-to-one correspondence between reflexive and transitive single-valued neutrosophic approximation spaces and quasidiscrete ordinary singlevalued neutrosophic topological spaces. In the future work, we shall present a more general single-valued neutrosophic topology such that it can be regarded as an extension of bifuzzy topology in [49]. We will also consider the corresponding single-valued neutrosophic rough sets related to the new single-valued neutrosophic topology. Furthermore, from Remark 1, we know that when restricting single-valued neutrosophic sets to Pythagorean fuzzy sets, we can define a model of Pythagorean fuzzy rough sets. It is well known that Pythagorean fuzzy sets and (fuzzy) rough sets have been applied in many fields, particularly in multiple attribute decision-making [9,16,[52][53][54][55]. erefore, in the future, we will also consider the potential application of Pythagorean fuzzy rough sets.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares no conflicts of interest.

Authors' Contributions
Qiu Jin and Kai Hu contributed the central idea, and all authors contributed to the writing and revisions.