On Single-Valued Neutrosophic Proximity Spaces

In this paper, the notion of single-valued neutrosophic proximity spaces which is a generalisation of fuzzy proximity spaces [Katsaras AK. Fuzzy proximity spaces. Anal and Appl. 1979;68(1):100–110.] and intuitionistic fuzzy proximity spaces [Lee SJ, Lee EP. Intuitionistic fuzzy proximity spaces. Int J Math Math Sci. 2004;49:2617–2628.] was introduced and some of their properties were investigated. Then, it was shown that a single-valued neutrosophic proximity on a set X induced a single-valued neutrosophic topology on X. Furthermore, the existence of initial single-valued neutrosophic proximity structure is proved. Finally, based on this fact, the product of single-valued neutrosophic proximity spaces was introduced.


Introduction
In 1998, F. Smarandache [1] introduced the concept of neutrosophic set which is a mathematical tool for handling problems involving incomplete, indeterminate and inconsistent information in real world. The neutrosophic sets are characterised by three membership functions independently: truth, indeterminacy and falsity, which are within the real standard or nonstandard unit interval ] − 0, 1 + [. Therefore, this notion is a generalisation of the theory of fuzzy sets [2] and intuitionistic fuzzy sets [3]. Salama and Alblowi [4] introduced and studied neutrosophic topological spaces and its continuous functions.
The neutrosophic set generalises the sets from a philosophical point of view. But, from a scientific or an engineering point of view, the neutrosophic set operators need to be specified. Because, it is not convenient to apply neutrosophic sets to practical problems in the real-life applications. So, Wang et al. [5] introduced the single-valued neutrosophic sets (SVNSs) by simplifying neutrosophic sets (NSs). SVNSs are also a generalisation of intuitionistic fuzzy sets, in which three membership functions are independent and their value belong to the unit interval [0, 1].
Neutrosophic set theory is widely studied by many researchers. It is used in many real application area, such as medical diagnosis [6], image processing [7], fault diagnosis [8] and multi-criteria decision making [9], which are over-cited research topics in various fields.
Proximity spaces were introduced by Efremovich during the first part of 1930s and later axiomatised [10,11]. He characterised the proximity relation 'A is near B' for subsets A and CONTACT S. Özkan ozkans@nevsehir.edu.tr B of any set X. Efremovich [11] defined the closure of a subset A of X to be the collection of all points of X 'close' A. In this way, he showed that a topology (completely regular) can be introduced in a proximity space. This theory was improved by Smirnov [12]. He was the first to discover relationship between proximities and uniformities.
In 2007, Peters [13] extended the standard spatial proximity space to a descriptive proximity space that examined the descriptive nearness of objects. The concept of descriptive proximity is useful in identifying, analysing and classifying the parts of a digital image. In recent years, several practical applications in many fields such as cytology (cell biology), criminology, digital image processing have been published [14,15].
The most comprehensive work on the proximity spaces theory was done by Naimpally and Warrack [16]. All preliminary information about proximity spaces can be found in this source.
The main objective of this paper is (1) to introduce the concept of single-valued neutrosophic proximity spaces and investigated some of their properties, (2) to show that a single-valued neutrosophic proximity on a set X induced a single-valued neutrosophic topology on X, and (3) to define the initial single-valued neutrosophic proximity structure and the product of single-valued neutrosophic proximity spaces.

Preliminaries
The following are some basic definitions and notations which we will use throughout the paper.

Definition 2.1 ([4]):
Let X be a nonempty fixed set. A neutrosophic set (NS for short) A is an object having the form x ∈ X} where the functions T A , I A and F A which represent the degree of membership (namely T A (x)), the degree of indeterminacy (namely I A (x)) and the degree of nonmembership (namely F A (x)) respectively, of each element x ∈ X to the set A with the condition

Remark 2.1 ([4]):
Every intuitionistic fuzzy set (IFS for short) A is a nonempty set in X is obviously on NS having the form

Definition 2.2 ([4]):
The neutrosophic set 0 N in X may be defined as The neutrosophic set 1 N in X may be defined as

Definition 2.3 ([4]):
Let A = x, T A , I A , F A be a NS on X, then the complement of the set A (C(A) for short) may be defined as three kinds of complements: One can define several relations and operations between neutrosophic sets follows:

Definition 2.4 ([4]):
Let X be a nonempty set, and neutrosophic sets A and B in the form x ∈ X}. Then we may consider two possible definitions for subsets.
A ⊆ B may be defined as

Proposition 2.5 ([4]):
For any neutrosophic set A, then the following conditions hold: A ∪ B may be defined as where ∨ and ∧ denote the maximum and minimum, respectively.
Since it is not convenient to apply neutrosophic sets to practical problems in the real applications, Wang et al. [5] introduced the concept of single-valued neutrosophic sets (SVNSs for short), which is an instance of a neutrosophic set. SVNSs can be used in real scientific and engineering applications.

Definition 2.7 ([5]):
Let X be a space of points (objects), with a generic element in X denoted by x. A single-valued neutrosophic set (SVNS) A in X is characterised by three membership functions, a truth-membership function T A , an indeterminacy-membership function I A , and a falsity-membership function F A . For each point x ∈ X, T A , A SVNS A can be denoted by

Remark 2.2:
For the sake of simplicity, we shall use the symbol

Remark 2.3:
In SVNSs, we consider the neutrosophic set which takes the value from the subset of the classical unit interval [0, 1] to apply neutrosophic set to science and technology. But the neutrosophic set generalises the sets from a philosophical point of view to deal with incomplete, indeterminate and inconsistent information in real world. Therefore, some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic components > 1 and some neutrosophic components < 0. respectively. We need to make distinction between workers who work overtime, and those who work full-time or part-time. That's why we need to associate a degree of membership greater than 1 to the overtime workers. Similarly, worker D was absent without pay for the whole week, and worker E did not come to work last week, but also caused damage which was estimated at a value half of the weekly salary. The membership degree of worker E has to be less than the worker D's. So the degrees of membership of workers D and E are 0/40 = 0 and −20/40 = −0.5 < 0, respectively. Consequently, the membership degrees can be off the interval [0, 1] in NSs.
An empty SVNS 0, a full SVNS 1 and the operators such as complement, containment, union, intersection in the single-valued neutrosophic sets can be defined in different forms as given in the above definitions for neutrosophic sets. We use the following definitions for these concepts: Empty SVNS 0 and full SVNS 1 are defined as Union and intersection operators are defined as Example 2.10:

Proposition 2.11: For any SVNSs A and B, then the following conditions hold:
Definition 2.12: Let X and Y be two nonempty sets and f : X → Y a function.
The concept of single-valued neutrosophic topological space is defined as follows:

Definition 2.13 ([17]):
A single-valued neutrosophic topology (SVNT for short) on a nonempty set X is a family τ of SVNSs in X satisfying the following axioms: In this case, the pair (X, τ ) is called a single-valued neutrosophic topological space (SVNTS for short). The elements of τ are called single-valued neutrosophic open sets (SVNOSs for short).

Example 2.14:
Definition 2.15: The complement of A of SVNOS is called a single-valued neutrosophic closed set (SVNCS for short) in X.
Single-valued neutrosophic interior and closure operations in SVNTSs are defined as follows:

Definition 2.17 ([17]):
Let (X, τ ) be an SVNTS and A be an SVNS in X. Then the single-valued neutrosophic interior and closure of A are defined by It can be also shown that int( A) is an SVNOS and cl( A) is an SVNCS in X.  (2) Similarly to (1).

Definition 2.19 ([16]):
A proximity (Efremovich proximity) space is a pair (X, δ), where X is a set and δ is a binary relation on the power set of X such that where A δ B means it is not true that A δ B.

Single-Valued Neutrosophic Proximity Spaces
In this section, we introduce the concept of neutrosophic proximity spaces as a generalisation of fuzzy proximity spaces [18] and intuitionistic fuzzy proximity spaces [19]. Definition 3.1: Let X be a nonempty set and t, i, f ∈ [0, 1]. The single-valued neutrosophic set x t,i,f is called a single-valued neutrosophic point (SVNP for short) in X given by    (X, δ), where δ is a relation on SVNS(X) such that where A δ B means it is not true that A δ B.

Definition 3.5: A map f : (X, δ) → (Y, δ ) between two single-valued neutrosophic proximity spaces is called a single-valued neutrosophic proximity mapping (or a p-map) if and only if
It can easily be shown that f is a p-map if and only if, for each C, D ∈ SVNS(Y), We have easily the following lemma, which follow directly from axioms ( P1), ( P2) and ( P4) of Definition 3.4.

then cl( A) ⊆ C( K) by the definition of closure and hence T cl( A) (x) ≤ T C( K) (x) = a < T cl( A) (x). This is a contradiction. Thus K δ A. Since K ⊆ C( E), A δ C( E).
This is a contradiction to the fact that Then L ∈ SVNS(X) and This is a contradiction to the fact that

(i) Suppose T cl( A∪ B) (x) > T cl( A) (x) ∨ T cl( B) (x).We may assume T cl( A) (x) ≥ T cl( B) (x). Let T cl( A∪ B) (x) = α. Then T cl( A) (x) < α and hence there exists an
Hence, by Proposition 2.11 (10) We may assume I cl We may assume This is a contradiction.

Theorem 3.8: For an SVNPS (X, δ), the family
is an SVN topology on X. This topology is called the SVN topology on X induced by the SVN proximity δ.

Initial Structures and Products
We prove the existences of initial single-valued neutrosophic proximity space. Then we define the product of SVNPSs.

Proof:
We first prove that δ is an SVN proximity on X.
. . , n and j = 1, . . . , m. Since for each α ∈ , δ α is an SVN proximity structure on X α , f α ( A i ) ∩ f α ( B j ) = 0. From this, it follows that If A δ B, then there exist finite covers { A i | i = 1, . . . , n} and { B j | j = 1, . . . , m} of A and B respectively such that f α ( A i ) δ α f α ( B j ) for some α = s ij ∈ , where i = 1, . . . , n and j = 1, . . . , m. Since each (X α , δ α ) is a single-valued neutrosophic proximity space, there It is clear that all mappings f α : (X, δ) → (X α , δ α ) (α ∈ ) are p-map. Let δ be another SVN proximity on X with respect to which each f α is a p-map. We shall show that δ is finer than δ, which will complete the proof. Suppose A δ B and consider any covers for each α ∈ . Hence, we get A δ B, i.e. δ is finer than δ.

Definition 4.2:
Let {(X α , δ α ) | α ∈ } be a family of single-valued neutrosophic proximity spaces, and X = α∈ X α . The product SVN proximity on X is defined to be the initial proximity structure δ = α∈ δ α on X with respect to which each projection map P α : (X, δ) → (X α , δ α ) (α ∈ ) is a p-map. In that case (X, δ) is said to be the product SVNPS. Proof: Let (Y, δ * ) be an SVNPS and f : (Y, δ * ) → (X, δ). It can easily be shown that if f is a p-map, then for each α ∈ , f α • f is a p-map.
Conversely, suppose that f α • f is a p-map for each α ∈ . We will show that f is a pmap. Let A, B ⊂ X, A δ * B and { A i | i = 1, . . . , n} and { B j | j = 1, . . . , m} be finite covers of f ( A) and f ( B) respectively. Then and f α • f is a p-map for each α ∈ , it follows that f α ( A i ) δ α f α ( B j ) for each α ∈ . This proves that f ( A) δ f ( B) so that f is a p-map.

Conclusion
Proximity and uniformity are important concepts close to topology and they have rich topological properties. For this reason, in recent years, these notions constitute a significant research area in the field of topological spaces. Also, these concepts have been studied by many authors on the fuzzy, soft and neutrosophic sets. In this paper, we introduced the single-valued neutrosophic proximity spaces and presented some of their properties. Then, we showed that each single-valued neutrosophic proximity determines a single-valued neutrosophic topology. Also, we introduced the initial single-valued neutrosophic proximity structure and hence we defined the products. We concluded that all the results of classical proximity spaces are still valid on the single-valued neutrosophic proximity spaces. We believe that these theoretical results will help the researchers to solve practical applications in various areas, to advance and promote other generalisations and the further studies on SVNPSs.
In future studies, the single-valued uniform spaces can be introduced and the relationships among the notions of single-valued uniform, proximity and topological spaces can be investigated. Also, various topological notions such as separation, closedness, connectedness and compactness may be characterised in the single-valued topological spaces. Furthermore, in [20][21][22], using new types of partial belong and total non-belong relations on soft separation axioms and decision-making problem were investigated. In a similar way, these can be used on the domain of single-valued neutrosophic topological spaces and proximity spaces.