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FUZZY INFORMATION AND ENGINEERING 2018, VOL. 10, NO. 4, 452–466 https://doi.org/10.1080/16168658.2020.1822082 Samed Özkan Department of Mathematics, Nevşehir Hacı Bektaş Veli University, Nevşehir, Turkey ABSTRACT ARTICLE HISTORY Received 5 December 2018 In this paper, the notion of single-valued neutrosophic proximity Revised 23 June 2020 spaces which is a generalisation of fuzzy proximity spaces [Katsaras Accepted 20 July 2020 AK. Fuzzy proximity spaces. Anal and Appl. 1979;68(1):100–110.] and intuitionistic fuzzy proximity spaces [Lee SJ, Lee EP. Intuitionistic KEYWORDS fuzzy proximity spaces. Int J Math Math Sci. 2004;49:2617–2628. ] was Single-valued neutrosophic; introduced and some of their properties were investigated. Then, it proximity space; initial was shown that a single-valued neutrosophic proximity on a set X structure; product induced a single-valued neutrosophic topology on X. Furthermore, AMS CLASSIFICATIONS the existence of initial single-valued neutrosophic proximity struc- 54E05; 54B10 ture is proved. Finally, based on this fact, the product of single-valued neutrosophic proximity spaces was introduced. 1. Introduction In 1998, F. Smarandache [1] introduced the concept of neutrosophic set which is a mathe- matical 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 stan- − + dard 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 intu- itionistic 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 © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 453 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 prox- imity 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 investi- gated 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. 2. 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 A ={x, T (x), I (x), F (x): x ∈ X} where the functions T , I A A A A A and F which represent the degree of membership (namely T (x)), the degree of indetermi- A A nacy (namely I (x)) and the degree of nonmembership (namely F (x)) respectively, of each A A element x ∈ X to the set A with the condition − + 0 ≤ T (x) + I (x) + F (x) ≤ 3 A A A A neutrosophic set A ={x, T (x), I (x), F (x): x ∈ X} can be identified to an ordered A A A − + triple T , I , F in ] 0, 1 [ (nonstandard unit interval) on X. A A A 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 A ={x, T (x),1 − (T (x) + F (x)), F (x): x ∈ X}. A A A A Definition 2.2 ([4]): The neutrosophic set 0 in X may be defined as (0 ) 0 ={x,0,0,1: x ∈ X} 1 N (0 ) 0 ={x,0,1,1: x ∈ X} 2 N (0 ) 0 ={x,0,1,0: x ∈ X} 3 N (0 ) 0 ={x,0,0,0: x ∈ X}. 4 N 454 S. ÖZKAN The neutrosophic set 1 in X may be defined as (1 ) 1 ={x,1,0,0: x ∈ X} 1 N (1 ) 1 ={x,1,0,1: x ∈ X} 2 N (1 ) 1 ={x,1,1,0: x ∈ X} 3 N (1 ) 1 ={x,1,1,1: x ∈ X}. 4 N Definition 2.3 ([4]): Let A =x, T , I , F be a NS on X, then the complement of the set A A A A (C(A) for short) may be defined as three kinds of complements: (C ) C(A) ={x,1 − T (x),1 − I (x),1 − F (x): x ∈ X} 1 A A A (C ) C(A) ={x, F (x), I (x), T (x): x ∈ X} 2 A A A (C ) C(A) ={x, F (x),1 − I (x), T (x): x ∈ X}. 3 A A A 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 A ={x, T (x), I (x), F (x): x ∈ X} and B ={x, T (x), I (x), F (x): x ∈ X}. Then we may A A A B B B consider two possible definitions for subsets. A ⊆ B may be defined as (1) A ⊆ B ⇐⇒ T (x) ≤ T (x), I (x) ≤ I (x) and F (x) ≥ F (x), ∀ x ∈ X A B A B A B (2) A ⊆ B ⇐⇒ T (x) ≤ T (x), I (x) ≥ I (x) and F (x) ≥ F (x), ∀ x ∈ X. A B A B A B Proposition 2.5 ([4]): For any neutrosophic set A, then the following conditions hold: (1) 0 ⊆ A, 0 ⊆ 0 N N N (2) A ⊆ 1 , 1 ⊆ 1 . N N N Definition 2.6 ([4]): Let X be a nonempty set, A =x, T (x), I (x), F (x) and B = A A A x, T (x), I (x), F (x) are neutrosophic sets. Then B B B A ∩ B may be defined as (1) A ∩ B =x, T (x) ∧ T (x), I (x) ∧ I (x), F (x) ∨ F (x) A B A B A B (2) A ∩ B =x, T (x) ∧ T (x), I (x) ∨ I (x), F (x) ∨ F (x) A B A B A B A ∪ B may be defined as (1) A ∪ B =x, T (x) ∨ T (x), I (x) ∨ I (x), F (x) ∧ F (x) A B A B A B (2) A ∪ B =x, T (x) ∨ T (x), I (x) ∧ I (x), F (x) ∧ F (x) A B A B A B 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 FUZZY INFORMATION AND ENGINEERING 455 (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 , an indeterminacy-membership function I , and a falsity-membership function F . For each point x ∈ X, T , I , F ∈ [0, 1]. A A A A A A SVNS A can be denoted by A ={x, T (x), I (x), F (x): x ∈ X}. A A A Remark 2.2: For the sake of simplicity, we shall use the symbol A =T , I , F for the A A A single-valued neutrosophic set A ={x, T (x), I (x), F (x): x ∈ X}. A A A 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 technol- ogy. 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. Example 2.8: In a factory a full-time worker works 40 hours per week. Considering the period of last week, worker A worked only 32 hours part-time, worker B worked full time 40 hours, and worker C worked 48 hours, working 8 hours overtime. So the degrees of membership of workers A, B and C are 32/40 = 0.8 < 1, 40/40 = 1and 48/40 = 1.2 > 1, 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 = 0and −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: Definition 2.9: Let A =T , I , F and B =T , I , F be SVNSs on a nonempty set X. Then A A A B B B Empty SVNS 0 and full SVNS 1 are defined as • 0 ={x,0,0,1: x ∈ X} • 1 ={x,1,1,0: x ∈ X} 456 S. ÖZKAN Complement of the SVNS A (C(A) for short) is defined as • C(A) ={x,1 − T (x),1 − I (x),1 − F (x): x ∈ X} A A A A ⊆ B is defined as • A ⊆ B ⇐⇒ T (x) ≤ T (x), I (x) ≤ I (x) and F (x) ≥ F (x), ∀x ∈ X. A B A B A B Union and intersection operators are defined as • A ∪ B ={x, T (x) ∨ T (x), I (x) ∨ I (x), F (x) ∧ F (x): x ∈ X} A B A B A B • A ∩ B ={x, T (x) ∧ T (x), I (x) ∧ I (x), F (x) ∨ F (x): x ∈ X}. A B A B A B Example 2.10: Let X ={x , x }, A ={x , 0.3, 1, 0.6, x , 0.8, 0.3, 0.5} and B ={x , 0.4, 0.5, 1 2 1 2 1 0.9, x , 0, 0.7, 0.2} be SVNSs on X. Then, C(A) ={x , 0.7, 0, 0.4, x , 0.2, 0.7, 0.5} 1 2 C(B) ={x , 0.6, 0.5, 0.1, x , 1, 0.3, 0.8} 1 2 A ∪ B ={x , 0.4, 1, 0.6, x , 0.8, 0.7, 0.2} 1 2 A ∩ B ={x , 0.3, 0.5, 0.9, x , 0, 0.3, 0.5}. 1 2 Proposition 2.11: For any SVNSs Aand B, then the following conditions hold: (1) 0 ⊆ A ⊆ 1 (2) A ∪ 0 = A, A ∪ 1 = 1 and A ∩ 0 = 0, A ∩ 1 = A (3) A ∪ B = B ∪ Aand A ∩ B = B ∩ A (4) A = B ⇐⇒ A ⊆ Band B ⊆ A (5) A ⊆ A ∪ B, B ⊆ A ∪ Band A ∩ B ⊆ A, A ∩ B ⊆ B (6) A ⊆ B ⇐⇒ A ∪ B = Band A ⊆ B ⇐⇒ A ∩ B = A (7) C(0) = 1 and C(1) = 0 (8) C(C(A)) = A (9) C(A ∪ B) = C(A) ∩ C(B) (10) C(A ∩ B) = C(A) ∪ C(B). Definition 2.12: Let X and Y be two nonempty sets and f : X → Y a function. (i) If B ={y, T (y), I (y), F (y): y ∈ Y} is an SVNS in Y,thenthepreimageof B under f is B B B defined by −1 −1 −1 −1 f (B) ={x, f (T )(x), f (I )(x), f (F )(x): x ∈ X}, B B B −1 −1 −1 where f (x) = y and f (T )(x) = T (f (x)), f (I )(x) = I (f (x)), f (F )(x) = F (f (x)). B B B B B B (ii) If A ={x, T (x), I (x), F (x): x ∈ X} is a SVNS in X, then the image of A under f is A A A defined by f (A) ={y, f (T )(y), f (I )(y), (1 − f (1 − F ))(y): y ∈ Y}, A A A FUZZY INFORMATION AND ENGINEERING 457 where −1 sup −1 T (x),if f (y) =∅ x∈f (y) f (T )(y) = , 0, otherwise −1 sup −1 I (x),if f (y) =∅ x∈f (y) f (I )(y) = 0, otherwise and −1 inf −1 F (x),if f (y) =∅ x∈f (y) A (1 − f (1 − F ))(y) = 0, otherwise. 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: ( τ ) 0, 1 ∈ τ ( τ ) G ∩ G ∈ τ for any G , G ∈ τ 2 1 2 1 2 ( τ ) G ∈ τ for every {G : i ∈ J}⊆ τ. 3 i i 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: Let X ={x , x , x } and 1 2 3 A ={x , 0.4, 0.2, 0.5, x , 0.5, 0.8, 0.3, x , 0.7, 0.5, 1} 1 2 3 B ={x , 0.8, 0.8, 0.3, x , 0.7, 1, 0.1, x , 0.9, 0.9, 0.7} 1 2 3 C ={x , 0.2, 0.1, 0.6, x , 0.3, 0.7, 0.4, x , 0.6, 0.3, 1} 1 2 3 D ={x , 0.6, 0.4, 0.4, x , 0.5, 1, 0.2, x , 0.8, 0.7, 0.8} 1 2 3 Then the family τ ={0, 1, A, B, C, D} of single-valued neutrosophic sets in X is SVNT on X. Definition 2.15: The complement of A of SVNOS is called a single-valued neutrosophic closed set (SVNCS for short) in X. Definition 2.16: Let (X, τ) and (Y, σ) be SVNTSs. A map f : X → Y is said to be continuous −1 −1 if f (B) is an SVNOS in X, for each SVNOS B in Y, or equivalently, f (B) is an SVNCS in X,for each SVNCS B in Y. Single-valued neutrosophic interior and closure operations in SVNTSs are defined as follows: 458 S. ÖZKAN 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 int(A) = {G: G is an SVNOS in X and G ⊆ A} cl(A) = {F : F is an SVNCS in X and A ⊆ F}. It can be also shown that int(A) is an SVNOS and cl(A) is an SVNCS in X. (1) A is SVNOS if and only if A = int(A) (2) A is SVNCS if and only if A = cl(A). Proposition 2.18: For any SVNS Ain (X, τ), we have (1) int(C(A)) = C(cl(A)) (2) cl(C(A)) = C(int(A)). Proof: Let A be an SVNS in (X, τ). (1) int(C(A)) = {G: G is an SVNOS in X and G ⊆ C(A)} = {G: G is an SVNOS in X and A ⊆ C(G)} = {C(F): C(F) is an SVNOS in X and A ⊆ F} = {C(F): F is an SVNCS in X and A ⊆ F} = C( {F : F is an SVNCS in X and A ⊆ F}) = C(cl(A)). (2) Similarly to (1). Definition 2.19 ([16]): A proximity (Efremovich proximity) space is a pair (X, δ), where X is asetand δ is a binary relation on the power set of X such that (P1) A δ B iff B δ A; (P2) A δ(B ∪ C) iff A δ B or A δ C; (P3) A δ B implies A, B =∅; (P4) A ∩ B =∅ implies A δ B; (P5) A δ B implies there is an E ⊆ X such that A δ E and (X − E) δ B, where A δ B means it is not true that A δ B. A function f: (X, δ) → (Y, δ ) between two proximity spaces is called a proximity mapping (or a p-map) if and only if f (A)δ f (B) whenever A δ B. It can easily be shown that f is a p-map −1 −1 if and only if, for subsets C and D of Y, f (C) δ f (D) whenever C δ D. FUZZY INFORMATION AND ENGINEERING 459 3. Single-Valued Neutrosophic Proximity Spaces In this section, we introduce the concept of neutrosophic proximity spaces as a generalisa- tion 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 is called a single-valued neutrosophic point (SVNP for short) in X given by t,i,f (t, i, f ) ,if x = x x ( x ) = t,i,f (0, 0, 1) ,if x = x for x ∈ X is called the support of x , where t denotes the degree of membership value, i p t,i,f denotes the degree of indeterminacy and f denotes the degree of non-membership value of x . t,i,f Theorem 3.2 ([17]): Let (X, τ) be an SVNTS, SVNS(X) denote the set of all single-valued neu- trosophic sets in X and cl : SVNS(X) → SVNS(X) the SVN closure in (X, τ). Then for any A, B ∈ SVNS(X) the following properties hold: (1) cl(0) = 0 (2) A ⊆ cl(A) (3) cl(cl(A)) = cl(A) (4) cl(A ∪ B) = cl(A) ∪ cl(B) (5) If A ⊆ B, then cl(A) ⊆ cl(B). Theorem 3.3 ([17]): Let (X, τ) be an SVNTS and the single-valued neutrosophic operator cl : SVNS(X) → SVNS(X) satisfies the properties (1) –(4) in Theorem 3.2. Then there exists a single-valued neutrosophic topology τ on X such that cl = cl. cl τ cl Definition 3.4: Let X be a nonempty set and SVNS(X) denote the set of all single-valued neutrosophic sets in X. A single-valued neutrosophic proximity space (SVNPS for short) is a pair (X, δ), where δ is a relation on SVNS(X) such that (P1) A δ B iff B δ A; (P2) A δ(B ∪ C) iff A δ B or A δ C; (P3) A δ B implies A = 0and B = 0; (P4) A ∩ B = 0 implies A δ B; (P5) A δ B implies there is an E ∈ SVNS(X) such that A δ E and C(E) δ B, where A δ B means it is not true that A δ B. Definition 3.5: Amap f: (X, δ) → (Y, δ ) between two single-valued neutrosophic proxim- ity spaces is called a single-valued neutrosophic proximity mapping (or a p-map) if and only if f (A)δ f (B) whenever A δ B. 460 S. ÖZKAN It can easily be shown that f is a p-map if and only if, for each C, D ∈ SVNS(Y), −1 −1 f (C) δ f (D) whenever C δ D. We have easily the following lemma, which follow directly from axioms (P1), (P2) and (P4) of Definition 3.4. Lemma 3.6: Let (X, δ) be a SVNPS. Then the following properties hold: (1) If A δ B, A ⊆ Cand B ⊆ D, then C δ D (2) A δ Aand A δ 1 for each A = 0 Theorem 3.7: Let (X, δ) be an SVNPS and define a map cl : SVNS(X) → SVNS(X) by cl(A) = {C(B) ∈ SVNS(X) | A δ B} for each A ∈ SVNS(X). Then the following properties hold: (1) cl(0) = 0 (2) A ⊆ cl(A) (3) cl(cl(A)) = cl(A) (4) cl(A ∪ B) = cl(A) ∪ cl(B) Proof: (1) cl(0) = 0 cl(0) = {C(B) | B δ 0}= (0, 0, 1) = 0 since 1 δ 0. (2) A ⊆ cl(A) Let A = (T , I , F ) ∈ SVNS(X).Takeany B = (T , I , F ) ∈ SVNS(X) such that A δ B. Then A A A B B B A ∩ B = 0 = (0, 0, 1) and hence min{T , T }= 0, min{I , I }= 0and max{F , F }= A B A B A B 1. So T + T ≤ 1, I + I ≤ 1and F + F ≥ 1. Thus T ≤ 1 − T , I ≤ 1 − I and A B A B A B A B A B F ≥ 1 − F . Hence C(B) = (1 − T ,1 − I ,1 − F ) ⊇ (T , I , F ) = A. Therefore A ⊆ A B B B B A A A {C(B) | A δ B}= cl(A). (3) cl(cl(A)) = cl(A) It is sufficient to show that cl(A) δ B iff A δ B by the definition of closure. If A δ B, then cl(A)δ B obviously. Conversely, suppose that A δ B and cl(A)δ B. Then there exists an E ∈ SVNS(X) such that B δ E and C(E) δ A.Since cl(A)δ B and B δ E, cl(A) E and T T or I I or F F . cl(A) E cl(A) E cl(A) E So there exists an x ∈ X such that (i) T (x)> T (x) or (ii) I (x)> I (x) or (iii) F (x)< cl(A) E cl(A) E cl(A) F (x). (i) If T (x)> T (x), we choose a ∈ [0, 1] such that T (x)< a < T (x). Define cl(A) E E cl(A) K : X −→ [0, 1] × [0, 1] × [0, 1] by (1 − a,0,1),if x = x K(x ) = (0, 0, 1),if x = x Then K ∈ SVNS(X) and K ⊆ C(E) since T (x)< T (x), I (x) ≤ I (x) and F (x) ≥ K C(E) K C(E) K F (x).If K δ A, then cl(A) ⊆ C(K) by the definition of closure and hence T (x) ≤ C(E) cl(A) T (x) = a < T (x). This is a contradiction. Thus K δ A.Since K ⊆ C(E), A δ C(E). C(K) cl(A) This is a contradiction to the fact that C(E) δ A. Hence T (x) ≤ T (x). cl(A) E FUZZY INFORMATION AND ENGINEERING 461 (ii) If I (x)> I (x), we choose b ∈ [0, 1] such that I (x)< b < I (x). Define L: X −→ cl(A) E E cl(A) [0, 1] × [0, 1] × [0, 1] by (0, 1 − b,1),if x = x L(x ) = (0, 0, 1),if x = x . Then L ∈ SVNS(X) and L ⊆ C(E) since T (x) ≤ T (x), I (x)< I (x) and F (x) ≥ L C(E) L C(E) L F (x).If L δ A, then cl(A) ⊆ C(L) by the definition of closure and hence I (x) ≤ C(E) cl(A) I (x) = b < I (x). This is a contradiction. Thus L δ A.Since L ⊆ C(E), A δ C(E).This C(L) cl(A) is a contradiction to the fact that C(E) δ A. Hence I (x) ≤ I (x). cl(A) E (iii) If F (x)< F (x), we choose c ∈ [0, 1] such that F (x)< c < F (x). Define cl(A) E cl(A) E M: X −→ [0, 1] × [0, 1] × [0, 1] by (0, 0, 1 − c),if x = x M(x ) = (0, 0, 1),if x = x . Then M ∈ SVNS(X) and M ⊆ C(E) since T (x) ≤ T (x), I (x) ≤ I (x) and F (x)> M C(E) M C(E) M F (x).If M δ A, then cl(A) ⊆ C(M) by the definition of closure and hence F (x) ≥ C(E) cl(A) F (x) = c > F (x). This is a contradiction. Thus M δ A.Since M ⊆ C(E), A δ C(E). C(M) cl(A) This is a contradiction to the fact that C(E) δ A. Hence F (x) ≥ F (x). cl(A) E (4) cl(A ∪ B) = cl(A) ∪ cl(B) Let A = (T , I , F ), B = (T , I , F ) ∈ SVNS(X).Since T ≤ T ∨ T , I ≤ I ∨ I and F ≤ A A A B B B A A B A A B A F ∨ F , A ⊆ A ∪ B. So, from Theorem 3.2 (5),wehave cl(A) ⊆ cl(A ∪ B). Similarly, cl(B) ⊆ A B cl(A ∪ B), and hence cl(A ∪ B) ⊇ cl(A) ∪ cl(B). On the other hand, suppose cl(A ∪ B) cl(A) ∪ cl(B). Then there exists an x ∈ X such that (i) T (x)> T (x) ∨ T (x) or (ii) I (x)> I (x) ∨ I (x) or cl(A∪B) cl(A) cl(B) cl(A∪B) cl(A) cl(B) (iii) F (x)< F (x) ∧ F (x). cl(A∪B) cl(A) cl(B) (i) Suppose T (x)> T (x) ∨ T (x).We may assume T (x) ≥ T (x).Let cl(A∪B) cl(A) cl(B) cl(A) cl(B) T (x) = α. Then T (x)<α and hence there exists an > 0suchthat T (x)< cl(A∪B) cl(A) cl(A) α − .Since T (x) = {1 − T (x) | C δ A},thereexistsa C ∈ SVNS(X) such that cl(A) C C δ A and 1 − T (x)<α − .Notethat1 − T (x) ≥ T (x) ≥ T (x)> T (x) − C C cl(A) cl(B) cl(B) /2, and hence T (x)< 1 − T (x) + /2. Since T (x) = {1 − T (x) | D δ B}, cl(B) C cl(B) D there exists a D ∈ SVNS(X) such that D δ B and 1 − T (x)< 1 − T (x) + /2. Since D C (C ∩ D) δ A and (C ∩ D) δ B,wehave (C ∩ D) δ(A ∪ B). So, from the definition of clo- sure, we have cl(A ∪ B) ⊆ C(C ∩ D).Also (1 − T (x)) ∨ (1 − T (x)) < 1 − T (x) + /2. C D C Hence, by Proposition 2.11(10), α = T (x) ≤ T (x) = T (x) ∨ T (x) = cl(A∪B) C(C∩D) C(C) C(D) (1 − T (x)) ∨ (1 − T (x)) < 1 − T (x) + /2 <α − + /2 = α − /2. C D C This is a contradiction. (ii) Suppose I (x)> I (x) ∨ I (x). cl(A∪B) cl(A) cl(B) We may assume I (x) ≥ I (x).Let I (x) = β. Then I (x)<β and hence cl(A) cl(B) cl(A∪B) cl(A) there exists an > 0suchthat I (x)<β − .Since I (x) = {1 − I (x) | C δ A}, cl(A) cl(A) C there exists a C ∈ SVNS(X) such that C δ A and 1 − I (x)<β − .Notethat, 1 − I (x) ≥ I (x) ≥ I (x)> I (x) − /2, and hence I (x)< 1 − I (x) + /2. C cl(A) cl(B) cl(B) cl(B) C 462 S. ÖZKAN Since I (x) = {1 − I (x) | D δ B},thereexistsa D ∈ SVNS(X) such that D δ B cl(B) D and 1 − I (x)< 1 − I (x) + /2. Since (C ∩ D) δ A and (C ∩ D) δ B,wehave (C ∩ D C D) δ(A ∪ B). So, from the definition of closure, we have cl(A ∪ B) ⊆ C(C ∩ D).Also (1 − I (x)) ∨ (1 − I (x)) < 1 − I (x) + /2. Hence β = I (x) ≤ I (x) = I (x) ∨ C D C cl(A∪B) C(C∩D) C(C) I (x) = (1 − I (x)) ∨ (1 − I (x)) < 1 − I (x) + /2 <β − + /2 = β − /2. C(D) C D C This is a contradiction. (iii) Suppose F (x)< F (x) ∧ F (x). cl(A∪B) cl(A) cl(B) We may assume F (x) ≤ F (x).Let F (x) = γ . Then F (x)>γ and cl(A) cl(B) cl(A∪B) cl(A) hence there exists an > 0suchthat F (x)>γ + .Since F (x) = {1 − cl(A) cl(A) F (x) | C δ A},thereexistsa C ∈ SVNS(X) such that C δ A and 1 − F (x)>γ + C C . Note that, 1 − F (x) ≤ F (x) ≤ F (x)< F (x) + /2, and hence F (x)> C cl(A) cl(B) cl(B) cl(B) 1 − F (x) − /2. Since F (x) = {1 − F (x) | D δ B},thereexistsa D ∈ SVNS(X) C cl(B) D such that D δ B and 1 − F (x)> 1 − F (x) − /2. Since (C ∩ D) δ A and (C ∩ D) δ B, D C we have (C ∩ D) δ(A ∪ B). So, from the definition of closure, we have cl(A ∪ B) ⊆ C(C ∩ D).Also (1 − F (x)) ∧ (1 − F (x)) > 1 − F (x) − /2. Hence γ = F (x) ≥ C D C cl(A∪B) F (x) = F (x) ∧ F (x) = (1 − F (x)) ∧ (1 − F (x)) > 1 − F (x) − /2 >γ C(C∩D) C(C) C(D) C D C + − /2 = γ + /2. This is a contradiction. Theorem 3.8: For an SVNPS (X, δ), the family τ(δ) ={A ∈ SVNS(X) | cl(C(A)) = C(A)} is an SVN topology on X. This topology is called the SVN topology on X induced by the SVN proximity δ. Proof: It follows from Theorems 3.3 and 3.7. Theorem 3.9: Let (X, δ ) and (Y, δ ) be two single-valued neutrosophic proximity spaces. An 1 2 SVN proximity mapping f : (X, δ ) → (Y, δ ) is continuous with respect to the SVN topologies 1 2 τ(δ ) and τ(δ ). 1 2 −1 −1 Proof: Let A ∈ τ(δ ). Then cl(C(A)) = C(A). We will show that cl(C(f (A))) = C(f (A)). −1 −1 Clearly C(f (A)) ⊆ cl(C(f (A))). −1 −1 −1 Conversely, let B δ C(A).Since f is a p-map, f (B) δ f (C(A)) = C(f (A)).So 2 1 −1 −1 −1 cl(C(f (A))) = {C(F) | F δ C(f (A))}⊆ C(f (B)). −1 −1 Hence for any B δ C(A), cl(C(f (A))) ⊆ C(f (B)).Thuswehave −1 −1 cl(C(f (A))) ⊆ {C(f (B)) | B δ C(A)} −1 = {f (C(B)) | B δ C(A)} −1 = f ( {C(B) | B δ C(A)}) −1 −1 = f (cl(C(A))) = f (C(A)) −1 = C(f (A)). FUZZY INFORMATION AND ENGINEERING 463 −1 −1 −1 So cl(C(f (A))) = C(f (A)). Hence f (A) is open. Therefore, f : (X, τ(δ )) → (Y, τ(δ )) is 1 2 continuous. 4. Initial Structures and Products We prove the existences of initial single-valued neutrosophic proximity space. Then we define the product of SVNPSs. Theorem 4.1: Let X be a set, {(X , δ ) | α ∈ } be a family of single-valued neutrosophic α α proximity spaces, and for each α ∈ ,letf : X → X be a p-map. For any A, B ∈ SVNS(X), α α define A δ B iff for every finite families {A | i = 1, ... , n} and {B | j = 1, ... , m} where A = A i j i i=1 and B = B (i.e. finite covers of Aand B respectively), there exist an A and a B such that j i j j=1 f (A )δ f (B ) for each α ∈ . α i α α j Then δ is the coarsest (initial) proximity structure of single-valued neutrosophic spaces on X for which all mappings f : (X, δ) → (X , δ ) (α ∈ )are p-map. α α α Proof: We first prove that δ is an SVN proximity on X. (P1) A δ B iff B δ A Since δ is an SVN proximity structure for each α ∈ ,itisclear that A δ B iff B δ A. (P2) A δ(B ∪ C) iff A δ B or A δ C If A δ B, then A δ D for each D ⊇ B. Because every cover of D is a cover of B. Therefore, A δ B or A δ C implies A δ(B ∪ C). Conversely, suppose A δ B and A δ C. Then, there exist finite covers {A | i = 1, ... , n} and {B | j = 1, ... , m} of A and B respectively such that f (A ) δ f (B ) for some j α i α α j α = s ∈ , where i = 1, ... , n and j = 1, ... , m. Likewise, there are finite covers ij {A | k = 1, ... , p} and {B | j = m + 1, ... , m + q} of A and C respectively such that k j f (A ) δ f (B ) for some α = t ∈ , where k = 1, ... , p and j = m + 1, ... , m + q. α k α α j kj Then, {A ∪ A | i = 1, ... , n; k = 1, ... , p} and {B | j = 1, ... , m + q} are finite covers i k j of A and B ∪ C, respectively. Hence, from the fact that f (A ∪ A ) δ f (B ) for α = s α i k α α j ij or α = t , we conclude that A δ(B ∪ C). kj (P3) A δ B implies A = 0and B = 0 It is obvious. (P4) A ∩ B = 0 implies A δ B We will show that if A δ B, then A ∩ B = 0. Suppose A δ B. Then, there exist finite covers {A | i = 1, ... , n} and {B | j = 1, ... , m} of A and B respectively such that i j f (A ) δ f (B ) for some α = s ∈ , where i = 1, ... , n and j = 1, ... , m. Since for α i α α j ij each α ∈ , δ is an SVN proximity structure on X , f (A ) ∩ f (B ) = 0. From this, it α α α i α j follows that n m f ( A ) ∩ f ( B ) = f (A) ∩ f (B) = 0. α i α j α α i=1 j=1 So we have A ∩ B = 0. (P5) A δ B implies there is an E ∈ SVNS(X) such that A δ E and C(E) δ B. 464 S. ÖZKAN If A δ B, then there exist finite covers {A | i = 1, ... , n} and {B | j = 1, ... , m} of A i j and B respectively such that f (A ) δ f (B ) for some α = s ∈ , where i = 1, ... , n and α i α α j ij j = 1, ... , m. Since each (X , δ ) is a single-valued neutrosophic proximity space, there α α m n −1 exist E such that f (A ) δ E and C(E ) δ f (B ).Set E = f (E ) and E = E , ij α i α ij ij α α j j ij j i=1 α j=1 n m m m −1 −1 −1 i.e. E = f (E ). It follows that f (E ) = f ( f (E )) ⊂ f (f (E )) ⊂ ij α j α ij α ij j=1 i=1 α i=1 α i=1 α E ⊂ E .Since f (E ) ⊂ E ,wehave f (A ) δ f (E ) for α = s ∈ ; that is, A δ E. ij ij α j ij α i α α j ij i=1 −1 −1 m n Let D = C(f (E )) = f (C(E )) and F = C(E ) = D . Then C(E) = F , ij ij ij j j ij j α α i=1 j=1 n m −1 −1 i.e. C(E) = C(f (E )).Since C(E ) δ f (B ) and D = f (C(E )),wehave ij ij α α j ij ij j=1 i=1 α α −1 −1 f (C(E )) δ f (f (B )), i.e. by P2 of Definition 3.4, D δ B for all i and j. This implies F δ B ij α j ij j j j α α for all j. Hence C(E) δ B for all j, showing that C(E) δ B. It is clear that all mappings f : (X, δ) → (X , δ ) (α ∈ )are p-map. Let δ be another α α α SVN proximity on X with respect to which each f 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 {A | i = 1, ... , n} and {B | j = 1, ... , m} of A and B respectively. Since (A ∪ ··· ∪ A )δ B,by P2of j 1 n Definition 3.4, there is an i ∈{1, ... , n} such that A δ B. Similarly, A δ (B ∪ ··· ∪ B ),by i i 1 m P2 of Definition 3.4, there is an j ∈{1, ... , m} such that A δ B . Since each f is a p-map with i j α respect to δ , it follows that f (A )δ f (B ) for each α ∈ . Hence, we get A δ B, i.e. δ is finer α i α α j than δ. Definition 4.2: Let {(X , δ ) | α ∈ } be a family of single-valued neutrosophic proximity α α spaces, and X = X . The product SVN proximity on X is defined to be the initial prox- α∈ imity structure δ = δ on X with respect to which each projection map P : (X, δ) → α α α∈ (X , δ ) (α ∈ )isa p-map. In that case (X, δ) is said to be the product SVNPS. α α ∗ ∗ Corollary 4.3: A mapping f from an SVNPS (Y, δ ) to (X = X , δ), i.e. f : (Y, δ ) → α∈ (X, δ),isa p-map if and only if the composition f ◦ f : (Y, δ ) → (X , δ ) is a p-map for every α α α α ∈ . ∗ ∗ 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 ◦ f is a p-map. Conversely, suppose that f ◦ f is a p-map for each α ∈ . We will show that f is a p- map. Let A, B ⊂ X, A δ B and {A | i = 1, ... , n} and {B | j = 1, ... , m} be finite covers of i j n m n −1 f (A) and f (B) respectively. Then A = A , B = B and we have A ⊆ f (A ), i j i i=1 j=1 i=1 m n m −1 ∗ −1 ∗ −1 B ⊆ f (B ).Since A δ B,weobtain f (A )δ f (B ) and by P2of j i j j=1 i=1 j=1 −1 ∗ −1 −1 Definition 3.4, there exist i, j such that f (A )δ f (B ).Since f ◦ f ◦ f (A ) ⊆ f (A ), i j α i α i −1 f ◦ f ◦ f (B ) ⊆ f (B ) and f ◦ f is a p-map for each α ∈ , it follows that f (A )δ f (B ) α j α j α α i α α j for each α ∈ .Thisprovesthat f (A)δ f (B) so that f is a p-map. 5. Conclusion Proximity and uniformity are important concepts close to topology and they have rich topo- logical 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, FUZZY INFORMATION AND ENGINEERING 465 we showed that each single-valued neutrosophic proximity determines a single-valued neutrosophic topology. Also, we introduced the initial single-valued neutrosophic prox- imity 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 applica- tions 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 relation- ships among the notions of single-valued uniform, proximity and topological spaces can be investigated. Also, various topological notions such as separation, closedness, connect- edness and compactness may be characterised in the single-valued topological spaces. Furthermore, in [20–22], 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. Acknowledgments I would like to thank the referees for their valuable and helpful suggestions that improved the paper. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on contributor Dr S. Özkan is an Assistant Professor in the Department of Mathematics, Nevşehir Hacı Bektaş Veli Uni- versity, Turkey. He received his BSc degree in Mathematics from Yıldız Technical University, İstanbul and his MSc and PhD degrees from the Graduate School of Natural and Applied Sciences, Erciyes Uni- versity, Kayseri. His research areas are General Topology, Category Theory, Proximity Structures and Generalizations, and Ordered Structures. ORCID Samed Özkan http://orcid.org/0000-0003-3063-6168 References [1] Smarandache F. Neutrosophy. Neutrosophic Probability. Rehoboth, USA: Set and Logic. Amer. Res. Press; 1998. [2] Zadeh L. Fuzzy sets. Inform Control. 1965;8:338–353. [3] Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986;20:87–96. [4] Salama AA, Alblowi SA. Neutrosophic set and neutrosophic topological spaces. IOSR J Math. 2012;3(4):31–35. [5] Wang H, Smarandache F, Zhang Y, et al. Single valued neutrosophic sets. Multispace Multistruct Neutr Trans. 2010;4:410–413. [6] Ansari AQ, Biswas R, Aggarwal S. Proposal for applicability of neutrosophic set theory in medical AI. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Oct 2, 2018
Keywords: Single-valued neutrosophic; proximity space; initial structure; product; 54E05; 54B10
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