aa r X i v : . [ m a t h . G M ] J un Neutrosophic Metric Spaces
Murat Kiri¸sci* and Necip S¸im¸sek a. Department of Mathematical Education, Hasan Ali Y¨ucel Education Faculty,Istanbul University-Cerrahpa¸sa, Vefa, 34470, Fatih, Istanbul, Turkeye-mail: [email protected]. Department of mathematics, Faculty of Arts and Sciences,Istanbul Commerce University, Istanbul, Turkeye-mail: [email protected]
Abstract:
In present paper, the definition of new metric space with neutrosophic num-bers is given. Several topological and structural properties have been investigated. Theanalogues of Baire Category Theorem and Uniform Convergence Theorem are given forNeutrosophic metric spaces.
Subject Classification:
Primary 03E72; Secondary 54E35, 54A40, 46S40.
Keywords:
Neutrosophic metric space, Baire Category Theorem, Uniform ConvergenceTheorem, nowhere dense, completeness, Hausdorffness.1.
Introduction
Fuzzy Sets (FSs) put forward by Zadeh [21] has influenced deeply all the scientific fieldssince the publication of the paper. It is seen that this concept, which is very important forreal-life situations, had not enough solution to some problems in time. New quests for suchproblems have been coming up. Atanassov [1] initiated Intuitionistic fuzzy sets (IFSs) forsuch cases. Neutrosophic set (NS) is a new version of the idea of the classical set which isdefined by Smarandache [15]. Examples of other generalizations are FS [21] interval-valuedFS [17], IFS [1], interval-valued IFS [2], the sets paraconsistent, dialetheist, paradoxist, andtautological [16], Pythagorean fuzzy sets [19] .Using the concepts Probabilistic metric space and fuzzy, fuzzy metric space (FMS) isintroduced in [11]. Kaleva and Seikkala [7] have defined the FMS as a distance between twopoints to be a non-negative fuzzy number. In [5] some basic properties of FMS studied andthe Baire Category Theorem for FMS proved. Further, some properties such as separability,countability are given and Uniform Limit Theorem is proved in [6]. Afterward, FMS hasused in the applied sciences such as fixed point theory, image and signal processing, medicalimaging, decision-making et al. After defined of the IFS, it was used in all areas where FStheory was studied. Park [13] defined IF metric space (IFMS), which is a generalization ofFMSs. Park used George and Veeramani’s [5] idea of applying t-norm and t-conorm to theFMS meanwhile defining IFMS and studying its basic features.Bera and Mahapatra defined the neutrosophic soft linear spaces (NSLSs) [3]. Later, neu-trosophic soft normed linear spaces(NSNLS) has been defined by Bera and Mahapatra [4]. In[4], neutrosophic norm, Cauchy sequence in NSNLS, convexity of NSNLS, metric in NSNLSwere studied.In present study, from the idea of neutrosophic sets, new metric space was defined whichis called Neutrosophic metric Spaces (NMS). We investigate some properties of NMS such
Murat Kiri¸sci and Necip S¸im¸sek as open set, Hausdorff, neutrosophic bounded, compactness, completeness, nowhere dense.Also we give Baire Category Theorem and Uniform Convergence Theorem for NMSs.2.
Preliminaries
Some definitions related to the fuzziness, intuitionistic fuzziness and neutrosophy aregiven as follows:The fuzzy subset F of R is said to be a fuzzy number(FN). The FN is a mapping F : R → [0 ,
1] that corresponds to each real number a to the degree of membership F ( a ).Let F is a FN. Then, it is known that [8] • If F ( a ) = 1, for a ∈ R , F is said to be normal, • If for each µ > F − { [0 , τ + µ ) } is open in the usual topology ∀ τ ∈ [0 , F is saidto be upper semi continuous, , • The set [ F ] τ = { a ∈ R : F ( a ) ≥ τ } , τ ∈ [0 ,
1] is called τ − cuts of F .Choose non-empty set F . An IFS in F is an object U defined by U = { < a, G U ( a ) , Y U ( a ) > : a ∈ F } where G U ( a ) : F → [0 ,
1] and Y U ( a ) : F → [0 ,
1] are functions for all a ∈ F such that0 ≤ G U ( a ) + Y U ( a ) ≤ U be an IFN. Then, • an IF subset of the R , • If G U ( a ) = 1 and, Y U ( a ) = 0 for a ∈ R , normal, • If G U ( λa + (1 − λ ) a ) ≥ min( G U ( a ) , G U ( a )), ∀ a , a ∈ R and λ ∈ [0 , G U ( a ) is called convex, • If Y U ( λa + (1 − λ ) a ) ≥ min( Y U ( a ) , Y U ( a )), ∀ a , a ∈ R and λ ∈ [0 , Y U ( a ) is concav, • G U is upper semi continuous and Y U is lower semi continuous • suppU = cl ( { a ∈ F : Y U ( a ) < } ) is bounded.An IFS U = { < a, G U ( a ) , Y U ( a ) > : a ∈ F } such that G U ( a ) and 1 − Y U ( a ) are FNs,where (1 − Y U )( a ) = 1 − Y U ( a ), and G U ( a ) + Y U ( a ) ≤ F is a space of points(objects). Denote the G U ( a ) is a truth-MF, B U ( a ) is an indeterminacy-MF and Y U ( a ) is a falsity-MF, where U is a set in F with a ∈ F .Then, if we take I =]0 − , + [ G U ( a ) : F → I,B U ( a ) : F → I,Y U ( a ) : F → I, There is no restriction on the sum of G U ( a ), B U ( a ) and Y U ( a ). Therefore,0 − ≤ sup G U ( a ) + sup B U ( a ) + sup Y U ( a ) ≤ + . The set U which consist of with G U ( a ), B U ( a ) and Y U ( a ) in F is called a neutrosophicsets(NS) and can be denoted by U = { < a, ( G U ( a ) , B U ( a ) , Y U ( a )) > : a ∈ F, G U ( a ) , B U ( a ) , Y U ( a ) ∈ ]0 − , + [ } (1)Clearly, NS is an enhancement of [0 ,
1] of IFSs.An NS U is included in another NS V , ( U ⊆ V ), if and only if,inf G U ( a ) ≤ inf G V ( a ) , sup G U ( a ) ≤ sup G V ( a ) , inf B U ( a ) ≥ inf B V ( a ) , sup B U ( a ) ≥ sup B V ( a ) , inf Y U ( a ) ≥ inf Y V ( a ) , sup Y U ( a ) ≥ sup Y V ( a ) . eutrosophic Metric Spaces 3 for any a ∈ F . However, NSs are inconvenient to practice in real problems. To cope withthis inconvenient situation, Wang et al [18] customized NS’s definition and single-valued NSs(SVNSs) suggested.To cope with this inconvenient situation, Wang et al [18] customized NS’s definition andsingle-valued NSs (SVNSs) suggested. Ye [20], described the notion of simplified NSs(SNSs),which may be characterized by three real numbers in the [0 , U , in (1), is U = { < a, ( G U ( a ) , B U ( a ) , Y U ( a )) > : a ∈ F } , which called an SNS. Especially, if F has only one element < G U ( a ) , B U ( a ) , Y U ( a ) > is saidto be an SNN. Expressly, we may see SNSs as a subclass of NSs.An SNS U is comprised in another SNS V ( U ⊆ V ), iff G U ( a ) ≤ G V ( a ), B U ( a ) ≥ B V ( a )and Y U ( a ) ≥ Y V ( a ) for any a ∈ F . Then, the following operations are given by Ye[20]: U + V = h G U ( a ) + G V ( a ) − G U ( a ) .G V ( a ) , B U ( a ) + B V ( a ) − B U ( a ) .B V ( a ) , Y U ( a ) + Y V ( a ) − Y U ( a ) .Y V ( a ) i ,U.V = h G U ( a ) .G V ( a ) , B U ( a ) .B V ( a ) , Y U ( a ) .Y V ( a ) i ,α.U = h − (1 − G U ( a )) α , − (1 − B U ( a )) α , − (1 − Y U ( a )) α i f or α > ,U α = h G αU ( a ) , B αU ( a ) , Y αU ( a ) i f or α > . Triangular norms (t-norms) (TN) were initiated by Menger [12]. In the problem of com-puting the distance between two elements in space, Menger offered using probability distri-butions instead of using numbers for distance. TNs are used to generalize with the prob-ability distribution of triangle inequality in metric space conditions. Triangular conorms(t-conorms) (TC) know as dual operations of TNs. TNs and TCs are very significant forfuzzy operations(intersections and unions).
Definition 2.1.
Give an operation ◦ : [0 , × [0 , → [0 , . If the operation ◦ is satisfyingthe following conditions, then it is called that the operation ◦ is continuous TN : For s, t, u, v ∈ [0 , , i. s ◦ s ii. If s ≤ u and t ≤ v , then s ◦ t ≤ u ◦ v , iii. ◦ is continuous, iv. ◦ is commutative and associative. Definition 2.2.
Give an operation • : [0 , × [0 , → [0 , . If the operation • is satisfyingthe following conditions, then it is called that the operation • is continuous TC : i. s • s , ii. If s ≤ u and t ≤ v , then s • t ≤ u • v , iii. • is continuous, iv. • is commutative and associative. Form above definitions, we note that if we choose 0 < ε , ε < ε > ε , then thereexist 0 < ε , ε < , ε ◦ ε ≥ ε , ε ≥ ε • ε . Further, if we choose ε ∈ (0 , ε , ε ∈ (0 ,
1) such that ε ◦ ε ≥ ε and ε • ε ≤ ε .3. Neutrosophic Metric Spaces
Definition 3.1.
Take F be an arbitrary set, N = { < a, G ( a ) , B ( a ) , Y ( a ) > : a ∈ F } be aNS such that N : F × F × R + → [0 , . Let ◦ and • show the continuous TN and continuous Murat Kiri¸sci and Necip S¸im¸sek
TC, respectively. The four-tuple ( F, N , ◦ , • ) is called neutrosophic metric space(NMS) whenthe following conditions are satisfied. ∀ a, b, c ∈ F , i. 0 ≤ G ( a, b, λ ) ≤ , ≤ B ( a, b, λ ) ≤ , ≤ Y ( a, b, λ ) ≤ ∀ λ ∈ R + , ii. G ( a, b, λ ) + B ( a, b, λ ) + Y ( a, b, λ ) ≤ , (for λ ∈ R + ), iii. G ( a, b, λ ) = 1 (for λ > ) if and only if a = b , iv. G ( a, b, λ ) = G ( b, a, λ ) (for λ > ), v. G ( a, b, λ ) ◦ G ( b, c, µ ) ≤ G ( a, c, λ + µ ) ( ∀ λ, µ > , vi. G ( a, b, . ) : [0 , ∞ ) → [0 , is continuous, vii. lim λ →∞ G ( a, b, λ ) = 1 ( ∀ λ > , viii. B ( a, b, λ ) = 0 (for λ > ) if and only if a = b , ix. B ( a, b, λ ) = B ( b, a, λ ) (for λ > ), x. B ( a, b, λ ) • B ( b, c, µ ) ≥ B ( a, c, λ + µ ) ( ∀ λ, µ > , xi. B ( a, b, . ) : [0 , ∞ ) → [0 , is continuous, xii. lim λ →∞ B ( a, b, λ ) = 0 ( ∀ λ > , xiii. Y ( a, b, λ ) = 0 (for λ > ) if and only if a = b , xiv. Y ( a, b, λ ) = Y ( b, a, λ ) ( ∀ λ > , xv. Y ( a, b, λ ) • Y ( b, c, µ ) ≥ Y ( a, c, λ + µ ) ( ∀ λ, µ > , xvi. Y ( a, b, . ) : [0 , ∞ ) → [0 , is continuous, xvii. lim λ →∞ Y ( a, b, λ ) = 0 (for λ > ), xviii. If λ ≤ , then G ( a, b, λ ) = 0 , B ( a, b, λ ) = 1 and Y ( a, b, λ ) = 1 .Then N = ( G, B, Y ) is called Neutrosophic metric(NM) on F . The functions G ( a, b, λ ) , B ( a, b, λ ) , Y ( a, b, λ ) denote the degree of nearness, the degree ofneutralness and the degree of non-nearness between a and b with respect to λ , respectively. Example 3.2.
Let ( F, d ) be a MS. Give the operations ◦ and • as default (min) TN a ◦ b = min { a, b } and default(max) TC a • b = max { a, b } . G ( a, b, λ ) = λλ + d ( a, b ) , B ( a, b, λ ) = d ( a, b ) λ + d ( a, b ) Y ( a, b, λ ) = d ( a, b ) λ , ∀ a, b ∈ F and λ > . Then, ( F, N , ◦ , • ) is NMS such that N : F × F × R + → [0 , . ThisNMS is expressed as produced by a metric d the NM. Example 3.3.
Choose F as natural numbers set. Give the operations ◦ and • as TN a ◦ b = max { , a + b − } and TC a • b = a + b − ab . ∀ a, b ∈ F , λ > G ( a, b, λ ) = (cid:26) ab , ( a ≤ b ) , ba , ( b ≤ a ) ,B ( a, b, λ ) = (cid:26) b − ay , ( ax ≤ b ) , a − bx , ( b ≤ a ) ,Y ( a, b, λ ) = (cid:26) b − a , ( a ≤ b ) ,a − b , ( b ≤ a ) , Then, ( F, N , ◦ , • ) is NMS such that N : F × F × R + → [0 , . Remark. N = { < a, G ( a ) , B ( a ) , Y ( a ) > : a ∈ F } defined in Example 3.2 is not a NM withTN a ◦ b = max { , a + b − } and TC a • b = a + b − ab .It can also be said that N = { < a, G ( a ) , B ( a ) , Y ( a ) > : a ∈ F } defined in Example 3.3 isnot a NM with TN a ◦ b = min { a, b } and TC a • b = max { a, b } . eutrosophic Metric Spaces 5 Definition 3.4.
Give ( F, N , ◦ , • ) be a NMS, < ε < , λ > and a ∈ F . The set O ( a, ε, λ ) = { b ∈ F : G ( a, b, λ ) > − ε, B ( a, b, λ ) < ε, Y ( a, b, λ ) < ε } is said to be theopen ball (OB) (center a and radius ε with respect to λ ). Theorem 3.5.
Every OB O ( a, ε, λ ) is an open set (OS).Proof. Take O ( a, ε, λ ) be an OB (center a , radius ε ). Choose b ∈ O ( a, ε, λ ). Therefore, G ( a, b, λ ) > − ε, B ( a, b, λ ) < ε, Y ( a, b, λ ) < ε . There exists λ ∈ (0 , λ ) such that G ( a, b, λ ) > − ε, B ( a, b, λ ) < ε, Y ( a, b, λ ) < ε because of G ( a, b, λ ) > − ε . If wetake ε = G ( a, b, λ ), then for ε > − ε , ζ ∈ (0 ,
1) will exist such that ε > − ζ > − ε .Give ε and ζ such that ε > − ζ . Then, ε , ε , ε ∈ (0 ,
1) will exist such that ε ◦ ε > − ζ ,(1 − ε ) • (1 − ε ) ≤ ζ and (1 − ε ) • (1 − ε ) ≤ ζ . Choose ε = max { ε , ε , ε } . Considerthe OB O ( b, − ε , λ − λ ). We will show that O ( b, − ε , λ − λ ) ⊂ O ( a, ε, λ ). If we take c ∈ O ( b, − ε , λ − λ ), then G ( b, c, λ − λ ) > ε , B ( b, c, λ − λ ) < ε and Y ( b, c, λ − λ ) < ε .Then, G ( a, c, λ ) ≥ G ( a, b, λ ) ◦ G ( b, c, λ − λ ) ≥ ε ◦ ε ≥ ε ◦ ε ≥ − ζ > − ε,B ( a, c, λ ) ≤ B ( a, b, λ ) • B ( b, c, λ − λ ) ≤ (1 − ε ) • (1 − ε ) ≤ (1 − ε ) • (1 − ε ) ≤ ζ < ε,Y ( a, c, λ ) ≤ Y ( a, b, λ ) • Y ( b, c, λ − λ ) ≤ (1 − ε ) • (1 − ε ) ≤ (1 − ε ) • (1 − ε ) ≤ ζ < ε It shows that c ∈ O ( a, ε, λ ) and O ( b, − ε , λ − λ ) ⊂ O ( a, ε, λ ). (cid:3) Remark.
From the Definition 3.4 and Theorem 3.5, we can say that τ N = { A ⊂ F : there exist λ > and ε ∈ (0 , such that O ( a, b, λ ) ⊂ A, for each a ∈ A } is atopology on F . In that case, every NM N on F produces a topology τ N on F which has abase the family of OSs of { O ( a, ε, λ ) : a ∈ F, ε ∈ (0 , , λ > } . This can be proved in asimilar to the proof of Theorem 28 in [10] . Theorem 3.6.
Every NMS is Hausdorff.Proof.
Let ( F, N , ◦ , • ) be a NMS. Choose a and b as two distinct points in F . Hence,0 < G ( a, b, λ ) <
1, 0 < B ( a, b, λ ) <
1, 0 < Y ( a, b, λ ) <
1. Take ε = G ( a, b, λ ), ε = B ( a, b, λ ), ε = Y ( a, b, λ ) and ε = max { ε , − ε , − ε } . If we take ε ∈ ( ε, ε , ε , ε such that ε ◦ ε ≥ ε , (1 − ε ) • (1 − ε ) ≤ − ε and (1 − ε ) • (1 − ε ) ≤ − ε . Let ε = max { ε , ε , ε } . If we consider the OBs O ( a, − ε , λ ) and O ( b, − ε , λ ), then clearly O ( a, − ε , λ ) T O ( b, − ε , λ ) = ∅ . From here, if we choose c ∈ O ( a, − ε , λ ) T O ( b, − ε , λ ), then ε = G ( a, b, λ ) ≥ G ( a, c, λ ◦ G ( c, b, λ ≥ ε ◦ ε ≥ ε ◦ ε ≥ ε > ε ,ε = B ( a, b, λ ) ≤ B ( a, c, λ • B ( c, b, λ ≤ (1 − ε ) • (1 − ε ) ≤ (1 − ε ) • (1 − ε ) ≤ − ε < ε , and ε = Y ( a, b, λ ) ≤ Y ( a, c, λ • Y ( c, b, λ ≤ (1 − ε ) • (1 − ε ) ≤ (1 − ε ) • (1 − ε ) ≤ − ε < ε , which is a contradiction. Therefore, we say that NMS is Hausdorff. (cid:3) Definition 3.7.
Let ( F, N , ◦ , • ) be a NMS. A subset A of F is called Neutrosophic-bounded (NB), if there exist λ > and ε ∈ (0 , such that G ( a, b, λ ) > − ε , B ( a, b, λ ) < ε and Y ( a, b, λ ) < ε ( ∀ a, b ∈ A ). Definition 3.8. If A ⊆ ∪ U ∈C N U , a collection C N of OSs is said to be an open cover(OC)of A . A subspace A of a NMS is compact, if every OC of A has a finite subcover. Murat Kiri¸sci and Necip S¸im¸sek
If every sequence in A has a convergent subsequence to a point in A , then it is calledsequential compact. Theorem 3.9.
Every compact subset A of a NMS is NB.Proof. Firstly, choose a compact subset A of NMS F . Consider the OC { O ( a, ε, λ ) : a ∈ A } for λ > ε ∈ (0 , A is compact, then there exist a , a , . . . , a n ∈ A such that A ⊆ ∪ nk =1 O ( a k , ε, λ ). For some k, m and a, b ∈ A , a ∈ O ( a k , ε, λ ) and b ∈ O ( a m , ε, λ ). Thenwe can write, G ( a, a k , λ ) > − ε , B ( a, a k , λ ) < ε , Y ( a, a k , λ ) < ε and G ( b, a m , λ ) > − ε , B ( b, a m , λ ) < ε , Y ( b, a m , λ ) < ε . Let ρ = min { G ( a k , a m , λ ) : 1 ≤ k, m ≤ n } , σ = max { B ( a k , a m , λ ) : 1 ≤ k, m ≤ n } and ϕ = max { Y ( a k , a m , λ ) : 1 ≤ k, m ≤ n } . Then, ρ, σ, ϕ >
0. From here, for 0 < ζ , ζ , ζ < G ( a, b, λ ) ≥ G ( a, a k , λ ) ◦ G ( a k , a m , λ ) ◦ G ( a m , b, λ ) ≥ (1 − ε ) ◦ (1 − ε ) ◦ ρ > − ζ ,B ( a, b, λ ) ≤ B ( a, a k , λ ) • B ( a k , a m , λ ) • B ( a m , b, λ ) ≤ ε • ε • σ < ζ ,Y ( a, b, λ ) ≤ Y ( a, a k , λ ) • Y ( a k , a m , λ ) • Y ( a m , b, λ ) ≤ ε • ε • ϕ < ζ . If we take ζ = max { ζ , ζ , ζ } and λ = 3 λ , we have G ( a, b, λ ) > − ζ , B ( a, b, λ ) < ζ and Y ( a, b, λ ) < ζ for all a, b ∈ A . This result leads us to the conclusion that the set A isNB. (cid:3) If (
F X, N , ◦ , • ) is NMS produces by a metric d on X and A ⊂ F , then A is NB if andonly if it is bounded. Consequently, with Theorems 3.6 and 3.9, we can write: Corollary 3.10.
In a NMS, every compact set is closed and bounded.
Theorem 3.11.
Take ( F, N , ◦ , • ) is FMS and τ N be the topology on F produced bythe FM. Then for a sequence ( a n ) in F , the sequence a n is convergent to a if and only if G ( a n , a, λ ) → , B ( a n , a, λ ) → and Y ( a n , a, λ ) → as n → ∞ .Proof. Take λ >
0. Assume that a n → a . If 0 < ε <
1, then there exist N ∈ N suchthat a n ∈ O ( a, ε, λ ), ( ∀ n ≥ N ). Therefore, 1 − G ( a n , a, λ ) < ε , B ( a n , a, λ ) < ε and Y ( a n , a, λ ) < ε . In that case, we can write G ( a n , a, λ ) → B ( a n , a, λ ) → Y ( a n , a, λ ) → n → ∞ .Conversely, G ( a n , a, λ ) → B ( a n , a, λ ) → Y ( a n , a, λ ) → n → ∞ , for each λ >
0. Then, for 0 < ε <
1, there exist N ∈ N such that 1 − G ( a n , a, λ ) < ε , B ( a n , a, λ ) < ε and Y ( a n , a, λ ) < ε ∀ N ∈ N . Then, G ( a n , a, λ ) > − ε , B ( a n , a, λ ) < ε and Y ( a n , a, λ ) < ε , ∀ N ∈ N . Then, a n ∈ O ( a, ε, λ ) ∀ n ≥ N . This is the desired result. (cid:3) Definition 3.12.
Take ( F, N , ◦ , • ) to be a NMS. A sequence ( a n ) in F is called Cauchy if for each ε > and each λ > , there exist N ∈ N such that G ( a n , a m , λ ) > − ε , B ( a n , a m , λ ) < ε , Y ( a n , a m , λ ) < ε for all n, m ≥ N . ( F, N , ◦ , • ) is called complete ifevery Cauchy sequence is convergent with respect to τ N . Theorem 3.13.
Take ( F, N , ◦ , • ) to be a NMS. Let’s every Cauchy sequence in F has aconvergent subsequences. Then the NMS ( F, N , ◦ , • ) is complete.Proof. Let the sequence ( a n ) be a Cauchy and let ( a i n ) be a subsequence of ( a n ) and a n → a . Let λ > µ ∈ (0 , < ε < − ε ) ◦ (1 − ε ) ≥ − µ , ε • ε ≤ µ . It is known that the sequence ( a n ) is Cauchy. Then there is N ∈ N such that G ( a m , a n , λ ) > − ε , B ( a m , a n , λ ) < ε and Y ( a m , a n , λ ) < ε ∀ m, n ∈ N . Since a n i → a ,there is positive integer i p such that i p > N , G ( a i p , a, λ ) > − ε , B ( a i p , a, λ ) < ε and eutrosophic Metric Spaces 7 Y ( a i p , a, λ ) < ε . Therefore, if n ≥ N , G ( a n , a, λ ) ≥ G ( a n , a i p , λ ◦ G ( a i p , a, λ > (1 − ε ) ◦ (1 − ε ) ≥ − µ,B ( a n , a, λ ) ≤ B ( a n , a i p , λ • B ( a i p ,a , λ < ε • ε ≤ µ,Y ( a n , a, λ ) ≤ Y ( a n , a i p , λ • Y ( a i p ,a , λ < ε • ε ≤ µ. Thus, we have a n → a . This is the desired result. (cid:3) Theorem 3.14.
Let ( F, N , ◦ , • ) is NMS and let A be a subset of F with the subspace NM ( G A , B A , Y A ) = ( G | A × (0 , ∞ ) , B | A × (0 , ∞ ) , Y | A × (0 , ∞ ) ) . Then ( A, N A , ◦ , • ) is complete if andonly if A is closed subset of F .Proof. Assume that A is a closed subset of F . Choose the sequence ( a n ) be a Cauchy in( A, N A , ◦ , • ). Since ( a n ) is a Cauchy in F , then there is a point a in F such that a n → a .From here, a ∈ A = A and so ( a n ) converges to A .Contrarily, consider the ( A, N A , ◦ , • ) is complete. Further, assume that A is not closed.Choose a ∈ A/A . Therefore, there exist a sequence ( a n ) of points in A that converges to a and so ( a n ) is a Cauchy. Hence, for n, m ≥ N , each 0 < µ <
1, each λ >
0, there is N ∈ N such that G ( a n , a m , λ ) > − µ , B ( a n , a m , λ ) < µ and Y ( a n , a m , λ ) < µ . Now, wecan write G ( a n , a m , λ ) = G A ( a n , a m , λ ), B ( a n , a m , λ ) = B A ( a n , a m , λ ) and Y ( a n , a m , λ ) = Y A ( a n , a m , λ ) because of the sequence ( a n ) is in A . Therefore ( a n ) is a Cauchy in A . Sincewe know that ( F, N , ◦ , • ) is complete, then there is a b ∈ A such that a n → b . Hence, thereis N ∈ N such that G A ( b, a n , λ ) > − µ , B A ( b, a n , λ ) < µ and Y A ( b, a n , λ ) < µ for n ≥ N ,each 0 < µ < λ >
0. Since the sequence ( a n ) is in A and b ∈ A , we can write G ( b, a n , λ ) = G A ( b, a n , λ ), B ( b, a n , λ ) = B A ( b, a n , λ ) and Y ( b, a n , λ ) = Y A ( b, a n , λ ). Thisgives us the conclusion that the sequence ( a n ) converges to both a and b in ( F, N , ◦ , • ).Since a A and b ∈ A , we have a = b . This is a contradiction and thus the desired resultis achieved. (cid:3) In proof of Lemma 3.15 and Theorem 3.16, used similar proof techniques of Propositions4.3 and 4.4 in [9].
Lemma 3.15.
Let ( F, N , ◦ , • ) is NMS. If λ > and ε , ε ∈ (0 , such that (1 − ε ) ◦ (1 − ε ) ≥ (1 − ε ) and ε • ε ≤ ε , then O ( a, ε , λ ) ⊂ O ( a, ε , λ ) .Proof. Let b ∈ O ( a, ε , λ ) and let O ( b, ε , λ ) be an OB with center a and radius ε . Since O ( b, ε , λ ) ∩ O ( a, ε , λ ) = ∅ , there is a c ∈ O ( b, ε , λ ) ∩ O ( a, ε , λ ). Then, we obtain G ( a, b, λ ) ≥ G ( a, c, λ ◦ G ( b, c, λ > (1 − ε ) ◦ (1 − ε ) ≥ − ε ,B ( a, b, λ ) ≤ B ( a, c, λ • B ( b, c, λ < ε • ε ≤ ε ,Y ( a, b, λ ) ≤ Y ( a, c, λ • Y ( b, c, λ < ε • ε ≤ ε . Hence, c ∈ O ( a, ε , λ ) and thus O ( a, ε , λ ) ⊂ O ( a, ε , λ ). (cid:3) Theorem 3.16.
A subset A of a NMS ( F, N , ◦ , • ) is nowhere dense if and only if everynonempty OS in F includes an OB whose closure is disjoint from A .Proof. Let γ be a nonempty open subset of F . Then there exist a nonempty OS δ suchthat δ ⊂ γ , δ ∩ A = ∅ . If we take a ∈ δ , then there exist ε ∈ (0 , λ > O ( a, ε , λ ) ⊂ δ . Now we take ε ∈ (0 ,
1) such that (1 − ε ) ◦ (1 − ε ) ≥ − ε and Murat Kiri¸sci and Necip S¸im¸sek ε • ε ≤ ε . Using the Lemma 3.15, we have O ( a, ε , λ ) ⊂ O ( a, ε , λ ). In that case, we canwrite O ( a, ε , λ ) ⊂ γ and O ( a, ε , λ ) ∩ A = ∅ .Conversely, assume that A is not nowhere dense. Therefore, int ( A ) = ∅ , so there existsa nonempty OS γ such that γ ⊂ A . Take O ( a, ε , λ ) be an OB such that O ( a, ε , λ ) ⊂ γ .Then, O ( a, ε , λ ) ∩ A = ∅ . This result indicates that there is a contradiction. (cid:3) Now, we will prove Baire Category Theorem for NMS:
Theorem 3.17.
Let { γ n : n ∈ N } be a sequence of dense open subsets of a complete NMS ( F, N , ◦ , • ) . Then ∩ n ∈ N γ n is also dense in F .Proof. Choose δ be nonempty OS of F . Since γ is dense in F , δ ∩ γ = ∅ . Let a ∈ δ ∩ γ .Since δ ∩ γ is open, then there exist ε ∈ (0 , λ > O ( a , ε , λ ) ⊂ δ ∩ γ .Take ε ∗ < ε and λ ∗ = min { λ , } such that O ( a , ε ∗ , λ ∗ ) ⊂ δ ∩ γ . Since γ is dense in F , O ( a , ε ∗ , λ ∗ ) ∩ γ = ∅ . Let a ∈ O ( a , ε ∗ , λ ∗ ) ∩ γ . Since O ( a , ε ∗ , λ ∗ ) ∩ γ is open, then thereexist ε ∈ (0 , /
2) and λ > O ( a , ε , λ ) ⊂ O ( a , ε ∗ , λ ∗ ) ∩ γ . Take ε ∗ < ε and λ ∗ = min { λ , / } such that O ( a , ε ∗ , λ ∗ ) ⊂ O ( a , ε ∗ , λ ∗ ) ∩ γ . If we continue this way, wehave a sequence ( a n ) in F and a sequence ( λ ∗ n ) such that 0 < λ ∗ n < /n and O ( a n , ε ∗ n , λ ∗ n ) ⊂ O ( a n − , ε ∗ n − , λ ∗ n − ) ∩ γ n Now, we show that the sequence ( a n ) is a Cauchy sequence. For λ > µ >
0, take N ∈ N such that 1 /N < λ and 1 /N < µ . Hence, for n ≥ N , m ≥ n , G ( a n , a m , λ ) ≥ G ( a n , a m , /n ) ≥ − /n > − µ,B ( a n , a m , λ ) ≤ B ( a n , a m , /n ) ≤ /n ≤ µ,Y ( a n , a m , λ ) ≤ Y ( a n , a m , /n ) ≤ /n ≤ µ. Therefore, the sequence ( a n ) is a Cauchy. We know that F is complete. Then there exists a ∈ F such that a n → a . Since a k ∈ O ( a n , ε ∗ n , λ ∗ n ) for k ≥ n , then we have a ∈ O ( a n , ε ∗ n , λ ∗ n ).Hence a ∈ O ( a n , ε ∗ n , λ ∗ n ) ⊂ O ( a n − , ε ∗ n − , λ ∗ n − ) ∩ γ n , ∀ n . Then, δ ∩ ( ∩ n ∈ N γ n ) = ∅ . Then ∩ n ∈ N γ n is dense in F . (cid:3) Definition 3.18.
Let ( F, N , ◦ , • ) be a NMS. A collection ( D n ) ( n ∈ N ) is said to haveneutrosophic diameter zero (NDZ) if for each < ε < and each λ > , then there exists N ∈ N such that G ( a, b, λ ) > − ε , B ( a, b, λ ) < ε and Y ( a, b, λ ) < ε for all a, b ∈ D N . Theorem 3.19.
The NMS ( F, N , ◦ , • ) is complete if and only if every nested sequence ( D n ) n ∈ N of nonempty closed sets with NDZ have nonempty intersection.Proof. Firstly consider the given condition is satisfied. We will show that ( F, N , ◦ , • ) iscomplete. Choose the Cauchy sequence ( a n ) in F . If we define the E n = { a k : k ≥ n } and D n = E n , then we can say that ( D n ) has NDZ. For given ζ ∈ (0 ,
1) and λ >
0, we take ε ∈ (0 ,
1) such that (1 − ε ) • (1 − ε ) • (1 − ε ) > − ζ and ε • ε • ε < ζ . Since the sequence( a n ) is Cauchy, then there exist N ∈ N such that G ( a n , a m , λ ) > − ε , B ( a n , a m , λ ) < ε and Y ( a n , a m , λ ) < ε , ( ∀ m, n ≥ N ). Then, G ( a, b, λ ) > − ε , B ( a, b, λ ) < ε and Y ( a, b, λ ) < ε , ( ∀ m, n ≥ E N ). eutrosophic Metric Spaces 9 Choose a, b ∈ D N . There exist the sequences ( a ∗ n ) and ( b ∗ n ) such that a ∗ n → a and b ∗ n → b .Thus, for sufficiently large n , a ∗ n ∈ O ( a, ε, λ ) and b ∗ n ∈ O ( b, ε, λ ). Now, we have G ( a, b, λ ) ≥ G ( a, a ∗ n , λ ◦ G ( a ∗ n , b ∗ n , λ ◦ G ( b ∗ n , b, λ > (1 − ε ) ◦ (1 − ε ) ◦ (1 − ε ) > − ζ,B ( a, b, λ ) ≤ B ( a, a ∗ n , λ • B ( a ∗ n , b ∗ n , λ • B ( b ∗ n , b, λ < ε • ε • ε < ζ,Y ( a, b, λ ) ≤ Y ( a, a ∗ n , λ • Y ( a ∗ n , b ∗ n , λ • Y ( b ∗ n , b, λ < ε • ε • ε < ζ. From here, G ( a, b, λ ) > − ζ , B ( a, b, λ ) < ζ and Y ( a, b, λ ) < ζ ( ∀ a, b ∈ D N ). There-fore, ( D N ) has NDZ and so by the hypothesis ∩ n ∈ N D n is nonempty. Take a ∈ ∩ n ∈ N D n .For ε ∈ (0 ,
1) and λ >
0, then there exist N ∈ N such that G ( a n , a, λ ) > − ε , B ( a n , a, λ ) < ε and Y ( a n , a, λ ) < ε ( ∀ n ≥ N ). Therefore, for each λ > G ( a n , a, λ ) → B ( a n , a, λ ) → Y ( a n , a, λ ) → n → ∞ . Hence, a n → a , that is ( F, N , ◦ , • ) is com-plete.Conversely, assume that ( F, N , ◦ , • ) is complete. Let’s ( D n ) n ∈ N is nested sequence ofnonempty closed sets with NDZ. For each n ∈ N , take a point a n ∈ D n . We will show thatthe sequence ( a n ) is Cauchy. Since ( D n ) has NDZ, for λ > < ε <
1, then there exist N ∈ N such that G ( a, b, λ ) > − ε , B ( a, b, λ ) < ε and Y ( a, b, λ ) < ε ( ∀ a, b ∈ D N ). Sincethe sequence ( D n ) is nested, then G ( a n , a m , λ ) > − ε , B ( a n , a m , λ ) < ε and Y ( a n , a m , λ ) <ε ( ∀ m, n ≥ N ). Hence the sequence ( a n ) is Cauchy. Since ( F, N , ◦ , • ) is complete, then a n → a for some a ∈ F . Therefore, a ∈ D n = D n for every n , and so a ∈ ∩ n ∈ N D n . (cid:3) Theorem 3.20.
Every separable NMS is second countable.Proof.
Give the separable NMS ( F, N , ◦ , • ). Let A = { a n : n ∈ N } be a countable densesubset of F . Establish the family O = { O ( a k , /m, /m ) : k, m ∈ N } . It can be easily seen, O is countable. We will show that O is base for the family of all OSs in F . Let γ be anyOS in F , a ∈ γ . Then there exist λ >
0, 0 < ε < O ( a, ε, λ ) ⊂ γ . Since0 < ε <
1, we can choose a 0 < ζ < − ζ ) ◦ (1 − ζ ) > − ε and ζ • ζ < ε .Take t ∈ N such that 1 /t < min { ζ, λ/ } . Since it is known that A is dense in F , there exist a k ∈ A such that a k ∈ O ( a, /t, /t ). If b ∈ O ( a k , /t, /t ), we have G ( a, b, λ ) ≥ G ( a, a k , λ ◦ G ( b, a k , λ ≥ G ( a, a k , t ) ◦ G ( b, a k , t ) ≥ (1 − t ) ◦ (1 − t ) ≥ (1 − ζ ) ◦ (1 − ζ ) > − ε,B ( a, b, λ ) ≤ B ( a, a k , λ • B ( b, a k , λ ≤ B ( a, a k , t ) • B ( b, a k , t ) ≤ t • t ≤ ζ • ζ < ε,Y ( a, b, λ ) ≤ Y ( a, a k , λ • Y ( b, a k , λ ≤ Y ( a, a k , t ) • Y ( b, a k , t ) ≤ t • t ≤ ζ • ζ < ε, Then, b ∈ O ( a, ε, λ ) ⊂ γ and so O is a base. (cid:3) Note that the second countability implies separability and the second countability isinheritable property. Then, we can say that every subspace of a separable NMS is separable.
Definition 3.21.
Let F be any nonempty set and ( H, N , ◦ , • ) be a NMS. The sequence offunctions ( f n ) : F → G is called converge uniformly to a function f : F → G , if given λ > , ε ∈ (0 , , then there exists N ∈ N such that G ( f n ( a ) , f ( a ) , λ ) > − ε , B ( f n ( a ) , f ( a ) , λ ) < ε , Y ( f n ( a ) , f ( a ) , λ ) < ε ∀ n ≥ N and ∀ a ∈ F . Now, we will give Uniform Convergence Theorem for NMS: Murat Kiri¸sci and Necip S¸im¸sek
Theorem 3.22.
Let f n : F → H be a sequence of continuous functions from a topologicalspace F to a NMS ( H, N , ◦ , • ) . If ( f n ) converges uniformly to f : F → H , then f iscontinuous.Proof. Take δ be OS of H and let a ∈ f − ( δ ). Since δ is open, then there exist λ > ε ∈ (0 ,
1) such that O ( f ( a ) , ε, λ ) ⊂ δ . Since ε ∈ (0 , ζ ∈ (0 ,
1) such that(1 − ζ ) ◦ (1 − ζ ) ◦ (1 − ζ ) > − ε and ζ • ζ • ζ < ε . Since ( f n ) converges uniformly to f , then, for λ > ζ ∈ (0 , N ∈ N such that G ( f n ( a ) , f ( a ) , λ ) > − ζ , B ( f n ( a ) , f ( a ) , λ ) < ζ and Y ( f n ( a ) , f ( a ) , λ ) < ζ ∀ n ≥ N and ∀ a ∈ F . Since f n continuous ∀ n ∈ N , then there exist a neighborhood γ of a such that f n ( γ ) ⊂ O ( f n ( a ) , ζ, λ ). Hence G ( f n ( a ) , f n ( a ) , λ ) > − ζ , B ( f n ( a ) , f n ( a ) , λ ) < ζ and Y ( f n ( a ) , f n ( a ) , λ ) < ζ for all a ∈ γ . Now G ( f ( a ) , f ( a ) , λ ) ≥ G ( f ( a ) , f n ( a ) , λ ◦ G ( f n ( a ) , f n ( a ) , λ ◦ G ( f n ( a ) , f ( a ) , λ ≥ (1 − ζ ) ◦ (1 − ζ ) ◦ (1 − ζ ) > − ε,B ( f ( a ) , f ( a ) , λ ) ≤ B ( f ( a ) , f n ( a ) , λ • B ( f n ( a ) , f n ( a ) , λ • B ( f n ( a ) , f ( a ) , λ ≤ ζ • ζ • ζ < ε,Y ( f ( a ) , f ( a ) , λ ) ≤ Y ( f ( a ) , f n ( a ) , λ • Y ( f n ( a ) , f n ( a ) , λ • Y ( f n ( a ) , f ( a ) , λ ≤ ζ • ζ • ζ < ε. Therefore, f ( a ) ∈ O ( f ( a ) , ε, λ ) ⊂ δ for all a ∈ γ . Hence f ( γ ) ⊂ δ and so f is continuous. (cid:3) Conclusion
The aim of this study is to define a neutrosophic metric spaces and examine some proper-ties. The structural characteristic properties of NMSs such as open ball, open set, Hausdorff-ness, compactness, completeness, nowhere dense in NMS have been established. Analoguesof Baire Category Theorem and Uniform Convergence Theorem are given for NMS.
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