# Neutrosophic Triplet \(v\)-Generalized Metric Space
## 3 Neutrosophic Triplet \(v\)-Generalized Metric Space  
**Definition 4**. : _An NTVGM on an NTS \((X,\,\#)\) is a function \(d_{v}\):\(XxX\)\(\rightarrow\)\(\mathbb{R}\) such that all \(a,b\), \(u_{1}\), \(u_{2}\),..., \(u_{v}\in X\);_  
_(i)_: \(a\#b\in X\)__ (ii)_: \(0\leq d_{v}(a,\,b)\)__ (iii)_: _if_ \(a=b\)_, then_ \(d_{v}(a,\,b)=0\)__ (iv)_: \(d_{v}(a,\,b)=d_{v}(b,\,a)\)__ (v)_: _If there exists elements_ \(a\)_,_ \(b\)_,_ \(u_{1}\)_,..., \(u_{v}\in X\) _such that_  
\(d_{v}(a,\,b)\leq d_{v}(a,\,b\#neut(u_{v}))\)_,_  
\(d_{v}(a,\,u_{2})\leq d_{v}(a,\,u_{2}\#neut(u_{1}))\)_,_  
\(d_{v}(u_{1},\,u_{3})\leq d_{v}(u_{1},\,u_{3}\#neut(u_{2}))\)_,_  
\(\ldots\)_,_  
\(d_{v}(u_{v-1},\,b)\leq d_{v}(u_{v-1},\,b\#neut(u_{v}))\)_;_  
_then_ \(d_{v}(a,\,b\*neut(u_{v}))\leq d_{v}(a,\,u_{1})+d_{v}(u_{1},\,u_{2})+\ldots+d_{v} (u_{v-1},\,u_{v})+d_{v}(u_{v},\,b)\)_, where_ \(x\)_,_ \(u_{1}\)_,..., \(u_{v}\)_,_ \(y\) _are all different._  
_Also, \(((X,\,\#),\,d_{v})\) space is called NTVGMS._  
**Example 1**. : _Let \(X=\{\emptyset,\,\{k\},\{l\},\,\{k,l\}\}\) be a set and \(n(A)\) be number of elements in \(K\in X.\) It is clear that for \(K\in X,\)\(K\cup K=K.\) Hence, we have_  
\(neut(K)=K,\)\(anti(K)=K\) _for all \(K\in X.\) Also, (\(X,\,\cup\)) is an NTS. Then let_  
\(d_{v}\)_: \(XxX\)\(\rightarrow\)\(X\) _be a function such that_ \(d_{v}(K,M)=\Big{|}n(K)\ +\ 2^{n(K)}-\ (\ n(M)+\ 2^{n(M)})\Big{|},\) _where_ \(M\in X.\)__  
_Now we will show that_ \(d_{v}\) is an NTVGM_._  
_It is clear that_ \(K\cup M\in X\) _for_ \(K,\,M\in X.\)__  
_It is clear that_ \(d_{v}(K,M)=\Big{|}n(K)\ +\ 2^{n(K)}-\ (\ n(M)+\ 2^{n(M)})\Big{|}\geq 0.\)__  
_If_ \(K=M,\)  
Hence, (X), \(\cup\)), \(d_{v}\)) is an NT2 GMS._  
**Corollary 1**. : _NTVGM is different from the classical metric because of triangle inequality and \"#\" binary operation._  
**Corollary 2**. : _The NTVGM is different from NTM because of triangle inequality._  
**Theorem 1**. : _In definition of NTVGM, if \(u_{1}=u_{2}=\ldots\ =u_{v}\), then each neutrosophtic triplet \(v\)-generalized metric is a neutrosophtic triplet metric._  
**Proof of Theorem 1. ** ((X, #), \(d_{v}\)) be an NTVGMS. It is clear that i, ii, iii and iv conditions are equal in NTVGMS and NTMS. Then for condition v, we take \(u_{1}=u_{2}=\ldots\ =u_{v}\). By definition of NTVGM,  
If there exists elements \(u_{1},\ldots,u_{v}\in X\) such that  
\(d_{v}(x,y)\leq d_{v}(x,\)\(\#neut(u_{v}))\),  
\(d_{v}(x,\)\(u_{2})\leq d_{v}(x,\)\(u_{2}\#neut(u_{1}))\),  
\(d_{v}(u_{1},\)\(u_{3})\leq d_{v}(u_{1},\)\(u_{3}\#neut(u_{2}))\),  
\(\ldots\),  
\(d_{v}(u_{v-1},\)\(y)\leq d_{v}(u_{v-1},\)\(\#neut(u_{v}))\);  
then \(d_{v}(x,\)\(\#neut(u_{v}))\leq d_{v}(x,\)\(u_{1})+d_{v}(u_{1},\)\(u_{2})+\ldots+d_{v}(u_{v-1},\)\(u_{v})+d_{v}(u_{v},\)y)\). We can take \(u_{1}=u_{3}=\ldots\)\(=u_{v}=u_{2}\) since for \(u_{1}=u_{2}=\ldots=u_{v}\). Thus, we can take  
\(d_{v}(x,\)\(\#neut(u_{v}))=d_{v}(x,\)\(\#neut(u_{2}))\) and  
\(d_{v}(x,\)\(u_{1})+d_{v}(u_{1},\)\(u_{2})+\ldots\)\(+d_{v}(u_{v-1},\)\(u_{v})+d_{v}(u_{v},\)\(y)=\)  
\(d_{v}(x,\)\(u_{2})+d_{v}(u_{2},\)\(u_{2})+\ldots\)\(+d_{v}(u_{2},\)\(u_{2})+d_{v}(u_{2},\)\(y)\). Hence,  
\(d_{v}(x,\)\(\#neut(u_{2}))\leq d_{v}(x,\)\(u_{1})+d_{v}(u_{1},\)\(u_{2})+\ldots\)\(+d_{v}(u_{v-1},\)\(u_{v})+d_{v}(u_{v},\)\(y)=\)  
\(d_{v}(x,\)\(u_{2})+d_{v}(u_{2},\)\(u_{2})+\ldots\)\(+d_{v}(u_{2},\)\(u_{2})+d_{v}(u_{2},\)\(y)=\)  
\(d_{v}(x,\)\(u_{2})+0+\ldots\)\(+0+d_{v}(u_{2},\)\(y)=d_{v}(x,\)\(u_{2})+d_{v}(u_{2},\)\(y)\). Therefore, \((X,\ \#),\ d_{v})\) is a neutrosophtic triplet metric space. **Corollary 3**. : _From Theorem 1, we can define an NTIGM with each NTM. Also, Each NTM is an NTIGM._  
**Definition 5**. : _Let \(((X,\#),\)\(d_{v})\) be an NTVGMS and \(\{x_{n}\}\) be a sequence in NTVGMS and \(a\in X\). If there exist \(N\)\(\in N\) for every \(\epsilon>0\) such that_  
\(d_{v}\ (a,\ \{xn\}\ )(a,\ \{xn\})<\epsilon\)_,_  
_then \(\{x_{n}\}\) converges to \(a\in X\), where \(n\geq M\).