On Product of Smooth Neutrosophic Topological Spaces

In this paper, we develop the notion of the basis for a smooth neutrosophic topology in a more natural way. As a sequel, we define the notion of symmetric neutrosophic quasi-coincident neighborhood systems and prove some interesting results that fit with the classical ones, to establish the consistency of theory developed. Finally, we define and discuss the concept of product topology, in this context, using the definition of basis.


Introduction
The idea of neutrosophy was initiated and developed by Smarandache [1] in 1999. In recent decades the theory was used at various junctions of mathematics. More precisely, the theory made an outstanding advancement in the field of topological spaces. Salama et al. and Hur et al. [2][3][4][5][6] are some who posted their works of neutrosophic topological spaces, following the approach of Chang [7] in the context of fuzzy topological spaces. One can easily observe that the fuzzy topology introduced by Chang is a crisp collection of fuzzy subsets.
Šostak [8] observed that Chang's approach is crisp in nature and so he redefined the notion of fuzzy topology, often referred as smooth fuzzy topology, as a function from the collection of all fuzzy subsets of X to [0, 1]; Fang Jin-ming et al. and Vembu et al. [9,10] are some who discussed the concept of basis as a function from a suitable collection of fuzzy subsets of X to [0, 1]. Yan, Wang, Nanjing, Liang and Yan [11,12] developed a parallel theory in the context of intuitionistic I-fuzzy topological spaces.
The notion of a single-valued neutrosophic set was proposed by Wang [13] in 2010. In 2016, Gayyar [14] introduced the concept of smooth neutrosophic topological spaces. The notion of the basis for an ordinary single-valued neutrosophic topology was defined and discussed by Kim [15]. Salama, Alblowi, Shumrani, Muhammed Gulisten, Smarandache, Saber, Alsharari, Zhang and Sunderraman [4,16,17] are some others who posted their work in the context of single-valued neutrosophic topological spaces.
In Section 2, we give all basic definitions and results, which are important prerequisites that are needed to go through the theory developed in this paper. In Section 3, we define the notion of the basis and subbasis for a smooth neutrosophic topology; further, we develop the theory using the concept of neutrosophic quasi-coincident neighborhood systems. In addition, we prove some results which are similar to the classical ones, to establish the consistency of theory developed. Finally, in Section 4, we define and discuss the product of smooth neutrosophic spaces using our definition of basis.

Preliminaries
In this section, we give all basic definitions and results which we need to go through our work. As usual R and Q denote the sets of all real numbers and rationals respectively. First we give the definition of a neutrosophic set [1,4]. Definition 1. Let X be a non-empty set. A neutrosophic set in X is an object having the form T N : X → − 0, 1 + , I N : X → − 0, 1 + , F N : X → − 0, 1 + represent the degree of membership (namely, T N (x)), the degree of indeterminacy (namely, I N (x)) and the degree of non-membership (namely, F N (x)), for all x ∈ X to the set to the set N.
Here − 0 = 1 − and 1 + = 1 + where is infinitesimal number and > 0; further, 1 and denote standard part and non-standard part of 1 + ; 0 and denote the standard part and non-standard part of 0 − . While dealing with scientific and engineering problems in real life applications, it is difficult to use a neutrosophic set with values from − 0, 1 + . In order to overcome this draw back, Wang et al. [13] defined the single-valued neutrosophic set, which is a particular case of the neutrosophic set.

Definition 2.
[13] Let X be a space of points (objects) with a generic element in X denoted by x. Then N is called a single-valued neutrosophic set in X if N has of the form N = T N , In this case, T N , I N , F N are called the truth membership function, indeterminancy membership function and falsity membership function, respectively.
For conventional reasons and as there is no ambiguity, we refer a single-valued neutrosophic set simply as a neutrosophic set throughout this paper; we also restate the definition, in order to view it explicitly as a function from a non-empty set X to ζ = [0, 1] 3 , in the following way: Let X be a nonempty set and I = [0, 1]. A neutrosophic set N on X is a mapping defined as We denote the set of all neutrosophic sets of X by ζ X and the neutrosophic sets 0, 1, 1 and 1, 0, 0 by 0 X and 1 X respectively. Let (r, s, t), (l, m, n) ∈ ζ; then

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The complement of N denoted by N c is given by The union of N and M denoted by N M is an neutrosophic set in X given by The intersection of N and M denoted by N M is an neutrosophic set in X given by The product of N and M denoted by N × M is given by For an any arbitrary collection {N i } i∈J ⊆ ζ X of neutrosophic sets the union and intersection are given by Definition 4. Let X be a nonempty set and x ∈ X. If r ∈ (0, 1], s ∈ [0, 1) and t ∈ [0, 1), then a neutrosophic point x r,s,t in X given by We say x r,s,t ∈ N if x r,s,t N. To avoid the ambiguity, we denote the set of all neutrosophic points by pt(ζ X ).

Definition 5.
A neutrosophic set N is said to be quasi-coincident with another neutrosophic set M, denoted by N[q]M, if there exists an element x ∈ X such that If M is not quasi-coincident with N, then we write M[ q]N. Definition 6. [14] Let X be a nonempty set. Then a neutrosophic set T = T T , I T , F T : ζ X → ζ is said to be a smooth neutrosophic topology on X if it satisfies the following conditions: The pair (X, T) is called a smooth neutrosophic topological space.

The Basis for a Smooth Neutrosophic Topology
The main objective of this section is to define and discuss the concept of basis for a neutrosophic topology. Many fundamental classical statements and theories describe ways to obtain a topology from a basis; every topology is a basis for itself; characterizations of a set to form a basis; comparison of two topologies is a way to get a basis from a subbasis; quasi-neighborhood systems are discussed. Though the structural development of the theory is same as the ones followed in the context of classical and fuzzy topological spaces, the strategies following the proofs of the statements are entirely different. We start with the definition of a basis for a smooth neutrosophic topology. Definition 7. Let B : ζ X → ζ be a function that satisfies: B1 If x ∈ X and , δ > 0, then there exists M ∈ ζ X such that B2 If x ∈ X, M, N ∈ ζ X and , δ > 0, then there exists L ∈ ζ X such that L M N, Then B is called a basis for a smooth neutrosophic topology on X.
Any function S : ζ X → ζ satisfying B1 is called a subbasis of a smooth neutrosophic topology on X. A collection {M λ } λ∈Λ of neutrosophic sets is said to be an inner cover for a neutrosophic set M if M = M λ . Definition 8. Let B be a basis for a smooth neutrosophic topology on X. Then the smooth neutrosophic topology T : ζ X → ζ generated by B is defined as follows: Then B is a basis for a smooth neutrosophic topology T on X. We note that B(ξ {(a,1)} ) = (0, 1, 1), whereas Theorem 1. Let B be a basis and T be as defined in Definition 8; then T is a smooth neutrosophic topology on X.
Proof. From the definition of T it directly follows that T(0 X ) = (1, 0, 0). Next we wish to show that T(1 X ) = (1, 0, 0). Indeed, let x ∈ X and δ, > 0; then by the definition of a basis for a smooth neutrosophic topology, there exists M x,δ, ∈ ζ X such that M x,δ, (x) 1 X (x) − (δ, 0, 0) and If we let L = {M x,δ, } x,δ , then it is easy to see that L is an inner cover for 1 X . However, Thus for every > 0, there exists an inner cover and hence T(1 X ) = (1, 0, 0). Next we claim that T(M N) T(M) ∧ T(N) for any two neutrosophic sets M, N in ζ X . Suppose M N = 0 X ; then there is nothing to prove. Let M N = 0 X and let > 0. Then there exist inner covers {M λ } λ∈Λ 1 and {N γ } γ∈Λ 2 , such that 1,1) and N(x) = (0, 1, 1); then by the definition of an inner cover there exist M λ 0 and N γ 0 in the corresponding inner covers, such that Then the collection {D λ,γ,x,δ } x,δ is an inner cover for L λ,γ and hence the collection {D λ,γ,x,δ } λ,γ, x,δ is an inner cover for M N.
Additionally, we have, Since this is true for every > 0 and Definition 9. Let (X, T) be smooth neutrosophic topological space. For all x r,s,t ∈ pt(ζ X ) and N ∈ ζ X , the mapping Q T x r,s,t : ζ X → ζ is defined as follows: x r,s,t ∈ pt(ζ X )} is called a neutrosophic quasi-coincident neighborhood system. Further, a neutrosophic quasi-coincident neighborhood system Q T is said to be symmetric if for any x r,s,t , y l,m,n ∈ pt(ζ X ), Theorem 2. Let (X, T) be neutrosophic topological space. Then for all M, N ∈ ζ X , Proof. As (i), (ii) and (iii) follow directly from the definition of Q T x r,s,t , we skip their proof. To prove (iv), first we observe that To prove the reverse inequality, consider To prove (v), for any N ∈ ζ X with x r,s,t [q]N M, we have Q T x r,s,t (N) T(N), and therefore, Hence, we have Proof. Let B be a basis for given smooth neutrosophic topology; then clearly B T. Let M ∈ ζ X and x r,s,t ∈ pt(ζ X ); then Q T x r,s,t Thus it follows that, This implies that Conversely, let x ∈ X and , δ > 0; then clearly x δ,0,0 ∈ pt(ζ X ). However, since Let > 0; then there exists L such that Then there exists L ∈ L x such that Thus, B2 of Definition 8 follows in both cases.
Here we note that, "If (X, T) is a smooth neutrosophic topological space, then T is a basis for a smooth fuzzy topology on X and the smooth fuzzy topology generated by T is itself." In the following, we give certain theorems which can be proved in a similar fashion to Theorems 3.8, 3.9 and 3.10 in [10]. To end this section, we present a theorem which gives a way to get a basis from a subbasis, from which a smooth neutrosophic topology can be generated. Now let us define a collection of neutrosophic sets L k 1 X , for k = 1, 2, . . . , n + m, as If we let L = n+m k=1 L k , then L = M N and therefore Thus it follows that

Product of Neutrosophic Topologies
In this section, we first define the concept of a finite product of smooth neutrosophic topologies, using the notion of basis defined in the previous section. We present a way to obtain the product topology from the given bases; in the following we present a subbasis for a product topology. Later, we generalize the discussed contents in the context of an arbitrary product of smooth neutrosophic topologies. Definition 10. Let (X, T 1 ) and (Y, T 2 ) be smooth neutrosophic topological spaces. Let B : ζ X×Y → ζ be defined as follows: Then B is a basis for the smooth neutrosophic topology called the smooth neutrosophic product topology on X × Y.
Example 2. Let X 1 = X 2 = R and let M 1 and N 1 be defined by for all x ∈ X 2 .
Let T 1 : ζ X 1 → ζ and T 2 : ζ X 2 → ζ be the functions defined by Then clearly T 1 and T 2 are smooth neutrosophic topologies on X 1 and X 2 . From the above definition, we get B : ζ X 1 ×X 2 → ζ given by which is a basis for a smooth neutrosophic topology T on X 1 × X 2 and the smooth neutrosophic topology (product topology) generated by B is given by and hence B2 of Definition 7 follows in this case also. Theorem 9. Let B 1 , B 2 be bases for the smooth neutrosophic topologies T 1 , T 2 respectively. Define B : ζ X×Y → ζ as follows: If A ∈ ζ X×Y cannot be written as M × N for any M ∈ ζ X and N ∈ ζ Y , then define B(A) = (0, 1, 1).
Then B is a basis for the product topology on X × Y.
Proof. First we claim that B is a basis for a smooth neutrosophic topology on X × Y. Let (x, y) ∈ X × Y, δ > 0 and > 0. Now since B 1 and B 2 are bases for the smooth neutrosophic topologies T 1 and T 2 , there exist M ∈ ζ X and N ∈ ζ Y such that Thus B1 of Definition 7 follows. To prove B2, let (x, y) ∈ X × Y, M, N ∈ ζ X×Y and δ, > 0. If any one of M and N, say M, cannot be written as M 1 × M 2 for any M 1 ∈ ζ X and M 2 ∈ ζ Y , then by letting L = M N, as in the above theorem, B2 of Definition 7 follows. On the other hand, suppose both M and N can be written as A × A and B × B for some A, B ∈ ζ X and A , B ∈ ζ Y ; then by definition of B, there exist M 1 , N 1 ∈ ζ X , and M 2 , Here it is easy to see that there exists L 1 ∈ ζ X such that and Thus B2 of Definition 7 follows in this case also. Hence B is a basis for a smooth neutrosophic topology on X × Y. Thus, proving that the smooth neutrosophic topology generated by this basis coincides with the smooth neutrosophic product topology remains. Let T be the smooth fuzzy topology generated by B. Let T p be the product topology on X × Y and B p be the basis for T p as described in Definition 10. Now we prove that T p = T. Let A ∈ ζ X×Y ; then where {L Λ } Λ∈Γ is the collection of all inner covers L Λ = {A λ } λ∈Λ of A. Now we divide the collection {L Λ } Λ∈Γ , say L, into two subcollections L and L where L is the collection all possible inner covers {A λ } λ∈Λ of A so that for all λ ∈ Λ, A λ is of the form M λ × N λ for at least one M λ ∈ ζ X and one N λ ∈ ζ Y , and L is the complement of L in L.
If an inner cover L Λ = {A λ } λ∈Λ of A is in L , then for at least one λ 0 ∈ Λ, A λ 0 is not of the form M × N for any M ∈ ζ X and N ∈ ζ Y ; hence B p (A λ 0 ) = (0, 1, 1) and therefore If L = ∅, then T p (A) = T(A) = (0, 1, 1) and hence it is enough to consider the case L = ∅. Now consider This implies that, T p T.
To prove the reverse inequality, let A ∈ ζ X×Y , > 0 and L, L , L be as above. Let L Λ = {A λ } λ∈Λ be an inner cover for A. As above it is enough to consider the case L = ∅. Now let L Λ ∈ L . Then for all λ ∈ Λ, we have A λ = M × N for at least one M ∈ ζ X and one N ∈ ζ Y . Fix a λ ∈ Λ. Let B λ denote the set of all pairs (M, N) such that A λ = M × N. Let (M, N) ∈ B λ . Since B 1 , B 2 are bases for T 1 , T 2 , by Theorem 5, for any x ∈ X, y ∈ Y and δ > 0 there exist M x,δ ∈ ζ X and N y,δ ∈ ζ Y such that Clearly the collection {M x,δ } x∈X,δ>0 is an inner cover for M and the collection {N y,δ } y ∈ ζ Y , δ > 0 is an inner cover for N. Therefore, the collection {M x,δ × N y,δ } x∈X,y∈Y,δ>0 is an inner cover for M × N which is equal to A λ . Thus for any pair (M, N) ∈ B λ with M × N = A λ , we have an inner cover {M x,δ × N y,δ } x∈X, y ∈Y, δ>0 of A λ such that and B 2 (N y,δ ) + ( , 0, 0) T 2 (N) (2) for all x ∈ X, y ∈ X and δ > 0. Now since using (1) and (2), we have Since this is true for every > 0, it follows that T p (A) T(A) and hence we get T p T as desired.
Theorem 10. Let (X, T 1 ) and (Y, T 2 ) be smooth neutrosophic topological spaces. Let Then S is a subbasis for the smooth neutrosophic product topology on X × Y.
Proof. Since S(1 X×Y ) = (1, 0, 0), by letting M = 1 X×Y , it clearly follows that S is a subbasis for a smooth neutrosophic topology on X × Y. Thus all that remains is to show the smooth neutrosophic topology induced by this subbasis is the same as the product topology on X × Y. We do this by proving that the basis induced by this subbasis is the same as the basis defined in Definition 10. Let B be the basis generated by S. Then for any A in ζ X×Y , we have . Let A = A 1 A 2 · · · A n be any representation of A as a finite intersection of neutrosophic sets of X × Y. First we claim that A i is neither of the form Without loss of generality, let us assume that A i = (M i × 1 Y ) for i = 1, 2, . . . , m and A i = (1 X × N i ) for i = m + 1, m + 2, . . . , n, then we have for i = m + 1, m + 2, . . . , n. Then, Let M = M 1 · · · M m and N = N m+1 · · · N n . Then we have Since this is true for any representation of A as a finite intersection of neutrosophic sets in ζ X×Y , we have B(A) B (A).
To prove the reverse inequality, let > 0; then by Definition 10, there exist M ∈ ζ X and N ∈ ζ Y such that A = M × N and ; thus, we have, Definition 11. Let {(X i , T i )} i∈J be a collection of smooth neutrosophic topological spaces, for some indexing set J. Now define a function B : ζ Π i∈J X i → ζ as follows: Then B is a basis for a smooth neutrosophic topology called the smooth product topology on Π i∈J X i .
Thus, B2 of Definition 7 follows in this case also, and hence B is a basis for a smooth neutrosophic topology on ΠX i .
Let S : ζ Π i∈J X i → ζ be defined as follows: Then S is a subbasis for a smooth neutrosophic product topology on ΠX i .
Proof. Since S(1 ΠX i ) = (1, 0, 0), B1 of Definition 7 follows. Thus S is a subbasis for a smooth neutrosophic topology on Π X i . Thus, proving that the smooth neutrosophic topology generated from S is the smooth neutrosophic product topology on Π X i needs proving. Now let B be the basis generated by S and let B be the basis for the smooth neutrosophic product topology defined in Definition 11. To prove the topologies generated by B and B are same, we prove the stronger result that B = B .
As B(1 ΠX i ) = B (1 ΠX i ) = (1, 0, 0) follows trivially, we prove the other cases. Let A ∈ ζ ΠX i and let A A be the collection of all {ΠA i } such that A = ΠA i , A i ∈ ζ X i and A i = 1 X i except for finitely many i ∈ J. If A A = ∅, then by the definition of B, we have B(A) = (0, 1, 1). Now to compute B (A), let A = A 1 A 2 · · · A n ; we claim that there must exist at least one A k which is not of the form Π A ki where A ki ∈ ζ X i and A ki = 1 X i except for finitely many i ∈ J. Suppose not; instead, let A j = ΠA ji where A ji ∈ ζ X i and A ji = 1 X i except for finitely many i ∈ J, for all j = 1, 2, . . . , n. Then using these finitely many A ji 's, A can be written in the form ΠA i where A i ∈ ζ X i and A i = 1 X i except for finitely many i ∈ J, which is a contradiction to our assumption that A A = ∅. Thus there exists at least one A k which is not of the form Π A ki where A ki ∈ ζ X i and A ki = 1 X i except for finitely many i ∈ J and hence S(A k ) = (0, 1, 1). Thus we have {S(A j )/j = 1, 2, . . . , n} = (0, 1, 1).
Since this is true for any possible finite representation A 1 A 2 · · · A n of A, we have B (A) = (0, 1, 1) and hence B (A) = B(A) in this case.
If A A = ∅, then there must exist a representation A 1 A 2 · · · A n of A such that A A j = ∅ for all j = 1, 2, . . . , n, where A A j is the collection of all {ΠA ji } such that A j = ΠA ji , A ji ∈ ζ X i and A ji = 1 X i except for finitely many i ∈ J. Let > 0. Then for each A j we can find a collection {A ji } i∈J such that A j = ΠA ji where A ji = 1 X i except for finitely many i ∈ J and T i (A ji ) S(A j ) − ( , 0, 0). Now since Since this is true for any representation of A as a finite intersection of neutrosophic sets in ζ ΠX i , we have

B(A) B (A).
To prove the reverse inequality, let > 0. Since A A = ∅, we can find a collection {A i } i∈J such that

Conclusions
In this paper, we have defined the notion of a basis and subbasis for a neutrosophic topology as a neutrosophic set from a suitable collection of neutrosophic sets of X to [0, 1] 3 . Using this idea of considering a basis as a neutrosophic set, we developed a theory of smooth neutrosophic topological spaces that fits exactly with the theory of classical and fuzzy topological spaces. Next, we introduced and investigated the concept of quasi-coincident neighborhood systems in this context. Finally, we defined and discussed the notion of both finite and infinite products of smooth neutrosophic topologies.

A Discussion for Future Works
The theory can extended in the following natural ways. One may

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Study the properties of neutrosophic metric topological spaces using the concept of basis defined in this paper; • Investigate the products of Hausdorff, regular, compact and connected spaces in the context of neutrosophic topological spaces.
Author Contributions: Conceptualization, K.C. and S.S.S.; writing, original draft, K.C. and S.S.S.; writing, review and editing, K.C. and S.S.S.; Supervision and Validation F.S. and S.J. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.