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Neutrosophic Supra Topological Applications in Data Mining Process.

1 Introduction

The concept of fuzzy set was introduced by A. Zadeh [1] in 1965 which is a generalization of crisp set to analyse imprecise mathematical information. Adlassnig [2] applied fuzzy set theory to formalize medical relationships and fuzzy logic to computerized diagnosis system. This theory [3, 4, 5] has been used in the fields of artificial intelligence, probability, biology, control systems and economics. C.L Chang [6] introduced the fuzzy topological spaces and further the properties of fuzzy topological spaces are studied by R. Lowen [7]. By relaxing one topological axiom, Mashhour et al. [8] introduced supra topological space in 1983 and discussed its properties. Abd-Monsef and Ramadan [9] introduced fuzzy supra topological spaces and its continuous mappings. K. Atanassov [10] considered the degree of non-membership of an element along with the degree of membership and introduced intuitionistic fuzzy sets. Dogan Coker [11] introduced intuitionistic fuzzy topology. Saadati [12] further studied the basic concept of intuitionistic fuzzy point. S.K.De et al. [13] was the first one to develop the applications of intuitionistic fuzzy sets in medical diagnosis. Several researchers [14, 15, 16] further studied intuitionistic fuzzy sets in medical diagnosis. Hung and Tuan [17] noted that the approach in [13] has some questionable results on false diagnosis of patients' symptoms. Generally it is recognized that the available information about the patient and medical relationships is inherently uncertain. There may be indeterminacy components in real life problems for data mining and neutrosophic logic can be used in this regard. Neutrosophic logic is a generalization of fuzzy, intuitionistic, boolean, paraconsistent logics etc. Compared to all other logics, neutrosophic logic introduces a percentage of "indeterminacy" and this logic allows each component t true, i indeterminate, f false to "boil over" 100 or "freeze" under 0. Here no restriction on T, I, F, or the sum n = t + i + f, where t, i, f are real values from the ranges T, I, F. For instance, in some tautologies t > 100, called "overtrue". As a generalization of Zadeh's fuzzy set and Atanassov's intuitionistic fuzzy set, Florentin Smarandache [18] introduced neutrosophic set. Neutrosophic set A consists of three independent objects called truth-membership [[mu].sub.A](x), indeterminacy-membership [[sigma].sub.A](x) and falsity-membership [[gamma].sub.A](x) whose values are real standard or non--standard subset of unit interval][.sup.-]0, [1.sup.+][. In data analysis, many methods have been introduced [19, 20, 21] to measure the similarity degree between fuzzy sets. But these are not suitable for the similarity measures of neutrosophic sets. The single-valued neutrosophic set is a neutrosophic set which can be used in real life engineering and scientific applications. The single valued neutrosophic set was first initiated by Smarandache [22] in 1998 and further studied by Wang et al. [23]. Majumdar and Samanta [24] defined some similarity measures of single valued neutrosophic sets in decision making problems. Recently many researchers [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] introduced several similarity measures and single-valued neutrosophic sets in medical diagnosis. The notion of neutrosophic crisp sets and topological spaces were introduced by A. A. Salama and S. A. Alblowi [46,47].

In section 2 of this paper, we present some basic preliminaries of fuzzy, intuitionistic, neutrosophic sets and topological spaces. The section 3 introduces the neutrosophic subspaces with its properties. In section 4, we define the concept of neutrosophic supra topological spaces. In section 5, we introduce neutrosophic supra continuity, S* -neutrosophic continuity and give some contradicting examples in fuzzy supra topological spaces [9]. As a real life application, a common method for data analysis under neutrosophic supra topological environment is presented in section 6. In section 7, we solve numerical examples of above proposed method and the last section states the conclusion and future work of this paper.

2 Preliminary

This section studies some of the basic definitions of fuzzy, intuitionistic, neutrosophic sets and respective topological spaces which are used for further study.

Definition 2.1. [1] Let X be a non empty set, then A = {(x, [[mu].sub.A] (x)) : x [member of] X} is called a fuzzy set on X, where [[mu].sub.A](x) [member of] [0, 1] is the degree of membership function of each x [member of] X to the set A. For X, [I.sup.X] denotes the collection of all fuzzy sets of X.

Definition 2.2. [10] Let X be a non empty set, then A = {(x, [[mu].sub.A](x), [[gamma].sub.A](x)) : x [member of] X} is called an intuitionistic set on X, where 0 [less than or equal to] [[mu].sub.A](x) + [[gamma].sub.A](x) [less than or equal to] 1 for all x [member of] X, [[mu].sub.A](x),[[gamma].sub.A](x) [member of] [0, 1] are the degree of membership and non membership functions of each x [member of] X to the set A respectively. The set of all intuitionistic sets of X is denoted by I(X).

Definition 2.3. [23] Let X be a non empty set, then A = {(x, [[mu].sub.A](x), [[sigma].sub.A](x), [[gamma].sub.A](x)) : x [member of] X} is called a neutrosophic set on X, where [.sup.-]0 [less than or equal to] [[mu].sub.A](x) + [[sigma].sub.A](x) + [[gamma].sub.A](x) [less than or equal to] [3.sup.+] for all x [member of] X, [[mu].sub.A](x),[[sigma].sub.A](x) and [[gamma].sub.A](x) [member of]][.sup.-]0, [1.sup.+][are the degree of membership (namely [[mu].sub.A](x)), the degree of indeterminacy (namely [[sigma].sub.A](x)) and the degree of non membership (namely [[gamma].sub.A](x)) of each x [member of] X to the set A respectively. For X, N(X) denotes the collection of all neutrosophic sets of X.

Definition 2.4. [18] The following statements are true for neutrosophic sets A and B on X:

(i) [[mu].sub.A](x) [less than or equal to] [[mu].sub.B](x), [[sigma].sub.A](x) [less than or equal to] [[sigma].sub.B](x) and [[gamma].sub.A](x) [greater than or equal to] [[gamma].sub.B](x) for all x [member of] X if and only if A [??] B.

(ii) A [??] B and B [??] A if and only if A = B.

(iii) A [intersection] B = {(x,min{[[mu].sub.A](x), [[mu].sub.B](x)},min{[[sigma].sub.A](x), [[sigma].sub.B](x)},max{[[gamma].sub.A](x),[[gamma].sub.B](x)}) : x [member of] X}.

(iv) A [union] B = {(x,max{[[mu].sub.A](x), [[mu].sub.B](x)},max{[[sigma].sub.A](x), [[sigma].sub.B](x)},min{[[gamma].sub.A](x), [[gamma].sub.B](x)}) : x [member of] X}.

More generally, the intersection and the union of a collection of neutrosophic sets [{[A.sub.i]}.sub.i[member of][LAMBDA]], are defined by [mathematical expression not reproducible] and [mathematical expression not reproducible].

Notation 2.5. Let X be a non empty set. We consider the fuzzy, intuitionistic, neutrosophic empty set as [empty set] = {(x, 0) : x [member of] X}, [empty set] = {(x, 0, 1) : x [member of] X}, [empty set] = {(x, 0, 0, 1) : x [member of] X} respectively and the fuzzy, intuitionistic, neutrosophic whole set as X = {(x, 1) : x [member of] X}, X = {(x, 1, 0) : x [member of] X}, X = {(x, 1, 1, 0) : x [member of] X} respectively.

Definition 2.6. [24] A neutrosophic set A = {(x,[[mu].sub.A](x), [[sigma].sub.A](x), [[gamma].sub.A](x)) : x [member of] X} is called a single valued neutrosophic set on a non empty set X, if [[mu].sub.A](x), [[sigma].sub.A](x) and [[gamma].sub.A](x) [member of] [0, 1] and 0 [less than or equal to] [[mu].sub.A](x) + [[sigma].sub.A](x) + [[gamma].sub.A](x) [less than or equal to] 3 for all x [member of] X to the set A. For each attribute, the single valued neutrosophic score function (shortly SVNSF) is defined as SVNSF = [1/3m][[[SIGMA].sup.m.sub.i=1][2 + [[mu].sub.i] - [[sigma].sub.i] - [[gamma].sub.i]]].

Definition 2.7. [6] Let X be a non empty set. A subcollection [[tau].sub.f] of [I.sup.X] is said to be fuzzy topology on X if the sets X and [empty set] belong to [[tau].sub.f], [[tau].sub.f] is closed under arbitrary union and [[tau].sub.f] is closed under finite intersection. Then (X, [[tau].sub.f]) is called fuzzy topological space (shortly fts), members of [[tau].sub.f] are known as fuzzy open sets and their complements are fuzzy closed sets.

Definition 2.8. [11] Let X be a non empty set and a subfamily [[tau].sub.i] of I (X) is called intuitionistic fuzzy topology on X if X and [empty set] [member of] [[tau].sub.i], [[tau].sub.i] is closed under arbitrary union and [[tau].sub.i] is closed under finite intersection. Then (X, [[tau].sub.i]) is called intuitionistic fuzzy topological space (shortly ifts), elements of [[tau].sub.i] are called intuitionistic fuzzy open sets and their complements are intuitionistic fuzzy closed sets.

Definition 2.9. [46, 47] Let X be a non empty set. A neutrosophic topology on X is a subfamily [[tau].sub.n] of N (X) such that X and [empty set] belong to [[tau].sub.n], [[tau].sub.n] is closed under arbitrary union and [[tau].sub.n] is closed under finite intersection. Then (X, [[tau].sub.n]) is called neutrosophic topological space (shortly nts), members of [[tau].sub.n] are known as neutrosophic open sets and their complements are neutrosophic closed sets. For a neutrosophic set A of X, the interior and closure of A are respectively defined as: [int.sub.n](A) = [union]{G : G [??] A, G [member of] [[tau].sub.n]} and [cl.sub.n](A) = [intersection]{F : A [??] F, [F.sup.c] [member of] [[tau].sub.n]}.

Corollary 2.10. [18] The following statements are true for the neutrosophic sets A, B, C and D on X:

(i) A [intersection] C [??] B [intersection] D and A [union] C [??] B [union] D, if A [??] B and C [??] D.

(ii) A [??] B [intersection] C, if A [??] B and A [??] C. A [union] B [??] C, if A [??] C and B [??] C.

(iii) A [??] C, if A [??] B and B [??] C.

Definition 2.11. [48] Let A = {(x,[[mu].sub.A](x), [[sigma].sub.A](x), [[gamma].sub.A](x)) : x [member of] X}, B = {(y, [[mu].sub.B](y), [[sigma].sub.B](y), [[gamma].sub.B](y)) : y [member of] Y} be two neutrosophic sets and f : X [right arrow] Y be a function.

(i) [f.sup.-l](B) = {(x, [f.sup.-l]([[mu].sub.B])(x), [f.sup.-l]([[sigma].sub.B])(x), [f.sup.-l] ([[gamma].sub.B)(x)) : x [member of] X} is a neutrosophic set on X called the pre-image of B under f.

(ii) f (A) = {(V, f ([[mu].sub.A])(y), f ([[sigma].sub.A])(y), (1 - f (1 - [[gamma].sub.A]))(y)) : y [member of] Y} is a neutrosophic set on Y called the image of A under f, where

[mathematical expression not reproducible]

For the sake of simplicity, let us use the symbol [f.sub.-]([[gamma].sub.A]) for (1 - f (1 - [[gamma].sub.A])).

3 Neutrosophic Subspaces

This section introduce differences of two fuzzy, intuitionistic and neutrosophic sets on X. We also introduce neutrosophic subspaces with its proprties.

Definition 3.1. The difference of neutrosophic sets A and B on X is a neutrosophic set on X, defined as A \ B = {(x, |[[mu].sub.A](x) - [[mu].sub.B](x)|, |[[sigma].sub.A](x) - [[sigma].sub.B](x)|, 1 - |[[gamma].sub.A](x) - [[gamma].sub.B](x)|) : x [member of] X}. Clearly [X.sup.c] = X \ X = (x, 0, 0, 1) = [empty set] and [[empty set].sup.c] = X \ [empty set] = (x, 1, 1, 0) = X.

Definition 3.2. Let A, B be two intuitionistic fuzzy sets of X, then the difference of A and B is a intuitionistic fuzzy set on X, defined as A \ B = {(x, |[[mu].sub.A](x) - [[mu].sub.B](x)|, 1 - |[[gamma].sub.A](x) - [[gamma].sub.B](x)|) : x [member of] X}. Clearly [X.sup.c] = X \ X =(x, 0, 1) = [empty set] and [[empty set].sup.c] = X [empty set] = (x, 1, 0) = X.

Definition 3.3. Let A, B be two fuzzy sets of X, then the difference of A and B is a fuzzy set on X, defined as A \ B = {(x, |[[mu].sub.A] (x) - [[mu].sub.B] (x)|) : x [member of] X}. Clearly [X.sup.c] = X \ X = (x, 0) = [empty set] and [[empty set].sup.c] = X\[empty set] = (x, 1) = X.

Corollary 3.4. The following statements are true for the neutrosophic sets [{A}.sup.[infinity].sub.i=1], A, B on X:

(i) [([[intersection].sub.i[member of][LAMBDA]] [A.sub.i]).sup.c] = [[union].sub.i[member of][LAMBDA]] [A.sup.c.sub.i], [([[union].sub.i[member of][LAMBDA]] [A.sub.i]).sup.c] = [[intersection].sub.i[member of][LAMBDA]] [A.sup.c.sub.i]

(ii) [([A.sup.c]).sup.c] = A. [B.sup.c] [??] [A.sup.c], if A [??] B.

Proof. : Part(i): [mathematical expression not reproducible]. Similarly we can prove [([[union].sub.i[member of][LAMBDA]] [A.sub.i]).sup.c] = [[intersection].sub.i[member of][LAMBDA]] [A.sup.c.sub.i] and part(ii).

Generally, in the sense of Chang [6] every fuzzy topology is intuitionistic fuzzy topology as well as neutrosophic topology. The following lemmas show that every intuitionistic fuzzy topology [[tau].sub.i] induce two fuzzy topologies on X and every neutrosophic topology [[tau].sub.n] induce three fuzzy topologies on X.

Lemma 3.5. In an intuitionistic fuzzy topological space (X, [[tau].sub.i]), each of the following collections form fuzzy topologies on X:

(i) [mathematical expression not reproducible]

(ii) [mathematical expression not reproducible]

Proof. : Here we shall prove part (ii) only and similarly we can prove part (i). Clearly [empty set] = (x, 0) and X = (x, 1) are belong to [mathematical expression not reproducible], then [mathematical expression not reproducible] and [mathematical expression not reproducible]. Therefore [mathematical expression not reproducible] and so [mathematical expression not reproducible]. If [{[A.sub.j]}.sup.m.sub.j=1] [member of] [mathematical expression not reproducible], then [mathematical expression not reproducible] and [mathematical expression not reproducible]. Therefore [mathematical expression not reproducible] and so [mathematical expression not reproducible].

Lemma 3.6. In a neutrosophic topological space (X, [[tau].sub.n]), each of the following collections form fuzzy topologies on X:

(i) [mathematical expression not reproducible]

(ii) [mathematical expression not reproducible]

(iii) [mathematical expression not reproducible]

Proof. : Part (i): Clearly [empty set] = (x, 0) and X = (x, 1) are belong to [mathematical expression not reproducible]. If [{[A.sub.j]}.sub.j[member of][LAMBDA]] [member of] [mathematical expression not reproducible], then {([mathematical expression not reproducible] [member of] [[tau].sub.n] and [mathematical expression not reproducible] [member of] [[tau].sub.n]. Therefore [mathematical expression not reproducible] and so [mathematical expression not reproducible]. If [{[A.sub.j]}.sup.m.sub.j=1] [member of] [mathematical expression not reproducible], then [mathematical expression not reproducible] [member of] [[tau].sub.n] and [mathematical expression not reproducible] [member of] [[tau].sub.n]. Therefore [mathematical expression not reproducible] and so [[intersection].sup.m.sub.j=1] [A.sub.j] [member of] [mathematical expression not reproducible]. In similar manner we can prove part (ii) and (iii).

Corollary 3.7. Let A be a neutrosophic set of (X, [[tau].sub.n), then the collection ([[tau].sub.n).sub.A] ={A [intersection] O :O [member of] [[tau].sub.n]} is a neutrosophic topology on A, called the induced neutrosophic topology on A and the pair (A, [([[tau].sub.n]).sub.A]) is called neutrosophic subspace of nts (X, [[tau].sub.n]). The elements of [([[tau].sub.n]).sub.A]) are called [([[tau].sub.n]).sub.A])-open sets and their complements are called [([[tau].sub.n]).sub.A])-closed sets.

Proof. : Obviously [empty set] = (x,min([[mu].sub.A](x), 0),min([[sigma].sub.A](x), 0) ,max([[gamma].sub.A](x), 1)) = A [intersection] [empty set] [member of] [([[tau].sub.n]).sub.A]) and A = (x,min([[mu].sub.A](x), 1),min([[sigma].sub.A](x), 1),max([[gamma].sub.A](x), 0)) = A [intersection] X [member of] [([[tau].sub.n]).sub.A]). Take [{[A.sub.j]}.sub.j[member of][LAMBDA]] [member of] [([[tau].sub.n]).sub.A]), there exist [O.sub.j] [member of] [[tau].sub.n], j [member of] A, such that [A.sub.j] = A [intersection] [O.sub.j] for each j [member of] A. Then [mathematical expression not reproducible] Therefore [([[tau].sub.n]).sub.A]) is closed under arbitrary union on A. If we take [{[A.sub.j]}.sup.m.sub.j=1] [member of] [([[tau].sub.n]).sub.A]), there exist [O.sub.j] [member of] [[tau].sub.n], j = l, 2,..., m, such that [A.sub.j] = A[intersection][O.sub.j] for each j [member of] A. Then [mathematical expression not reproducible]. Therefore [([[tau].sub.n]).sub.A]) is a neutrosophic topology on A.

Corollary 3.8. Let A be a fuzzy set (resp. intuitionistic fuzzy set) of fts (X, [[tau].sub.f]) (resp. ifts (X, [[tau].sub.i])), then the collection [([[tau].sub.f]).sub.A]) = {A [intersection] O : O [member of] [[tau].sub.f]} (resp. [([[tau].sub.i]).sub.A]) = {A [intersection] O : O [member of] [[tau].sub.i]}) is a fuzzy topology (resp. intuitionistic fuzzy topology) on A, called the induced fuzzy topology (resp. induced intuitionistic fuzzy topology) on A and the pair (A, [([[tau].sub.f]).sub.A])) (resp. (A, [([[tau].sub.i]).sub.A]))) is called fuzzy subspace (resp. intuitionistic fuzzy subspace).

Proof. : Proof follows from the above corollory.

Lemma 3.9. Let (A, [([[tau].sub.n]).sub.A])) be a neutrosophic subspace of nts (X, [[tau].sub.n]) and B [??] A. If B is [([[tau].sub.n]).sub.A])-open in (A, [([[tau].sub.n]).sub.A])) and A is neutrosophic open in nts (X, [[tau].sub.n]), then B is neutrosophic open in (X, [[tau].sub.n]).

Proof. : Since B is [([[tau].sub.n]).sub.A])-open in (A, [([[tau].sub.n]).sub.A])), B = A [intersection] O for some neutrosophic open set O in (X, [[tau].sub.n]) and so B is neutrosophic open in (X, [[tau].sub.n]).

Lemma 3.10. Let (A, [([[tau].sub.f]).sub.A])) (resp. (A, [([[tau].sub.i]).sub.A]))) be a fuzzy subspace (resp. intuitionistic fuzzy subspace) of fts (X, [[tau].sub.f]) (resp. of ifts (X, [[tau].sub.i])) and B [??] A. If B is [([[tau].sub.f]).sub.A])-open (resp. [([[tau].sub.i]).sub.A])-open) in (A, [([[tau].sub.f]).sub.A])) (resp. (A, [([[tau].sub.i]).sub.A]))) and A is fuzzy open (resp. intuitionistic fuzzy open) in fts (X, [[tau].sub.f]) (resp. ifts (X, [[tau].sub.i])), then B is fuzzy open (resp. intuitionistic fuzzy open) in (X, [[tau].sub.f]) (resp. ifts (X, [[tau].sub.i])).

Proof. : Proof is similar as above lemma.

Remark 3.11. In classical topology, we know that if (A, [[tau].sub.A]) is a subspace of (X, [tau]) and B [??] A, then

(i) B = A [intersection] F, where F is closed in X if and only if B is closed in A.

(ii) B is closed in X, if B is closed in A and A is closed in X.

The following examples illustrate that these are not true in fuzzy, intuitionistic fuzzy and neutrosophic topological spaces.

Example 3.12. Let X = {a,b,c} with [[tau].sub.n] = {[empty set],X, ((1,1,1), (0,0,0), (0.7,0.7,0.7)), ((0.6,0.6,0.6), (0,0,0), (0, 0, 0)), ((1,1,1), (0, 0, 0), (0, 0, 0)), ((0.6, 0.6,0.6), (0, 0, 0), (0.7,0.7, 0.7))}. Then [([[tau].sub.n]).sup.c] = {X, [empty set], ((0,0, 0), (1,1,1), (0.3,0.3, 0.33)), ((0.4, 0.4, 0.4), (1,1,1), (1,1,1)), ((0, 0, 0), (1,1,1), (1,1,1)), ((0.4,0.4, 0.4), (1,1,1), (0.3,0.3, 0.3))}. Let A = ((0.6, 0.6, 0.2), (1, 0,1), (0.8, 0.7, 0.6)), then [([[tau].sub.n]).sub.A]) = {[empty set], A, ((0.6, 0.6, 0.2), (0, 0, 0), (0.8,0.7, 0.7)), ((0.6, 0.6, 0.2), (0,0, 0), (0.8, 0.7, 0.6))} and [([([[tau].sub.n]).sub.A]).sup.c] = {A, [empty set], ((0, 0, 0), (1, 0,1), (1,1, 0.9)), ((0, 0, 0), (1, 0,1), (1,1,1))}. Clearly B = ((0, 0, 0), (1, 0,1), (1,1, 0.9)) is [([[tau].sub.n]).sub.A])-closed in (A, [([[tau].sub.n]).sub.A])) and B [not equal to] A [intersection] F for every neutrosophic closed set F in (X, [[tau].sub.n]). Since A = ((0, 0, 0), (1, 1, 1), (0.3, 0.3, 0.3)) is neutrosophic closed in (X, [[tau].sub.n]), then [([[tau].sub.n]).sub.A]) = {[empty set], A, ((0, 0, 0), (0, 0, 0), (0.7, 0.7, 0.7)), ((0, 0,0), (0, 0, 0), (0.3,0.3, 0.3))} and [([([[tau].sub.n]).sub.A]).sup.c] = {A, [empty set], ((0, 0, 0), (1,1,1), (0.6, 0.6, 0.6)), ((0, 0, 0), (1,1,1), (1,1,1))}. Clearly B = ((0, 0, 0), (1,1,1), (0.6, 0.6, 0.6)) is [([[tau].sub.n]).sub.A])-closed in (A, [([[tau].sub.n]).sub.A])), but it is not neutrosophic closed in (X, [[tau].sub.n]).

Example 3.13. Let X = {a, b, c} with [[tau].sub.i] = {[empty set], X, ((0.4, 0.4, 0.3), (0.6, 0.6, 0.7)), ((0.3, 0.8, 0.1), (0.7, 0.2, 0.9)), ((0.3, 0.4, 0.1), (0.7, 0.6, 0.9)), ((0.4,0.8, 0.3), (0.6, 0.2, 0.7))}. Then [([[tau].sub.i]).sup.c] = {X, [empty set], ((0.6, 0.6, 0.7), (0.4,0.4, 0.3)), ((0.7, 0.2, 0.9), (0.3, 0.8,0.1)), ((0.7, 0.6, 0.9), (0.3, 0.4, 0.1)), ((0.6, 0.2, 0.7), (0.4, 0.8, 0.3))}. Since A = ((0.7, 0.2,0.9), (0.3, 0.8, 0.1)) is intuitionistic fuzzy closed in (X,[[tau].sub.i]), then [([[tau].sub.i]).sub.A]) = {9, A, ((0.4, 0.2, 0.3), (0.6, 0.8, 0.7)), ((0.3, 0.2, 0.1), (0.7, 0.8,0.9))} and [([([[tau].sub.n]).sub.A]).sup.c] = {A, [empty set], ((0.3, 0, 0.6), (0.7,1, 0.4)), ((0.4, 0, 0.8), (0.6,1, 0.2))}. Clearly B = ((0.3, 0, 0.6), (0.7,1, 0.4)) is [([[tau].sub.i]).sub.A])-closed in (A, [([[tau].sub.i]).sub.A])), but B [not equal to] A [intersection] F for every intuitionistic fuzzy closed set F in (X, [[tau].sub.i]) and B is not intuitionistic fuzzy closed in (X, [[tau].sub.i]).

Example 3.14. Let X = {a, b, c} with[[tau].sub.f] = {[empty set],X, (0.2,0.3,0.1), (0.7,0.1,0.8), (0.2,0.1,0.1), (0.7,0.3,0.8)}. Then [([[tau].sub.f]).sup.c] = {X, [empty set], (0.8, 0.7, 0.9), (0.3, 0.9, 0.2), (0.8, 0.9, 0.9), (0.3,0.7, 0.2)}. Since A = (0.8, 0.7,0.9) is fuzzy closed in (X,[[tau].sub.f]), then [([[tau].sub.f]).sub.A]) = {[empty set], A, (0.2, 0.3,0.1), (0.7, 0.1, 0.8), (0.2, 0.1,0.1), (0.7, 0.3, 0.8)} and [([([[tau].sub.f]).sub.A]).sup.c] = {A, [empty set], (0.6, 0.4, 0.8), (0.1, 0.6,0.1), (0.6, 0.6, 0.8), (0.1, 0.4, 0.1)}. Clearly B = (0.6, 0.6, 0.8) is [([[tau].sub.f]).sub.A])-closed in (A, [([[tau].sub.f]).sub.A])), but B [not equal to] A [intersection] F for every fuzzy closed set F in (X, [[tau].sub.f]) and B is not fuzzy closed in (X,[[tau].sub.f]).

4 Neutrosophic Supra Topological Spaces

In this section, we introduce neutrosophic supra topological spaces and also establish its properties.

Definition 4.1. A subcollection [[tau]*.sub.n] of neutrosophic sets on a non empty set X is said to be a neutrosophic supra topology on X if the sets [empty set], X [member of] [[tau]*.sub.n] and [[intersection].sup.[infinity].sub.i=1] [A.sub.i] [member of] [[tau]*.sub.n], for [{[A.sub.i]}.sup.[infinity].sub.i=1] [member of] [[tau]*.sub.n]. Then (X, [[tau]*.sub.n]) is called neutrosophic supra topological space on X (for short nsts). The members of [[tau]*.sub.n] are known as neutrosophic supra open sets and its complement is called neutrosophic supra closed. A neutrosophic supra topology [[tau]*.sub.n] on X is said to be an associated neutrosophic supra topology with neutrosophic topology [[tau].sub.n] if [[tau].sub.n] [??] [[tau]*.sub.n]. Every neutrosophic topology on X is neutrosophic supra topology on X.

Remark 4.2. The following table illustrates the combarison of fuzzy supra topological spaces, intuitionistic supra topological spaces, neutrosophic supra topological spaces.
Comparison Tabic

S. No   Fuzzy supra topological spaces

1       It deals with fuzzy sets
2       A subcollection [[tau]*.sub.f] of fuzzy sets on a non empty set
        X is said to be a fuzzy supra topology on X if the sets
        [empty set], X [member of] [[tau]*.sub.f] and
        [[union].sup.[infinity].sub.i=1] [A.sub.i] [member of]
        [[tau]*.sub.f], for [{[A.sub.i]}.sup.[infinity].sub.i=1]
        [member of] [[tau]*.sub.f]
3       A non empty set X together with the collection [[tau]*.sub.f],
        is called fuzzy supra topological space on X (for short fsts)
        denoted by the ordered pair (X, [[tau]*.sub.n]).
4       The members of [[tau]*.sub.f] are known as fuzzy supra open
        sets.
5       It is a generalization of classical supra topological spaces.
6       Every fuzzy topology is fuzzy supra topology.

S. No   Intuitionistic supra topological spaces

1       It deals with intuitionistic sets
2       A subcollection [[tau]*.sub.i] of intuitionistic sets on a non
        empty set X is said to be a intuitionistic supra topology on X
        if the sets [empty set], X [member of] [[tau]*.sub.i] and
        [[union].sup.[infinity].sub.i=1] [A.sub.i] [member of]
        [tau].*sub.i], for [{[A.sub.i]}.sup.[infinity].sub.i=1]
        [member of] [[tau]*.sub.i]
3       A non empty set X together with the collection [[tau]*.sub.i] is
        called intuitionistic supra topological space on X (for short
        ists) denoted by the ordered pair (X, [[tau]*.sub.i]).
4       The members of [[tau]*.sub.i] are known as intuitionistic supra
        open sets.
5       It is a generalization of fuzzy supra topological spaces.
6       Every intuitionistic topology is intuitionistic supra topology.

S. No   Neutrosophic supra topological spaces

1       It deals with neutrosophic sets
2       A subcollection [[tau]*.sub.n] of neutrosophic sets on a non
        empty set X is said to be a neutrosophic supra topology on X if
        the sets [empty set],X [member of] [[tau]*.sub.n] and
        [[union].sup.[infinity].sub.i=1] [member of] [[tau]*.sub.n], for
        [{[A.sub.i]}.sup.[infinity].sub.i=1] [member of] [[tau]*.sub.n].
3       A non empty set X together with the collection [[tau]*.sub.n] is
        called neutrosophic supra topological space on X (for short
        nsts) denoted by the ordered pair (X, [[tau]*.sub.n]).
4       The members of [[tau]*.sub.n] are known as neutrosophic supra
        open sets.
5       It is a generalization of intuitionistic supra topological
        spaces.
6       Every neutrosophic topology is neutrosophic supra topology.


Proposition 4.3. The collection [([[tau]*.sub.n]).sup.c] of all neutrosophic supra closed sets in (X, [[tau]*.sub.n]) satisfies: [empty set], X [member of] [([[tau]*.sub.n]).sup.c] and [([[tau]*.sub.n]).sup.c] is closed under arbitrary intersection.

Proof. : The proof is obvious.

Lemma 4.4. As Proposition 3.4, every neutrosophic supra topology [[tau]*.sub.n] induce three fuzzy supra topologies [mathematical expression not reproducible] and [mathematical expression not reproducible] on X.

Definition 4.5. The neutrosophic supra topological interior [mathematical expression not reproducible] and closure [mathematical expression not reproducible] operators of a neutrosophic set A are respectively defined as: [mathematical expression not reproducible] and [mathematical expression not reproducible] and [F.sup.c] [member of] [T*.sub.n]}.

Theorem 4.6. The following are true for neutrosophic sets A and B of nsts (X, [[tau]*.sub.n]):

(i) [mathematical expression not reproducible] if and only if A is neutrosophic supra closed.

(ii) [mathematical expression not reproducible] if and only if A is neutrosophic supra open.

(iii) [mathematical expression not reproducible], if A [??] B.

(iv) [mathematical expression not reproducible], if A [??] B.

(v) [mathematical expression not reproducible] (A [union] B).

(vi) [mathematical expression not reproducible] (A [union] B).

(vii) [mathematical expression not reproducible] (A [intersection] B).

(viii) [mathematical expression not reproducible] (A [intersection] B).

(ix) [mathematical expression not reproducible].

Proof. : Here we shall prove parts (iii), (v) and (ix) only. The remaining parts similarly follows. Part (iii): [mathematical expression not reproducible]. Thus, [mathematical expression not reproducible]. Part (v): Since A [union] B [??] A, B, then [mathematical expression not reproducible]. Part (ix): [mathematical expression not reproducible] is a neutrosophic supra open in X and [mathematical expression not reproducible]. Thus, [mathematical expression not reproducible].

Remark 4.7. In neutrosophic topological space, we have [mathematical expression not reproducible] and [mathematical expression not reproducible]. These equalities are not true in neutrosophic supra topological spaces as shown in the following examples.

Example 4.8. Let X = {a,b,c} with neutrosophic topology [[tau]*.sub.n] = {[empty set], X, ((0.5, l, 0), (0.5, l, 0), (0.5,0, 1)), ((0.25, 0, 1), (0.25, 0, 1), (0.75, l, 0)), ((0.5, l, 1), (0.5, l, 1), (0.5, 0, 0))}. Then [([[tau]*.sub.n]).sup.c] = {X, [empty set], ((0.5, 0, 1), (0.5, 0, 1), (0.5, l, 0)), ((0.75, l, 0), (0.75, l, 0), (0.25, 0, 1)), ((0.5, 0, 0), (0.5, 0, 0), (0.5, l, 1))}. Let C = ((0.5, 0.5, 0), (0.5, 0.5,0), (0.5, 0.5, 1)) and D = ((0.5, 0, 0.5), (0.5, 0, 0.5), (0.5, l, 0.5)), then [mathematical expression not reproducible] and [mathematical expression not reproducible], so [mathematical expression not reproducible]. But C[union]D = ((0.5, 0.5, 0.5), (0.5, 0.5, 0.5), (0.5, 0.5, 0.5)) and [mathematical expression not reproducible]. Therefore [mathematical expression not reproducible].

Let E = ((0.5,1, 0.25), (0.5,1, 0.25), (0.5, 0, 0.75)) and F = ((0.5, 0.5,1), (0.5,0.5,1), (0.5, 0.5, 0)). Then [mathematical expression not reproducible] and [mathematical expression not reproducible], so [mathematical expression not reproducible]. But E[intersection]F = ((0.5, 0.5, 0.25), (0.5, 0.5, 0.25), (0.5,0.5,0.75)) and [mathematical expression not reproducible]. Therefore [mathematical expression not reproducible].

5 Mappings of Neutrosophic Spaces

In this section, we define and establish the properties of some mappings in neutrosophic supra topological spaces and neutrosophic subspaces.

Definition 5.1. Let [[tau]*.sub.n] and [[sigma]*.sub.n] be associated neutrosophic supra topologies with respect to [[tau].sub.n] and [[sigma].sub.n]. A mapping f from a nts (X, [[tau].sub.n]) into nts (Y, [[sigma].sub.n]) is said to be S*-neutrosophic open if the image of every neutrosophic open set in (X, [[tau].sub.n]) is neutrosophic supra open in (Y, [[sigma]*.sub.n]) and f : X [right arrow] Y is said to be S*-neutrosophic continuous if the inverse image of every neutrosophic open set in (Y, [[sigma].sub.n]) is neutrosophic supra open in (X, [[tau]*.sub.n]).

Definition 5.2. Let [[tau]*.sub.n] and [[sigma]*.sub.n] be associated neutrosophic supra topologies with respect to nts's [[tau].sub.n] and [[sigma].sub.n]. A mapping f from a nts (X, [[tau].sub.n]) into a nts (Y, [[sigma].sub.n]) is said to be supra neutrosophic open if the image of every neutrosophic supra open set in (X, [[tau]*.sub.n]) is a neutrosophic supra open in (Y, [[sigma]*.sub.n]) and f : X [right arrow] Y is said to be supra neutrosophic continuous if the inverse image of every neutrosophic supra open set in (Y, [[sigma]*.sub.n]) is neutrosophic supra open in (X, [[tau]*.sub.n]).

A mapping f of nts (X, [[tau].sub.n]) into nts (Y, [[sigma].sub.n]) is said to be a mapping of neutrosophic subspace (A, [([[tau].sub.n]).sub.A])) into neutrosophic subspace (B, [([[sigma].sub.n]).sub.B]) if f (A) [subset] B.

Definition 5.3. A mapping f of neutrosophic subspace (A, [([[tau].sub.n]).sub.A])) of nts (X, [[tau].sub.n]) into neutrosophic subspace (B, [([[sigma].sub.n]).sub.B]) of nts (Y, [[sigma].sub.n]) is said to be relatively neutrosophic continuous if [f.sup.-1](O) [intersection] A [member of] [([[tau].sub.n]).sub.A]) for every O [member of] [([[sigma].sub.n]).sub.B]. If f (O') [member of] [([[sigma].sub.n]).sub.B] for every O' [member of] [([[tau].sub.n]).sub.A]), then f is said to be relatively neutrosophic open.

Theorem 5.4. If a mapping f is neutrosophic continuous from nts (X, [[tau].sub.n]) into nts (Y, [[sigma].sub.n]) and f (A) [subset] B. Then f is relatively neutrosophic continuous from neutrosophic subspace (A, [([[tau].sub.n]).sub.A])) of nts (X, [[tau].sub.n]) into neutrosophic subspace (B, [([[sigma].sub.n]).sub.B]) of nts (Y, [[sigma].sub.n]).

Proof. : Let O [member of] [([[sigma].sub.n]).sub.B], then there exists G [member of] [[sigma].sub.n] such that O = B [intersection] G and [f.sup.-1](G) [member of] [[tau].sub.n]. Therefore [f.sup.-1] (O) [intersection] A = [f.sup.-1](B) [intersection] [f.sup.-1] (G) [intersection] A = [f.sup.-1](G) [intersection] A [member of] [([[tau].sub.n]).sub.A]).

Remark 5.5. (i) Every neutrosophic continuous (resp. neutrosophic open) mapping is S*-neutrosophic continuous (resp. S*-neutrosophic open), but converse need not be true.

(ii) Every supra neutrosophic continuous (resp. supra neutrosophic open) mapping is S*-neutrosophic continuous (resp. S*-neutrosophic open), but converse need not be true.

(iii) Supra neutrosophic continuous and neutrosophic continuous mappings are independent each other.

(iv) Supra neutrosophic open and neutrosophic open mappings are independent each other.

Proof. : The proof follows from the definition, the converse and independence are shown in the following example.

Example 5.6. Let Y = {x,y,z}, X = {a,b,c} with neutrosophic topologies [[tau].sub.n] = {[empty set],X, ((0.5, 0.5, 0), (0.5, 0.5,0), (0.5, 0.5, 1))} and [[sigma].sub.n] = {0,Y, ((0.5, 0.25,0.5), (0.5, 0.25, 0.5), (0.5, 0.75, 0.5))}. Let [[tau]*.sub.n] = {[empty set], X, ((0.5, 0.5, 0), (0.5, 0.5, 0), (0.5, 0.5, 1)), ((0.5, 0.5, 0.5), (0.5, 0.5, 0.5), (0.5, 0.5,0.5)), ((0.5, 0.25, 0.5), (0.5,0.25, 0.5), (0.5, 0.75, 0.5))} and [[sigma]*.sub.n] = [[sigma].sub.n] be associated neutrosophic supra topologies with respect to [[tau].sub.n] and [[sigma].sub.n]. Define a mapping f : X [right arrow] Y by f(c) = z,f(b) = y,f(a) = x, then [f.sup.-1](((0.5,0.25,0.5), (0.5,0.25,0.5), (0.5, 0.75, 0.5))) = ((0.5,0.25, 0.5), (0.5, 0.25, 0.5), (0.5, 0.75, 0.5)) [member of] [[tau]*.sub.n]. Clearly f is supra neutrosophic continuous and S* -neutrosophic continuous but not neutrosophic continuous.

Let Y = {x, y, z}, X = {a, b, c} with neutrosophic topologies [[tau].sub.n] = {[empty set], X, ((0.5, 0.5, 0), (0.5, 0.5, 0), (0.5, 0.5, 1)), ((0.5, 0.25, 0), (0.5, 0.25, 0), (0.5, 0.75, 1))} and [[sigma].sub.n] = {[empty set],Y, ((0.5, 0.25, 0), (0.5, 0.25, 0), (0.5, 0.75, 1))}. Let [[tau]*.sub.n] = {[empty set], X, ((0.5, 0.5, 0), (0.5, 0.5, 0), (0.5, 0.5, 1)), ((0.5, 0.25, 0), (0.5, 0.25, 0), (0.5, 0.75, 1)), ((1, 0.5, 0), (1, 0.5, 0), (0, 0.5, 1))} and [[sigma]*.sub.n] = {[empty set], Y, ((0.5, 0.25, 0), (0.5, 0.25, 0), (0.5, 0.75, 1)), ((0.3, 0.25, 0.5), (0.3, 0.25, 0.5), (0.7, 0.75, 0.5)), ((0.5, 0.25, 0.5), (0.5,0.25, 0.5), (0.5, 0.75, 0.5))} be associated neutrosophic supra topologies with respect to [[tau].sub.n] and [[sigma].sub.n]. Consider a mapping f : X [right arrow] Y by f(c) = z, f(b) = y, f(a) = x, then [f.sup.-1](((0.5, 0.25, 0.5), (0.5, 0.25, 0.5), (0.5, 0.75,0.5))) = ((0.5,0.25, 0.5), (0.5, 0.25, 0.5), (0.5, 0.75, 0.5)) [??] [[tau]*.sub.n]. Therefore f is neutrosophic continuous and S*-neutrosophic continuous but not supra neutrosophic continuous. If we consider a mapping g : Y [right arrow] X by g(z) = c, g(y) = b,g(x) = a, then g is neutrosophic open and S -neutrosophic open but not supra neutrosophic open.

Let Y = {x, y, z}, X = {a, b, c} with neutrosophic topologies [[tau].sub.n] = {0, X, ((1, 0.5, 0.3), (1, 0.5, 0.33), (0,0.5, 0.7))} and [[sigma].sub.n] = {[empty set],Y, ((1, 0.3,0.5), (1, 0.33, 0.5), (0, 0.7,0.5))}. Let [[sigma]*.sub.n] = {[empty set], Y, ((1, 0.3, 0.5), (1, 0.3, 0.5), (0, 0.7, 0.5)), ((1, 0.5, 0.3), (1 , 0.5, 0.3), (0, 0.5, 0.7)), ((1, 0.5, 0.5), (1, 0.5, 0.5), (0, 0.5, 0.5))} and [[tau]*.sub.n] = [[tau].sub.n] be associated neutrosophic supra topologies with respect to [[sigma].sub.n] and [[tau].sub.n]. Then f : X [right arrow] Y defined by f(c) = z, f(b) = y, f(a) = x is S*-neutrosophic open and supra neutrosophic open but not neutrosophic open.

Observation 5.7. The following are the examples of contradicting the statements of Abd-Monsef and Ramadan [9]. In fuzzy supra topological space, consider Y = {x,y,z}, X = {a,b,c} with fuzzy topologies [[tau].sub.f] = {[empty set],X, (0.5, 0.5, 0), (0.5, 0.25, 0)} and [[sigma].sub.f] = {[empty set],Y, (0.5, 0.25, 0)}. Let [[tau]*.sub.f] = {[empty set],X, (0.5, 0.5, 0), (0.5, 0.25, 0), (1, 0.5, 0)} and [[sigma]*.sub.f] = {[empty set],Y, (0.5, 0.25,0), (0.3, 0.25,0.5), (0.5, 0.25, 0.5)} be associated fuzzy supra topologies with respect to [[tau].sub.f] and [[sigma].sub.f]. Consider a mapping h : X [right arrow] Y by h(c) = z, h(b) = y, h(a) = x, then [h.sup.-1]((0.5, 0.25, 0.5)) = (0.5, 0.25, 0.5) [??] [[tau]*.sub.f]. Then h is fuzzy continuous but not supra fuzzy continuous. If we define a mapping g : Y [right arrow] X by g(z) = c, g(y) = b, g(x) = a, then g is fuzzy open but not supra fuzzy open.

Theorem 5.8. The following statements are equivalent for the mapping f of nts (X, [[tau].sub.n]) into nts (Y, [[sigma].sub.n]):

(i) The mapping f : X [right arrow] Y is S*-neutrosophic continuous.

(ii) The inverse image of every neutrosophic closed set in (Y, [[sigma].sub.n]) is neutrosophic supra closed in (X, [[tau]*.sub.n]).

(iii) For each neutrosophic set A in Y, [mathematical expression not reproducible].

(iv) For each neutrosophic set B in X, [mathematical expression not reproducible].

(v) For each neutrosophic set A in Y, [mathematical expression not reproducible].

Proof.: (i) [??] (ii): Let f be a S* -neutrosophic continuous and A be a neutrosophic closed set in (Y,[[sigma].sub.n]), [f.sup.-1](Y - A) = X - [f.sup.-1](A) is neutrosophic supra open in (X, [[tau].*sub.n]) and so [f.sup.-1] (A) is neutrosophic supra closed in (X, [[tau].*sub.n])..

(ii) [??] (Hi): [mathematical expression not reproducible] (A) is neutrosophic closed in (Y, [[sigma].sub.n]), for each neutrosophic set A in Y, then [mathematical expression not reproducible] is neutrosophic supra closed in (X,[[tau].*sub.n]). Thus [mathematical expression not reproducible].

(iii) [??] (iv): [mathematical expression not reproducible], for each neutrosophic set B in X and so [mathematical expression not reproducible].

(iv) [??] (ii): Let B = [f.sup.-1](A), for each neutrosophic closed set A in Y, then [mathematical expression not reproducible] and [mathematical expression not reproducible]. Therefore B = [f.sup.-1](A) is neutrosophic supra closed in X.

(ii) [??] (i): Let A be a neutrosophic open set in Y, then X - [f.sup.-1](A) = [f.sup.-1](Y - A) is neutrosophic supra closed in X, since Y - A is neutrosophic closed in Y. Therefore [f.sup.-1](A) is neutrosophic supra open in X.

(i) [??] (v): [mathematical expression not reproducible] is neutrosophic supra open in X, for each neutrosophic set A in Y and [mathematical expression not reproducible].

(v) [??] (i): [mathematical expression not reproducible], for each neutrosophic open set A in Y and so [f.sup.-1](A) is neutrosophic supra open in X.

Theorem 5.9. The following statements are equivalent for the mapping f of nts (X, [[tau].sub.n]) into nts (Y, [[sigma].sub.n]):

(i) A mapping f : (X, [[tau].*sub.n]) [right arrow] (Y, [[sigma]*.sub.n]) is neutrosophic supra continuous.

(ii) The inverse image of every neutrosophic supra closed set in (Y, [[sigma].*sub.n]) is neutrosophic supra closed in (X,[[tau].*sub.n]).

(iii) For each neutrosophic set A in Y, [mathematical expression not reproducible].

(iv) For each neutrosophic set B in X, [mathematical expression not reproducible].

(v) For each neutrosophic set A in Y, [mathematical expression not reproducible].

Proof. : The proof is straightforward from theorem 5.8.

Theorem 5.10. If f : X [right arrow] Y is S*-neutrosophic continuous and g : Y [right arrow] Z is neutrosophic continuous, then g [omicron] f : X [right arrow] Z is S*-neutrosophic continuous.

Proof. : The proof follows directly from the definition.

Theorem 5.11. If f : X [right arrow] Y is supra neutrosophic continuous and g : Y [right arrow] Z is S* -neutrosophic continuous (or neutrosophic continuous), then g [omicron] f : X [right arrow] Z is S*-neutrosophic continuous.

Proof. : It follows from the definition.

Theorem 5.12. If f : X [right arrow] Y and g : Y [right arrow] Z are supra neutrosophic continuous (resp. supra neutrosopic open) mappings, then g [omicron] f : X [right arrow] Z is supra neutrosophic continuous (resp. supra neutrosophic open).

Proof. : The proof follows obviously from the definition.

Remark 5.13. Abd-Monsef and Ramadan[9] stated that if g : X - Y is supra fuzzy continuous and h : Y [right arrow] Z is fuzzy continuous, then h [omicron] g : X [right arrow] Z is supra fuzzy continuous. But in general this is not true, for example consider Z = {p,q,r}, Y = {x,y,z}, and X = {a,b,c} with fuzzy topologies [[tau].sub.f] = {[phi],X, (1, 0.5, 0), (0.33, 0.33, 0)}, [[sigma].sub.f] = {[phi],Y, (0.5, 0.5, 0), (0.5,0.25, 0)} and [[eta].sub.f] = {[phi],Z, (0.5, 0.25, 0)} on X,Y and Zrespectively. Let [[tau]*.sub.f] = {[phi],X, (0.5, 0.5, 0), (0.5, 0.25, 0), (1, 0.5, 0), (0.3, 0.3, 0)}, [[sigma]*.sub.f] = {[phi], Y, (0.5, 0.5, 0), (0.5, 0.25, 0), (1, 0.5, 0)} and [[eta]*.sub.f] = {[phi], Z, (0.5, 0.25, 0), (0.3, 0.25, 0.5), (0.5, 0.25, 0.5)} be associated fuzzy supra topologies with respect to [[tau].sub.f], [[sigma].sub.f] and [[eta].sub.f]. Then the mapping g : X [right arrow] Y defined by g(c) = z,g(b) = y,g(a) = x is supra fuzzy continuous and the mapping h : Y [right arrow] Z by h(z) = r,h(y) = q,h(x) = p is fuzzy continuous. But h [omicron] g : X [right arrow] Z is not supra fuzzy continuous, since [(g [omicron] h).sup.-1]((0.3, 0.25, 0.5)) = (0.3, 0.25, 0.5) [??] [[tau]*.sub.f].

Remark 5.14. In general the composition of two supra neutrosophic continuous mappings is again supra neutrosophic continuous, but the composition of two S*-neutrosopic continuous mappings may not be S*neutrosophic continuous. Let Z = {p, q, r}, Y = {x, y, z}, and X = {a, b, c} with neutrosophic topologies [[tau].sub.n] = {[phi],X, ((0.5, 0.5, 0), (0.5, 0.5, 0), (0.5, 0.5, 1))}, [[sigma].sub.n] = {[phi],Y, ((0.5, 0.25,0.5), (0.5, 0.25, 0.5), (0.5, 0.75, 0.5))} and [[eta].sub.n] = {[phi], Z, ((0.3, 0.7, 0.5), (0.3, 0.7,0.5), (0.7, 0.3, 0.5))} on X, Y and Z respectively. Let [[tau].*sub.n] = {[phi], X, ((0.5, 0.5, 0), (0.5, 0.5, 0), (0.5, 0.5, 1)), ((0.5, 0.5, 0.5), (0.5, 0.5, 0.5), (0.5, 0.5, 0.5)), ((0.5, 0.25, 0.5), (0.5, 0.25, 0.5), (0.5,0.75, 0.5))} and [[sigma]*.sub.n] = {[phi],Y, ((0.5, 0.25,0.5), (0.5, 0.25, 0.5), (0.5, 0.75, 0.5)), ((0.3, 0.7, 0.5), (0.3, 0.7, 0.5), (0.7, 0.3, 0.5)), ((0.5, 0.7, 0.5), (0.5, 0.7, 0.5), (0.5, 0.3, 0.5))} be associated neutrosophic supra topologies with respect to [[tau].sub.n] and [[sigma].sub.n]. Then the mappings f : X [right arrow] Y and g : Y [right arrow] Z are defined respectively by f (c) = z,f (b) = y,f (a) = x and g(z) = r,g(y) = q, g(x) = p are S*-neutrosophic continuous. But g [omicron] f : X [right arrow] Z is not S*-neutrosophic continuous, since [(g [omicron] f).sup.-1](((0.3,0.7, 0.5), (0.3, 0.7, 0.5), (0.7, 0.3,0.5))) = ((0.3,0.7, 0.5), (0.3, 0.7, 0.5), (0.7, 0.3, 0.5)) [??] [[tau]*.sub.n].

Theorem 5.15. If mappings f : (A, [([[tau].sub.n]).sub.A]) [right arrow] (B, [([[sigma].sub.n]).sub.B]) from neutrosophic subspace (A, [([[tau].sub.n]).sub.A]) of nts (X,[[tau].sub.n]) into neutrosophic subspace (B, [([[sigma].sub.n]).sub.B]) of nts (Y,[[sigma].sub.n]) and g : (B, [([[sigma].sub.n]).sub.B]) - (C, [([[eta].sub.n]).sub.C]) from neutrosophic subspace (B, [([[sigma].sub.n]).sub.B]) of nts (Y, [[sigma].sub.n]) into neutrosophic subspace (C, [([[eta].sub.n])c.sub.B]) of nts (Z,[[eta].sub.n]) are relatively neutrosophic continuous (resp. relatively neutrosopic open) mappings, then g [omicron] f : (A, [([[tau].sub.n]).sub.A]) [right arrow] (C, [([[eta].sub.n]).sub.A]C]) is relatively neutrosophic continuous (resp. relatively neutrosophic open) from neutrosophic subspace (A, [([[tau].sub.n]).sub.A)] of nts (X, [[tau].sub.n]) into neutrosophic subspace (C, ([[eta].sub.n])c) of nts (Z, [[eta].sub.n]).

Proof. : Let O [member of] [([[eta].sub.n]).sub.C], then [g.sub.-1](O) [intersection] B [member of] ([[sigma].sub.n])B and [f.sup.-1]([g.sup.-1](O) [intersection] B) [intersection] A [member of] [([[tau].sub.n]).sub.A]. Since B [contains] f (A), then [(g [omicron] f).sup.-1](O) [intersection] A = [f.sup.-1]([g.sup.-1](O) [intersection] B) [intersection] A. Therefore g [omicron] f is relatively neutrosophic continuous. Let U [member of] [([[tau].sub.n]).sub.A], then f (U) [member of] [([[sigma].sub.n]).sub.B] and g(f (U)) = (g [omicron] f)(U) [member of] [([[eta].sub.n]).sub.C]. Therefore g [omicron] f is relatively neutrosophic open.

6 Neutrosophic Supra Topology in Data Mining

In this section, we present a methodical approach for decision-making problem with single valued neutrosophic information. The following necessary steps are proposed the methodical approach to select the proper attributes and alternatives in the decision-making situation. Step 1: Problem field selection:

Consider multi-attribute decision making problems with m attributes [A.sub.1],[A.sub.2]..., [A.sub.m] and n alternatives [C.sub.1], [C.sub.2],[C.sub.n] and p attributes [D.sub.1],[D.sub.2],...,[D.sub.p], (n [less than or equal to] p).
Here all the attributes [a.sub.ij] and [d.sub.ki] (i= 1,2,..., j =
1,2,...,n and k = 1,2,..., p) are single valued neutrosophic numbers.

            [C.sub.1]      [C.sub.2]      *   *   *   [C.sub.n]
[A.sub.1]   ([a.sub.11])   ([a.sub.12])   *   *   *   ([a.sub.1n])
[A.sub.2]   ([a.sub.21])   ([a.sub.22])   *   *   *   ([a.sub.2n])
*           *              *              *   *   *   *
*           *              *              *   *   *   *
*           *              *              *   *   *   *
[A.sub.m]   ([a.sub.m1])   ([a.sub.m2])   *   *   *   ([a.sub.mn])

            [A.sub.1]      [A.sub.2]      *   *   *   [A.sub.m]
[D.sub.1]   ([D.sub.11])   ([D.sub.12])   *   *   *   ([D.sub.1m])
[D.sub.2]   ([D.sub.21])   ([D.sub.22])   *   *   *   ([D.sub.2m])
*           *              *              *   *   *
*           *              *              *   *   *
*           *              *              *   *   *
[D.sub.p]   [D.sub.p1]     [D.sub.p2]     *   *   *   [D.sub.pm]


Step 2: Form neutrosophic supra topologies for ([C.sub.j]) and ([D.sub.k]):

(i) [[tau].*sub.j] = A [union] B, where A = {[1.sub.N], [0.sub.N], [a.sub.1j], [a.sub.2j],..., [a.sub.mj]} and B = {[a.sub.1j] [union] [a.sub.2j], [union] [a.sub.3j],[a.sub.m-1j] [union] [a.sub.mj]}.

(ii) [[nu].*sub.k] = C [union] D, where C = {[1.sub.N], [0.sub.N], [s.sub.k1], [d.sub.k2],..., [d.sub.km]} and D = {[d.sub.k1] [union] [d.sub.k2], [d.sub.k1] [union] [d.sub.k3],...,[d.sub.km-1] [union] [d.sub.km]}.

Step 3: Find Single valued neutrosophic score functions:

Single valued neutrosophic score functions (shortly SVNSF) of A, B, C, D, [C.sub.j] and [D.sub.k] are defined as follows.

(i) SVNSF(A) = [1/3(m+2)][[SIGMA].sup.m+2.sub.i=1][2+[[mu].sub.i] - [[sigma].sub.i] - [[gamma].sub.i]]], and SVNSF(B) = [1-3q][[SIGMA].sup.q.sub.i=1]2 + [[mu].sup.i] - [[sigma].sup.i] - [[gamma].sub.i]]], where q is the number of elements of B. For j = 1, 2,...,n,

[mathematical expression not reproducible]

(ii) SVNSF(C) = [1/[3(m+2)]][[SIGMA].sup.m+2.sub.i=1][2 + [[mu].sub.i] -[[sigma].sup.i] - [[gamma].sub.i]]] and SVNSF(D) = [[1/3r]] [[SIGMA].sup.i] [[sigma].sup.i] - [[sigma].sup.i] - [[gamma].sub.i]]], where r is the number of elements of D. For k = 1, 2,..., p,

[mathematical expression not reproducible]

Step 4: Final Decision

Arrange single valued neutrosophic score values for the alternatives [C.sub.1] [less than or equal to] [C.sub.2] [less than or equal to] ... [less than or equal to] [C.sub.n] and the attributes [D.sub.1] [less than or equal to] [D.sub.2] [less than or equal to] ... [less than or equal to] [D.sub.p]. Choose the attribute [D.sub.p] for the alternative [C.sub.1] and [D.sub.p-1] for the alternative [C.sub.2] etc. If n<p, then ignore [D.sub.k], where k = 1, 2,...,n - p.

7 Numerical Example

Medical diagnosis has increased volume of information available to physicians from new medical technologies and comprises of uncertainties. In medical diagnosis, very difficult task is the process of classifying different set of symptoms under a single name of a disease. In this section, we exemplify a medical diagnosis problem for effectiveness and applicability of above proposed approach.

Step 1: Problem field selection:

Consider the following tables giving informations when consulted physicians about four patients [P.sub.1],[P.sub.2],[P.sub.3], [P.sub.4] and symptoms are Temperature, Cough, Blood Plates, Joint Pain, Insulin. We need to find the patient and to find the disease such as Tuberculosis, Diabetes, Chikungunya, Swine Flu, Dengue of the patient. The data in Table 1 are explained by the membership, the indeterminacy and the non-membership functions. From Table 2, we can observe that for tuberculosis, cough is high ([micro] = 0.9, [sigma] = 0.1,[gamma] = 0.1), but for chikungunya, cough is low ([micro] = 0,[sigma] = 0.1,[gamma] = 0.9).

Step 2: Form neutrosophic supra topologies for ([C.sub.j]) and ([D.sub.k]):

(i) [[tau].*sub.1] = A [union] B, where A = {(1, 1, 0), (0,0, 1), (0.8, 0, 0.2), (0.1, 0.2, 0.7), (0.4, 0.2, 0.5), (0.3, 0.2, 0.5)} and B = {(0.8, 0.2, 0.2)}.

(ii) [[tau].*sub.2] = A [union] B, where A = {(1, 1, 0), (0,0, 1), (0.1, 0, 0.7), (0.1, 0.1, 0.8), (0.2, 0.1, 0.6), (0.4, 0.2, 0.5), (0.9, 0,0.1)} and B = {(0.1, 0.1, 0.7), (0.9, 0.1, 0.1), (0.9, 0.2, 0.1)}.

(iii) [[tau].*sub.3] = A [union] B, where A = {(1, 1, 0), (0,0, 1), (0.9, 0.1, 0), (0, 0.3, 0.7), (0.3, 0.1, 0.6), (0.9, 0, 0.1), (0.2, 0.1, 0.7)} and B = {(0.9, 0.3, 0), (0.3, 0.3, 0.6), (0.9, 0.3, 0.1), (0.2, 0.3, 0.7), (0.9, 0.1, 0.1)}.

(iv) [[tau].*sub.4] = A [union] B, where A = {(1, 1, 0), (0,0, 1), (0, 0.1, 0.9), (0.8, 0.1, 0.2), (0.3, 0.1, 0.6), (0.2, 0.2, 0.7), (0.4, 0.3, 0.2)} and B = {(0.8, 0.2, 0.2), (0.8, 0.3, 0.2), (0.3, 0.2, 0.6)}.

(i) [[nu].*sub.1]= C [union] D, where C = {(1, 1, 0), (0,0, 1), (0.6, 0.3, 0.1), (0.9, 0.1, 0.1), (0, 0.2, 0.8), (0, 0.1, 0.8), (0, 0.1, 0.9)} and D = {(0.9, 0.3, 0.1), (0.9,0.2, 0.1)}.

(ii) [[nu].*sub.2] = C [union] D, where C = {(1,1,0), (0, 0,1), (0.1,0.1, 0.8), (0.2, 0.2, 0.1), (0.1,0.4, 0.6), (0.9, 0, 0.1)} and D = {(0.9, 0.1, 0.1), (0.2, 0.4,0.1), (0.9, 0.2, 0.1), (0.9, 0.4,0.1)}.

(iii) [[nu].*sub.3] = C [union] D, where C = {(1,1, 0), (0, 0,1), (0.9, 0, 0.1), (0, 0.1, 0.9), (0.7,0.2, 0.1), (0.9, 0.1, 0.1), (0.2, 0, 0.8)} and D = {(0.9, 0.2,0.1), (0.2, 0.1, 0.8)}.

(iv) [[nu].*sub.4] = C [union] D, where C = {(1,1,0), (0, 0,1), (0.2,0.5, 0.3), (0.1, 0.4, 0.3), (0.2,0.4, 0.3), (0.2, 0.4, 0.1), (0.1, 0.3, 0.5)} and D = {(0.2, 0.5,0.1)}.

(v) [[nu].*sub.k] = C [union] D, where C = {(1,1,0), (0, 0,1), (0.9,0, 0.1), (0.2, 0.6, 0.4), (0.3,0.1, 0.6), (0.2, 0.1, 0.7)} and D = {(0.9, 0.6, 0.1), (0.9, 0.1,0.1), (0.3, 0.6, 0.4)}.

Step 3: Find Single valued neutrosophic score functions:

(i) SVNSF(A) = 0.5611 and SVNSF(B) = 0.8, where q = 1. SVNSF([C.sub.1]) = 0.6801.

(ii) SVNSF(A) = 0.5524 and SVNSF(B) = 0.7333, where q = 3. SVNSF([C.sub.2]) = 0.6428.

(iii) SVNSF(A) = 0.6 and SVNSF(B) = 0.6933, where q = 5. SVNSF([C.sub.3]) = 0.6466.

(iv) SVNSF(A) = 0.5381 and SVNSF(B) = 0.6888, where q = 3. SVNSF([C.sub.4]) = 0.6135.

(i) SVNSF(C) = 0.5238 and SVNSF(D) = 0.85, where r = 2. SVNSF([D.sub.1]) = 0.6869.

(ii) SVNSF(C) = 0.5555 and SVNSF(D) = 0.7833, where r = 4. SVNSF([D.sub.2]) = 0.6694.

(iii) SVNSF(C) = 0.6333 and SVNSF(B) = 0.65, where r = 2. SVNSF([D.sub.3]) = 0.6416.

(iv) SVNSF(C) = 0.4888 and SVNSF(B) = 0.5333, where r = 1. SVNSF([D.sub.4]) = 0.5111.

(v) SVNSF(C) = 0.5555 and SVNSF(B) = 0.6888, where r = 3. SVNSF([D.sub.5]) = 0.6222.

Step 4: Final Decision:

Arrange single valued neutrosophic score values for the alternatives [C.sub.1],[C.sub.2],[C.sub.3], [C.sub.4] and the attributes [D.sub.1], [D.sub.2],[D.sub.3],[D.sub.4],[D.sub.5] in acending order. We get the following sequences [C.sub.4] [less than or equal to] [C.sub.2] [less than or equal to] [C.sub.3] [less than or equal to] [C.sub.1] and [D.sub.4] [less than or equal to] [D.sub.5] [less than or equal to] [D.sub.3] [less than or equal to] [D.sub.2] [less than or equal to] [D.sub.1]. Thus the patient [P.sub.4] suffers from tuberculosis, the patient [P.sub.2] suffers from diabetes, the patient [P.sub.3] suffers from chikungunya and the patient [P.sub.1] suffers from dengue.

8 Conclusion and Future Work

Neutrosophic topological space is one of the research areas in general fuzzy topological spaces to deal the concept of vagueness. This paper introduced neutrosophic supra topological spaces and its real life application. Moreover we have discussed some mappings in neutrosophic supra topological spaces and derived some contradicting examples in fuzzy supra topological spaces. This theory can be develop and implement to other research areas of general topology such as rough topology, digital topology and so on.

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Received: Juanary 27 2019. Accepted: March 28, 2019

G. Jayaparthasarathy (1), V.F. Little Flower (2), M. Arockia Dasan (3,*)

(1,2,3) Department of Mathematics, St.Jude's College, Thoothoor, Kanyakumari-629176, Tamil Nadu, India. (Manonmaniam Sundaranar University Tirunelveli-627 012, Tamil Nadu, India).

(1) E-mail: jparthasarathy123@gmail.com

(2) E-mail: visjoy05796@gmail.com

(3) E-mail: dassfredy@gmail.com

(*) Correspondence: M.Arockia Dasan (dassfredy@gmail.com)
Table 1.

Patients/Symptoms  [P.sub.1]       [P.sub.2]       [P.sub.3]

Temperature        (0.8,0,0.2)     (0.1,0,0.7)     (0.9,0.1,0)
Cough              (0.1,0.2,0.7)   (0.1,0.1,0.8)   (0,0.3,0.7)
Blood Plates       (0.8,0,0.2)     (0.2,0.1,0.6)   (0.3,0.1,0.6)
Joint Pain         (0.4,0.2,0.5)   (0.4,0.2,0.5)   (0.9,0,0.1)
Insulin            (0.3,0.2,0.5)   (0.9,0,0.1)     (0.2,0.1,0.7)

Patients/Symptoms   [P.sub.4]

Temperature         (0,0.1,0.9)
Cough               (0.8,0.1,0.2)
Blood Plates        (0.3,0.1,0.6)
Joint Pain          (0.2,0.2,0.7)
Insulin             (0.4,0.3,0.2)

Table 2.

Symptoms/Diagnosis   Temperature     Cough           Blood Plates

Tuberculosis        (0.6,0.3,0.1)   (0.9,0.1,0.1)   (0,0.2,0.8)
Diabetes            (0.1,0,1,0.8)   (0.1,0.1,0.8)   (0.2,0.2,0.1)
Chikungunya         (0.9,0,0.1)     (0,0.1,0.9)     (0.7,0.2,0.1)
Swine Flu           (0.2,0.5,0.3)   (0.1,0.4,0.3)   (0.2,0.4,0.1)
Dengue              (0.9,0,0.1)     (0.2,0.6,0.4)   (0.2,0.6,0.4)

Symptoms/Diagnosis    Joint Pain      Insulin

Tuberculosis         (0,0.1,0.8)     (0,0.1,0.9)
Diabetes             (0.1,0.4,0.6)   (0.9,0,0.1)
Chikungunya          (0.9,0.1,0.1)   (0.2,0,0.8)
Swine Flu            (0.1,0.3,0.5)   (0.2,0.4,0.1)
Dengue               (0.3,0.1,0.6)   (0.2,0.1,0.7)
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Author:Jayaparthasarathy, G.; Flower, V.F. Little; Dasan, M. Arockia
Publication:Neutrosophic Sets and Systems
Date:Aug 27, 2019
Words:11563
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