Further Theory of Neutrosophic Triplet Topology and Applications

In this paper we study and develop the Neutrosophic Triplet Topology (NTT) that was recently introduced by Sahin et al. Like classical topology, the NTT tells how the elements of a set relate spatially to each other in a more comprehensive way using the idea of Neutrosophic Triplet Sets. This article is important because it opens new ways of research resulting in many applications in different disciplines, such as Biology, Computer Science, Physics, Robotics, Games and Puzzles and Fiber Art etc. Herein we study the application of NTT in Biology. The Neutrosophic Triplet Set (NTS) has a natural symmetric form, since this is a set of symmetric triplets of the form , , where  and  are opposites of each other, while , being in the middle, is their axis of symmetry. Further on, we obtain in this paper several properties of NTT, like bases, closure and subspace. As an application, we give a multicriteria decision making for the combining effects of certain enzymes on chosen DNA using the developed theory of NTT.


Introduction
The main aim of the paper is to introduce the Neutrosophic Triplet Topology (NTT) in various fields of research, due to its great potential of applicability. However, in order to do so, we first study its theoretical properties, such as open and closed sets, base and subspace, all extended from classical topology and neutrosophic topology to (NTT). In daily life we are witnessing many situations in which the role of neutralities is very important. To control neutralities Smarandache initiated the theme of neutrosophic logic in 1995, which later on proved to be a very handy tool to capture uncertainty. Thus Smarandache [1], generalizes almost all the existing logics like, fuzzy logic, intutionistic fuzzy logic etc. After this many reserchers used neutrosophic sets and logic in algebra, such as Kandasamy et al. [2][3][4], Agboola et al. [5][6][7][8], Ali et al. [9][10][11][12], Gulistan et al. [13][14][15]. More recently Smarandache et al. [16,17] introduced the idea of NT group which open a new research direction. Zhang et al. [18], Bal et al. [19], Jaiyeola el al. [20], Gulistan et al. [21] used NT set in different directions.
Thus in this aricle, we further extended the theory of NT topology. We study some basic properties of NTT where we introduce NT base, NT closure and NT subspace and investigate these topological notions. Moreover, as an application, we give a multicriteria decision making for the combining effects of certain enzymes on chosen DNA.

Preliminaries
In this section we recall some helpful material from [1,16] and for basics of topology we refer the reader [22].   [16] "Let H be a set together with a binary operation . Then H T is called a NT set if for any ∈H, there exist a neutral of " " called neut( ), different from the classical algebraic unitary element, and an opposite of " " called anti( ), with neut( ) and anti( ) belonging to H, such that: neut( ) = neut( ) = and anti( ) = anti( ) = neut( )."

Neutrosophic Triplet Topology (NTT)
In this section, we study NTT in detail.

Definition 3. [30]
Let H T be a NT set and let H τ be a non-empty subset of P (H T ). If H τ satisfy the following conditions: The intersection of a finite number of sets in H τ is also in H τ , • The union of an arbitrary number of sets in H τ is also in H τ .
then H τ is called a NTT.
)} be the set of triplets of H. Then Example 6. Let H T be a NT set having more than one element as a triplet element then any topology on H T is finer than the NT indiscrete topology on H T and coarser than the NT discrete topology on H T .
The intersection of two NT topologies is always a NTT while the union of two NT topologies is not in general a NTT as shown in the following example.

Neutrosophic Triplet Bases of Neutrosophic Triplet Topology (NTT)
In this section, we define and study bases of a NTT for generating NT topologies.
Conversly, suppose that for each Thus Therefore Thus H (O) is a union of members of H (β) and therefore H (β) is a NT bases for τ.

Theorem 2.
A family H(β) of NT subsets of a neutrosophic triplet set(NTS) H T is a NT bases for some NTT on H T if and only if the following conditions are satisfied: (2) For any H( 1 ),H( 2 ) belonging to H(β) the intersection is a union of members of H(β) . Equivalently, for each Conversly, Suppose that both conditions (1) and (2) is also a union of members of H (β) . Hence Also, if we take the union of empty class of members of H (β) we note that φ ∈ H τ . Hence H τ is a topology on H T . Since each member of H τ is a union of members of H (β) by definition, H (β) is a NT basis for H τ .

Neutrosophic Triplet Closure
In this section, we define NT closure of neutrosophic triplet topological space.
Next we will find H( 2 ).
H T and H T are both closed sets and therefore H T = H T by Theorem 3.

Remark 4. The equality
does not hold in general.

Neutrosophic Triplet Subspace
In this section, we define the NT subspace.

Applications
In Mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. Like topology, the NTT tells how elements of a set relate spatially to each other in a more comprehensive way using the idea of Neutrosophic triplet sets. It has many application in different disciplines, Biology, Computer science, Physics, Robotics, Games and Puzzles and Fiber art etc.
Here we study the application of NTT in Biology.
Suppose that we have a certain type of DNA and we are going to discuss the combine effects of certain enzymes like, 1 , 2 , 3 on chosen DNA using the idea of NT sets. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects. Assume the set H= { 1 , 2 , 3 } and assume that their mutual effect on each other is shown in the following table * Here P (H T ) discuss the all possible outcomes of anti and neut. Consider the following two subsets of P (H T ).
Then τ 1 and τ 2 are NT topologies and stand for the combination of enzymes that effect the DNA. While 2 , 2 )},H T } is not NTT and stands for the combination of enzymes that does not effect the DNA as union of {( 3 , 2 , 3 )}, {( 2 , 2 , 2 )} does not belongs to τ 3 . As τ 1 and τ 2 neutrosophic triplet topologies so τ 1 ∩ τ 2 = τ 1 and τ 1 ∪ τ 2 = τ 2 is again a neutrosophic triplets topology which effects the DNA. The NTT ∅ stands for the combination of enzymes where we can not have any answer while neutrosophic triplet topology P (H T ) stands for the strongest case of combination of enzymes which effects the DNA. Now if we want more insight of this problem we may use other concepts like, NT neighborhoods etc.
On the other hand Leonhard Euler demonstrated problem that it was impossible to find a route through the town that would cross each of its seven bridges exactly once. This problem leads us towards the NT graph theory using the concept of NTT as the route does not depend upon the any physical scenario, but it depends upon the spatially connectivity between the bridges.
Similarly to classify the letters correctly and the hairy ball theorem of algebraic topology can be discussed in a more practical way using the concept of NTT.

Conclusions
In this article, we used the idea of NTT and introduced some of their properties, such as NT base, NT closure and NT subspace. At the end we discuss an application of multicriteria decision making problem with the help of NTT.