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
], Agboola et al. [5
], Ali et al. [9
], Gulistan et al. [13
]. More recently Smarandache et al. [16
] 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.
On the other hand Munkres [22
], studied topology in detail. Chang [23
] gave the concept of fuzzy topology in 1968. After this further study at fuzzy topology has been done by Thivagar [24
], Lowen [25
], Sarkar [26
] and Palaniappan [27
], Onasanya et al. [28
], Shumrani et al. [29
]. Sahin et al. [30
] presented the fresh idea of NTT.
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.
In this section we recall some helpful material from [1
] and for basics of topology we refer the reader [22
Definition 1.  A neutrosophic set is of the form
Definition 2.  “Let be a set together with a binary operation
★. Then is called a NT set if for any , there exist a neutral of “♭” called , different from the classical algebraic unitary element, and an opposite of “♭” called , with and belonging to , such that:
3. Neutrosophic Triplet Topology (NTT)
In this section, we study NTT in detail.
Definition 3.  Let be a NT set and let be a non-empty subset of . If satisfy the following conditions:
∅, in ,
The intersection of a finite number of sets in is also in ,
The union of an arbitrary number of sets in is also in .
then is called a NTT.
The pair (, ) is called a NT topological space. The elements of which are subsets of are called NT open sets of NT topological space (,).
Let be a NT set of and =,. Then is a topology for and it is called the NT trivial (or indiscrete) topology.
Let be a NT set of and =(. Then τ is a topology for and it is called the NT discrete topology.
Let be a NT set and be the collection of ∅ and those subsets of whose complements are finite. Then is called the neutrosophic triplet cofinite topology.
Example 4. Let with the binary operation defined by the following table
Then and are neutrosophic triplets of . Let be the set of triplets of . Then
Consider the following subsets
then and are NT topologies whileis not NTT.
Let (, ) be a topological space. A subset is said to be NT closed if and only if its complement is NT open.
Example 5. Let be as in Example 4 with the NTT
The NT closed sets of a NT topological space (, has the following properties,
Two NT topologies and of the NT set are said to be comparable if ⊂ or ⊂ . Further and are said to be equal if ⊂ and ⊂ . If ⊂ holds, then we say that is finer than and is coarser than .
Let be a NT set having more than one element as a triplet element then any topology on is finer than the NT indiscrete topology on and coarser than the NT discrete topology on .
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.
Example 7. Let be as in Example 4. Consider the two NT topologies
is not a NTT.
Let ( ) be a NT topological space. If for some and , we have , we say that M is a neighborhood of . A set is open if and only if for each there exists a neighborhood of contained in L.
Example 9. Let be as in Example 4. Consider the following NTT
Note that the NT has two neighborhoods, namely and while is the only neighborhood for both and .
4. Neutrosophic Triplet Bases of Neutrosophic Triplet Topology (NTT)
In this section, we define and study bases of a NTT for generating NT topologies.
Let (, be a NT topological space. A family is called a NT basis (or NT base) for if each NT open subset of is the union of members of . The members of are called basis open sets of the topology
Let be any NT set. Then the collection of all NT subsets of is a basis for the NT discrete topology on .
Example 11. Let be as in Example 4 with the NTT
Then is a NT basis for (.
Theorem 1. Let ( be a NT topological space. A familyis a NT basis forif and only if, for eachandthere is asuch that
is a NT base for NTT
By definition each
is a union of members of
is an arbitrary NT point of
belongs to at least one
in the union
Conversly, suppose that for each
there is a
Thus is a union of members of and therefore is a NT bases for □
A family of NT subsets of a neutrosophic triplet set(NTS) is a NT bases for some NTT on if and only if the following conditions are satisfied:
Each in is contained in somei.e.,
For any , belonging to the intersectionis a union of members of Equivalently, for each
there exist asuch that
Suppose that a family
of a NT subsets of NT set
is a NT basis for some NTT on
(is open), then by definition of NT basis,
can be written as union of members of
be members of
are NT sets and so is
By Theorem 1, for each
there is a
Conversly, Suppose that both conditions
be the family of NT subsets of
. Which are obtained by taking union of members of
. We claim that
is a NTT on
. We need to show that the conditions of NTT are satisfied by the member of
be a class of members of
is a union of members of
is also a union of members of
There are sets
be such that
which means that
. By (1)
So . Also, if we take the union of empty class of members of we note that Hence is a topology on . Since each member of is a union of members of by definition, is a NT basis for □
5. Neutrosophic Triplet Closure
In this section, we define NT closure of neutrosophic triplet topological space.
Definition 7. Let be a NT topological space and let be any NT subset of . A NT is said to be NT adherent to if each NT neighbourhood of contain a NT point of (which may be itself). The NT set of all NT points of adherent to is called the NT closure of and is denoted by in symbols,
Equivalently, NT closure of is the smallest NT closed super set of . Neutrosophic triplet closure of is denoted by or .
It is clear from the definition that
Let be as in Example 4 with the NTT . Let and . We will find and Since , we have .
Since the only neighborhood of is and , we have that Similarly, we have that . Therefore, .
Next we will find . Since is a neighborhood of and , we have that . Since the only neighborhood of is and , we have . Similarly, we have that . Hence, .
is NT closed if and only if .
Assume that is a NT closed. Then is a closed set containing . Therefore, . However, by definition . Hence, . Conversely, assume that . Since is the smallest NT superset of so is NT closed, which implies that is NT closed. □
Let () be a NT topological space and let and be arbitrary NT subsets of . Then
If , then
It is trivial.
and are both closed sets and therefore = by Theorem 3.
Then each NT neighbourhood
contains some point of
contains some point of
For the converse inclusion, we have, by definition
is a NT closed set containing
Hence by Theorem 3 we have
is a NT closed set and therefore by Theorem 3
We apply Theorem 3 to the NT closed set
Taking closures on both sides and applying (3) we have
Hence, . □
Remark 4. The equality
does not hold in general.
6. Neutrosophic Triplet Subspace
In this section, we define the NT subspace.
Definition 8. Let () be a NT topological space and , where Then
is a NTT on , called NT subspace topology. Open sets in consist of all intersections of open sets of with .
Let us check that the collection is a NTT on .
We shall show that satisfies the three properties of a NT topology on
: Suppose that
then, there are subsets
A NT open set in
This finite intersection of members of is again in
be an arbitrary family of members of
Then there exist a family
of member of
Since is a NTT on .
Thus, belongs to Hence arbitrary union of members of is also in
Hence, is a NTT on
Example 13. Let be as in Example 4 with the NTT
Taking intersection of each member of τ with Then
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,
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
and assume that their mutual effect on each other is shown in the following table
are neutrosophic triplets of
means that the enzymes
play the role of anti and neut of each other,
means that the enzyme
has no neut and anti and
are anti and neut of each other in different situations. Let
be the set of triplets of
Here discuss the all possible outcomes of anti and neut. Consider the following two subsets of . and . Then and are NT topologies and stand for the combination of enzymes that effect the DNA. While is not NTT and stands for the combination of enzymes that does not effect the DNA as union of does not belongs to . As and neutrosophic triplet topologies so and 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 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.
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.