Introduction to neutrosophic soft topological space

Abstract

The primary aim of this paper is to construct a topology on a neutrosophic soft set (NSS). The notion of neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighbourhood, neutrosophic soft boundary, regular NSS are introduced and some of their basic properties are studied in this paper. Then the base for neutrosophic soft topology and subspace topology on NSS have been defined with suitable examples. Some related properties have been developed, too. Moreover, the concept of separation axioms on neutrosophic soft topological space have been introduced along with investigation of several structural characteristics.

1 Introduction

The complexity generally arises from uncertainty in the form of ambiguity in real world. Researchers in economics, sociology, medical science and many other several fields deal daily with the complexities of modeling uncertain data. Classical methods do not give fruitful result always because the uncertain appearing in these domains may be of different kinds. The probability theory has been an age old and effective tool to handle uncertainty but it can be applied only on random process. After that, theory of evidence, theory of fuzzy set by Zadeh [1], intuitionistic fuzzy set theory by Atanassov [2] were introduced to solve uncertain problems. But each of these theories has it’s inherent difficulties as pointed out by Molodtsov [3]. The basic reason for these difficulties is inadequacy of parametrization tool of the theories.

Molodtsov [3] initiated the soft set theory as a new mathematical tool which is free from the parametrization inadequacy syndrome of different theory dealing with uncertainty in 1999. This makes the theory very convenient and easily applicable in practice. Molodtsov [3] successfully applied several directions for the applications of soft set theory, such as smoothness of functions, game theory, operation research, Riemann integration, Perron integration and probability etc. Now, soft set theory and it’s applications are progressing rapidly in different fields. Shabir and Naz [4] presented soft topological spaces and defined some concepts of soft sets on this spaces and separation axioms. Moreover, topological structure on fuzzy, fuzzy soft, intuitionistic fuzzy and intuitionistic fuzzy soft set was defined by Chang [5], Tanay and Kandemir [6], Coker [7], Li and Cui [8], Osmanoglu and Tokat [9], Bayramov and Gunduz [10, 11], Neog et al. [12], Varol and Aygun [13].

The concept of Neutrosophic Set (NS) was first introduced by Smarandache [14, 15] which is a generalisation of classical sets, fuzzy set, intuitionistic fuzzy set etc. Later, Maji [16] has introduced a combined concept Neutrosophic soft set (NSS). Using this concept, several mathematicians have produced their research works in different mathematical structures for instance Sahin et al. [17], Broumi [18], Deli and Broumi [19, 20], Maji [21], Broumi and Smarandache [22], Bera and Mahapatra [23–25], Deli [26, 27], Salama and Alblowi [28], Arockiarani et al. [29, 30], Saroja and Kalaichelvi [31] and others.

The primary aim of this paper is to construct a topology on an NSS. The notion of neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighbourhood, neutrosophic soft boundary, regular neutrosophic soft set are introduced and some of their basic properties are studied in this paper. The content of the paper is organised as following:

In Sect. 2, some basic definitions and preliminary results are given which will be used in rest of the paper. The notion of neutrosophic soft topological space has been introduced along with some related properties and several structural characteristics in Sect. 3. Section 4 gives the concept of base for neutrosophic soft topology with suitable examples and some related theorems. In Sect. 5, the idea of subspace topology on an NSS set is proposed along with some properties. Then, the concept of separation axioms of neutrosophic soft topological space has been introduced along with investigation of several structural characteristics in Sect. 6. Finally Sect. 7 presents the conclusion of our work.

2 Preliminaries

We recall some necessary definitions related to fuzzy set, soft set, neutrosophic set, neutrosophic soft set for completeness.

2.1 Definitions related to fuzzy set and soft set

This section gives some important definitions related to Fuzzy set, Soft Set [3, 32]:

1. A binary operation $*: [0,1]\times [0,1] \rightarrow [0,1]$ is continuous t - norm if $*$ satisfies the following conditions:

• (i) $*$ is commutative and associative.

• (ii) $*$ is continuous.

• (iii) $a*1 = 1*a = a, \, \,\forall a\in [0,1]$.

• (iv) $a*b \le c*d$ if $a\le c, \,b\le d$ with $a,b,c,d \in [0,1]$.

A few examples of continuous t-norm are $a*b = ab, \, \, a*b = min\{a,b\}, a*b = max\{a+b-1,0\}$.

2. A binary operation $\diamond: [0,1]\times [0,1] \rightarrow [0,1]$ is continuous t - conorm (s - norm) if $\diamond$ satisfies the following conditions:

• (i) $\diamond$ is commutative and associative.

• (ii) $\diamond$ is continuous.

• (iii) $a\diamond 0 = 0\diamond a = a, \, \,\forall a\in [0,1]$.

• (iv) $a\diamond b \le c\diamond d$   if   $a\le c, \,b\le d$   with   $a,b,c,d \in [0,1]$.

A few examples of continuous s-norm are     $a\diamond b = a+b-ab, \,\, a\diamond b = max\{a,b\}, a\diamond b = min\{a+b,1\}$.

3. Let U be an initial universe set and E be a set of parameters. Let P(U) denote the power set of U. Then for $A\subseteq E$, a pair (F, A) is called a soft set over U, where $F: A\rightarrow P(U)$ is a mapping.

2.2 Definitions related to neutrosophic set and neutrosophic soft set

Few relevant definitions [14, 16, 20] are given below:

1. A neutrosophic set (NS) on the universe of discourse U is defined as:

$$\begin{aligned} A= \left\{ \langle x,T_{A}(x),I_{A}(x),F_{A}(x) \rangle: x\in U\right\} , \end{aligned}$$

where $T, I, F: U\rightarrow ]^{-}0,1^{+}[$ and $^{-}0\le T_{A}(x)+I_{A}(x)+F_{A}(x)\le 3^{+}$.

From philosophical point of view, the neutrosophic set (NS) takes the value from real standard or nonstandard subsets of $]^{-}0,1^{+}[$. But in real life application in scientific and engineering problems, it is difficult to use NS with value from real standard or nonstandard subset of $]^{-}0,1^{+}[$. Hence we consider the NS which takes the value from the subset of [0,1].

2. Let U be an initial universe set and E be a set of parameters. Let P(U) denote the set of all NSs of U. Then for $A\subseteq E$, a pair (F, A) is called an NSS over U, where $F: A\rightarrow P(U)$ is a mapping.

This concept has been modified by Deli and Broumi as given below:

3. Let U be an initial universe set and E be a set of parameters. Let P(U) denote the set of all NSs of U. Then, a neutrosophic soft set N over U is a set defined by a set valued function $f_N$ representing a mapping $f_N: E\rightarrow P(U)$ where $f_N$ is called approximate function of the neutrosophic soft set N. In other words, the neutrosophic soft set is a parameterized family of some elements of the set P(U) and therefore it can be written as a set of ordered pairs,

$$\begin{aligned} N = \left\{ \left( e,\left\{ \langle x, T_{f_N(e)}(x), I_{f_N(e)}(x), F_{f_N(e)}(x)\rangle: x\in U \right\} \right): e\in E\right\} \end{aligned}$$

where $T_{f_N(e)}(x), I_{f_N(e)}(x), F_{f_N(e)}(x) \in [0,1]$, respectively called the truth-membership, indeterminacy-membership, falsity-membership function of $f_N(e)$. Since supremum of each T, I, F is 1 so the inequality $0\le T_{f_N(e)}(x)+I_{f_N(e)}(x)+F_{f_N(e)}(x)\le 3$ is obvious.

4. The complement of a neutrosophic soft set N is denoted by $N^c$ and is defined by:

$$\begin{aligned} N^c = \left\{ \left( e,\left\{ \langle x, F_{f_N(e)}(x), 1-I_{f_N(e)}(x), T_{f_N(e)}(x)\rangle: x\in U \right\} \right): e\in E\right\} \end{aligned}$$

5. Let $N_1$ and $N_2$ be two NSSs over the common universe (U, E). Then $N_1$ is said to be the neutrosophic soft subset of $N_2$ if

$$\begin{aligned} T_{f_{N_1}(e)}(x)\le T_{f_{N_2}(e)}(x); \, I_{f_{N_1}(e)}(x)\ge I_{f_{N_2}(e)}(x); \, F_{f_{N_1}(e)}(x)\ge F_{f_{N_2}(e)}(x); \, \quad \forall e\in E \hbox { and } x\in U. \end{aligned}$$

We write $N_1\subseteq N_2$ and then $N_2$ is the neutrosophic soft superset of $N_1$.

6. Let $N_1$ and $N_2$ be two NSSs over the common universe (U, E). Then their union is denoted by $N_1\cup N_2 = N_3$ and is defined by:

$$\begin{aligned} N_3 = \left\{ \left( e,\left\{ \langle x, T_{f_{N_3}(e)}(x), I_{f_{N_3}(e)}(x), F_{f_{N_3}(e)}(x)\rangle: x\in U \right\} \right): e\in E\right\} \end{aligned}$$

where

$$\begin{aligned} T_{f_{N_3}(e)}(x)& =\, {} T_{f_{N_1}(e)}(x) \diamond T_{f_{N_2}(e)}(x), \,\, I_{f_{N_3}(e)}(x) = I_{f_{N_1}(e)}(x)*I_{f_{N_2}(e)}(x),\\ F_{f_{N_3}(e)}(x)& =\, {} F_{f_{N_1}(e)}(x)*F_{f_{N_2}(e)}(x); \end{aligned}$$

7. Let $N_1$ and $N_2$ be two NSSs over the common universe (U, E). Then their intersection is denoted by $N_1\cap N_2 = N_3$ and is defined by:

$$\begin{aligned} N_3 = \left\{ \left( e,\left\{ \langle x, T_{f_{N_3}(e)}(x), I_{f_{N_3}(e)}(x), F_{f_{N_3}(e)}(x)\rangle: x\in U \right\} \right): e\in E\right\} \end{aligned}$$

where

$$\begin{aligned} T_{f_{N_3}(e)}(x)= T_{f_{N_1}(e)}(x) *T_{f_{N_2}(e)}(x), \,\, I_{f_{N_3}(e)}(x) = I_{f_{N_1}(e)}(x)\diamond I_{f_{N_2}(e)}(x), F_{f_{N_3}(e)}(x) = F_{f_{N_1}(e)}(x)\diamond F_{f_{N_2}(e)}(x); \end{aligned}$$

3 Neutrosophic soft topology

In this section, the concept of neutrosophic soft topology has been introduced. Some related basic properties have been developed in continuation.

Unless otherwise stated, E is treated as the parametric set through out this paper and  $e\in E$, an arbitrary parameter.

3.1 Definition

• 1. A neutrosophic soft set N over (U, E) is said to be null neutrosophic soft set if $T_{f_N(e)}(x)=0, I_{f_N(e)}(x)=1, F_{f_N(e)}(x)=1; \forall e\in E, \forall x\in U$. It is denoted by $\phi _u$.

• 2. A neutrosophic soft set N over (U, E) is said to be absolute neutrosophic soft set if $T_{f_N(e)}(x)=1, I_{f_N(e)}(x)=0, F_{f_N(e)}(x)=0; \forall e\in E, \forall x\in U$. It is denoted by $1_u$.

Clearly, $\phi ^c_u = 1_u$ and $1^c_u = \phi _u$.

3.2 Definition

Let $\textit{NSS}(U,E)$ be the family of all neutrosophic soft sets over U via parameters in E and $\tau _u \subset NSS(U,E)$. Then $\tau _u$ is called neutrosophic soft topology on (U, E) if the following conditions are satisfied.

• (i) $\phi _u, 1_u \in \tau _u$

• (ii) the intersection of any finite number of members of $\tau _u$ also belongs to $\tau _u$.

• (iii) the union of any collection of members of $\tau _u$ belongs to $\tau _u$.

Then the triplet $(U, E, \tau _u)$ is called a neutrosophic soft topological space. Every member of $\tau _u$ is called $\tau _u$-open neutrosophic soft set. An NSS is called $\tau _u$-closed iff it’s complement is $\tau _u$-open. There may be a number of topologies on (U, E). If $\tau _{u^1}$ and $\tau _{u^2}$ are two topologies on (U, E) such that $\tau _{u^1}\subset \tau _{u^2}$, then $\tau _{u^1}$ is called neutrosophic soft strictly weaker ( coarser) than $\tau _{u^2}$ and in that case $\tau _{u^2}$ is neutrosophic soft strict finer than $\tau _{u^1}$. Moreover $\textit{NSS}(U,E)$ is a neutrosophic soft topology on (U, E).

3.2.1 Example

1. Let $U=\{h_{1},h_{2}\}, \,\, E =\{e_{1}, e_{2}\}$ and $\tau _u = \{\phi _u, 1_u, N_1, N_2, N_3, N_4\}$ where $N_1, N_2, N_3, N_4$ being NSSs are defined as following:

$$\begin{aligned} f_{N_1}(e_{1})& =\, {} \left\{ \langle h_{1},(1,0,1)\rangle , \langle h_{2},(0,0,1)\rangle \right\} ;\\ f_{N_1}(e_{2})& =\, {} \left\{ \langle h_{1},(0,1,0)\rangle , \langle h_{2},(1,0,0)\rangle \right\} ;\\ f_{N_2}(e_{1})& =\, {} \left\{ \langle h_{1},(0,1,0)\rangle , \langle h_{2},(1,1,0)\rangle \right\} ; \\ f_{N_2}(e_{2})& =\, {} \left\{ \langle h_{1},(1,0,1)\rangle , \langle h_{2},(0,1,1)\rangle \right\} ; \\ f_{N_3}(e_{1})& =\, {} \left\{ \langle h_{1},(1,1,1)\rangle , \langle h_{2},(0,1,1)\rangle \right\} ; \\ f_{N_3}(e_{2})& =\, {} \left\{ \langle h_{1},(0,1,0)\rangle , \langle h_{2},(0,1,1)\rangle \right\} ; \\ f_{N_4}(e_{1})& =\, {} \left\{ \langle h_{1},(1,1,0)\rangle , \langle h_{2},(1,1,0)\rangle \right\} ; \\ f_{N_4}(e_{2})& =\, {} \left\{ \langle h_{1},(1,0,0)\rangle , \langle h_{2},(0,1,1)\rangle \right\} ; \end{aligned}$$

Here $N_1\cap N_1=N_1, N_1\cap N_2=\phi _u, N_1\cap N_3=N_3, N_1\cap N_4=N_3, N_2\cap N_2=N_2, N_{2} \cap N_{3}=\phi_{u}, N_2\cap N_4=N_2, N_3\cap N_3=N_3, N_3\cap N_4=N_3, N_4\cap N_4=N_4$; and $N_1\cup N_1=N_1, N_1\cup N_2=1_u, N_1\cup N_3=N_1, N_1\cup N_4=1_u, N_2\cup N_2=N_2, N_2\cup N_3=N_4, N_2\cup N_4=N_4, N_3\cup N_3=N_3, N_3\cup N_4=N_4, N_4\cup N_4=N_4$;

Corresponding t-norm and s-norm are defined as $a*b = max\{a+b-1,0\}$ and $a\diamond b = min\{a+b,1\}$. Then $\tau _u$ is a neutrosophic soft topology on (U, E) and so $(U, E ,\tau _u)$ is a neutrosophic soft topological space over (U, E).

2. Let $U=\{x_{1},x_{2},x_{3}\}, \,\, E =\{e_{1}, e_{2}\}$ and $\tau _u = \{\phi _u, 1_u, N_1, N_2, N_3\}$ where $N_1, N_2, N_3$ being NSSs over (U, E) are defined as follow:

$$\begin{aligned} f_{N_1}(e_{1})& =\, {} \left\{ \langle x_{1},(1.0,0.5,0.4)\rangle , \langle x_{2},(0.6,0.6,0.6)\rangle , \langle x_{3},(0.5,0.6,0.4)\rangle \right\} ; \\ f_{N_1}(e_{2})& =\, {} \left\{ \langle x_{1},(0.8,0.4,0.5)\rangle , \langle x_{2},(0.7,0.7,0.3)\rangle , \langle x_{3},(0.7,0.5,0.6)\rangle \right\} ; \\ f_{N_2}(e_{1})& =\, {} \left\{ \langle x_{1},(0.8,0.5,0.6)\rangle , \langle x_{2},(0.5,0.7,0.6)\rangle , \langle x_{3},(0.4,0.7,0.5)\rangle \right\} ; \\ f_{N_2}(e_{2})& =\, {} \left\{ \langle x_{1},(0.7,0.6,0.5)\rangle , \langle x_{2},(0.6,0.8,0.4)\rangle , \langle x_{3},(0.5,0.8,0.6)\rangle \right\} ; \\ f_{N_3}(e_{1})& =\, {} \left\{ \langle x_{1},(0.6,0.6,0.7)\rangle ,\langle x_{2},(0.4,0.8,0.8)\rangle , \langle x_{3},(0.3,0.8,0.6)\rangle \right\} ;\\ f_{N_3}(e_{2})& =\, {} \left\{ \langle x_{1},(0.5,0.8,0.6)\rangle , \langle x_{2},(0.5,0.9,0.5)\rangle , \langle x_{3},(0.2,0.9,0.7)\rangle \right\} ; \end{aligned}$$

The t-norm and s-norm are defined as $a*b = min\{a,b\}$ and $a\diamond b = max\{a,b\}$. Here $N_1\cap N_1=N_1, N_1\cap N_2=N_2, N_1\cap N_3=N_3, N_2\cap N_2=N_2, N_2\cap N_3=N_3, N_3\,\cap\,N_3=N_3$ and $N_1\cup N_1=N_1, N_1\cup N_2=N_1, N_1\cup N_3=N_1, N_2\cup N_2=N_2, N_2\cup N_3=N_2, N_3\cup N_3=N_3$. Then $\tau _u$ is a neutrosophic soft topology on (U, E) and so $(U, E, \tau _u)$ is a neutrosophic soft topological space over (U, E).

3. Let $\textit{NSS}(U,E)$ be the family of all neutrosophic soft sets over (U, E). Then $\{\phi _u, 1_u\}$ and $\textit{NSS}(U,E)$ are two examples of the neutrosophic soft topology over (U, E). They are called, respectively, indiscrete (trivial) and discrete neutrosophic soft topology. Clearly, they are the smallest and largest neutrosophic soft topology on (U, E), respectively.

3.3 Proposition

Let $(U, E, \tau _{u^1})$ and $(U, E, \tau _{u^2})$ be two neutrosophic soft topological spaces over (U, E). Suppose, $\tau _{u^1}\cap \tau _{u^2} = \{M\in \textit{NSS}(U,E): M\in \tau _{u^1}\cap \tau _{u^2} \}$; Then $\tau _{u^1}\cap \tau _{u^2}$ is also a neutrosophic soft topology on (U, E).

Proof

• (i) Clearly $\phi _u, 1_u \in \tau _{u^1}\cap \tau _{u^2}$;

• (ii) Let $M_1, M_2 \in \tau _{u^1}\cap \tau _{u^2}$

  $$\begin{aligned}& \Rightarrow M_1, M_2 \in \tau _{u^1} \, \, \text{ and } \, \, M_1, M_2 \in \tau _{u^2};\\ &\Rightarrow M_1\cap M_2 \in \tau _{u^1} \, \, \text{ and } \, \, M_1\cap M_2 \in \tau _{u^2};\\ &\Rightarrow M_1\cap M_2 \in \tau _{u^1}\cap \tau _{u^2} \end{aligned}$$

• (iii) Let $\{M_i: i\in \Gamma \} \in \tau _{u^1}\cap \tau _{u^2}$

  $$\begin{aligned}&\Rightarrow \left\{ M_i\right\} \in \tau _{u^1} \, \, \text{ and } \, \, \left\{ M_i\right\} \in \tau _{u^2}\\&\Rightarrow \cup _i M_i\in \tau _{u^1} \, \, \text{ and } \, \, \cup _i M_i\in \tau _{u^2}\\&\Rightarrow \cup _i M_i\in \tau _{u^1}\cap \tau _{u^2} \end{aligned}$$

  Thus $\tau _{u^1}\cap \tau _{u^2}$ is a neutrosophic soft topology on (U, E).

$\square$

3.3.1 Remark

The union of two neutrosophic soft topologies may not be so.

We consider $U=\{x_{1},x_{2},x_{3}\}, \,\, E =\{e_{1}, e_{2}\}$ and $\tau _{u^1} = \{\phi _u, 1_u, N_1, N_2, N_3\}$, $\tau _{u^2} = \{\phi _u, 1_u, N_4, N_5\}$ be two neutrosophic soft topologies on (U, E). Let $\tau _{u^1}$, t-norm and s-norm be as in example (2) of subsection [3.2.1]. We define $N_4 = N_2$ and $N_5$ as:

$$\begin{aligned} f_{N_5}(e_{1})& =\, {} \left\{ \langle x_{1},(0.7,0.5,0.8)\rangle , \langle x_{2},(0.4,0.8,0.6)\rangle , \langle x_{3},(0.4,0.9,0.7)\rangle \right\} ; \\ f_{N_5}(e_{2})& =\, {} \left\{ \langle x_{1},(0.6,0.7,0.8)\rangle , \langle x_{2},(0.5,0.9,0.6)\rangle , \langle x_{3},(0.3,0.8,0.8)\rangle \right\} ; \end{aligned}$$

Let $\tau _{u^3} = \tau _{u^1}\cup \tau _{u^2} = \{\phi _u, 1_u, N_1, N_2, N_3, N_5\}$. But $\tau _{u^3}$ is not a topology on (U, E), as $N_3\cap N_5\notin \tau _{u^3}$. Here $N_3\cap N_5 = M$, say, is given as following:

$$\begin{aligned} f_M(e_{1})& =\, {} \left\{ \langle x_{1},(0.6,0.6,0.8)\rangle , \langle x_{2},(0.4,0.8,0.8)\rangle , \langle x_{3},(0.3,0.9,0.7)\rangle \right\} ; \\ f_M(e_{2})& =\, {} \left\{ \langle x_{1},(0.5,0.8,0.8)\rangle , \langle x_{2},(0.5,0.9,0.6)\rangle , \langle x_{3},(0.2,0.9,0.8)\rangle \right\} ; \end{aligned}$$

3.4 Proposition (De-Morgan’s law)

Let $N_1, N_2$ be two neutrosophic soft sets over (U, E). Then,

• (i) $(N_1\cup N_2)^c = N_1{^c}\cap N_2{^c}$

• (ii) $(N_1\cap N_2)^c = N_1{^c}\cup N_2{^c}$

Proof

(i) For all $e\in E, x\in U$, we have,

$$\begin{aligned} N_1\cup N_2& =\, {} \left\{ \left\langle x, \left[ T_{f_{N_1}(e)}(x)\diamond T_{f_{N_2}(e)}(x), I_{f_{N_1}(e)}(x)*I_{f_{N_2}(e)}(x), F_{f_{N_1}(e)}(x)*F_{f_{N_2}(e)}(x)\right] \right\rangle \right\} \end{aligned}$$

Then,

$$\begin{aligned} (N_1\cup N_2)^c& =\, {} \left\{ \left\langle x, \left[ F_{f_{N_1}(e)}(x)*F_{f_{N_2}(e)}(x), 1- \left( I_{f_{N_1}(e)}(x)*I_{f_{N_2}(e)}(x)\right) , T_{f_{N_1}(e)}(x)\diamond T_{f_{N_2}(e)}(x)\right] \right\rangle \right\} \end{aligned}$$

Now,

$$\begin{aligned} N^c_1 = \left\{\left \langle x, \left[ F_{f_{N_1}(e)}(x), 1- I_{f_{N_1}(e)}(x), T_{f_{N_1}(e)}(x)\right] \right\rangle \right\} \end{aligned}$$

and

$$\begin{aligned} N^c_2 = \left\{\left \langle x, \left[ F_{f_{N_2}(e)}(x), 1- I_{f_{N_2}(e)}(x), T_{f_{N_2}(e)}(x)\right] \right\rangle \right\} . \end{aligned}$$

Then,

$$\begin{aligned} N^c_1\cap N^c_2& =\, {} \left\{ \left\langle x, \left[ F_{f_{N_1}(e)}(x)*F_{f_{N_2}(e)}(x), \left( 1- I_{f_{N_1}(e)}(x)\right) \diamond (1- I_{f_{N_2}(e)}(x)),T_{f_{N_1}(e)}(x)\diamond T_{f_{N_2}(e)}(x)\right] \right\rangle \right\} \\& =\, {} \left\{ \left\langle x, \left[ F_{f_{N_1}(e)}(x)*F_{f_{N_2}(e)}(x), 1- \left( I_{f_{N_1}(e)}(x)*I_{f_{N_2}(e)}(x)\right) , T_{f_{N_1}(e)}(x)\diamond T_{f_{N_2}(e)}(x)\right] \right\rangle \right\} \end{aligned}$$

Hence, $(N_1\cup N_2)^c = N_1{^c}\cap N_2{^c}$. $\square$

Note:- Here, $(1- I_{f_{N_1}(e)}(x))\diamond (1- I_{f_{N_2}(e)}(x)) = 1- (I_{f_{N_1}(e)}(x)*I_{f_{N_2}(e)}(x))$ holds for dual pairs of non-parameterized t-norms and s-norms e.g., $a*b = min\{a,b\}$ and $a\diamond b = max\{a,b\}$,   $a*b = max\{a+b-1,0\}$ and $a\diamond b = min\{a+b,1\}$ etc.

In a similar fashion, 2nd part can be established.

The theorems can be extended as: (i) $\{\cup _i N_i\}^c = \cap _i N^c_i$     (ii) $\{\cap _i N_i\}^c = \cup _i N^c_i$ for a family of NSSs $\{N_i\}_{i\in \Gamma }$ over (U, E).

3.5 Proposition

Let $(U, E, \tau _u)$ be a neutrosophic soft topological space over (U, E) and $\tau _u = \{N_i: N_i\in \textit{NSS}(U,E)\} = \{[e,f_{N_i}(e)]_{e\in E}: N_i\in \textit{NSS}(U,E)\}$ where $f_{N_i}(e) = \{\langle x, T_{f_{N_i}(e)}(x), I_{f_{N_i}(e)}(x), F_{f_{N_i}(e)}(x)\rangle : x\in U \}$; Then the collection $\tau _1= \{[T_{f_{N_i}(e)}(U)]_{e\in E}\}$, $\tau _2= \{[I_{f_{N_i}(e)}(U)]^c{_{e\in E}}\}$ and $\tau _3= \{[F_{f_{N_i}(e)}(U)]^c{_{e\in E}}\}$ define fuzzy soft topologies on (U, E).

Proof

• (i) $\phi _u, 1_u\in \tau _u$

  $$\begin{aligned} \Rightarrow 0, 1\in \tau _1; \, \, 1, 0\in \tau _2; \, \, 1, 0\in \tau _3; \end{aligned}$$

• (ii) $N_1, N_2\in \tau _u$ implies $N_1\cap N_2\in \tau _u$ i.e.,

  $$\begin{aligned}&\left\{ \left[ T_{f_{N_1}(e)}(U)*T_{f_{N_2}(e)}(U), \, I_{f_{N_1}(e)}(U)\diamond I_{f_{N_2}(e)}(U), \, F_{f_{N_1}(e)}(U)\diamond F_{f_{N_2}(e)}(U)\right] _{e\in E}\right\} \in \tau _u\\&\quad \Rightarrow \left\{ (T_{f_{N_1}(e)}(U)*T_{f_{N_2}(e)}(U))_{e\in E}\right\} \in \tau _1,\\&\quad \left\{ \left( \left[ I_{f_{N_1}(e)}(U)\right] ^c*\left[ I_{f_{N_2}(e)}(U)\right] ^c\right) _{e\in E} \right\} \in \tau _2,\\&\quad \left\{ \left( \left[ F_{f_{N_1}(e)}(U)\right] ^c*\left[ F_{f_{N_2}(e)}(U)\right] ^c\right) _{e\in E} \right\} \in \tau _3; \end{aligned}$$

• (iii) $\{N_i :i\in \Gamma \}\in \tau _u$ implies $\cup _i{N_i}\in \tau _u$

  $$\begin{aligned}&\Rightarrow \left\{ \left[ \diamond _i T_{f_{N_i}(e)}(U), \, *_i I_{f_{N_i}(e)}(U), \, *_i F_{f_{N_i}(e)}(U)\right] _{e\in E}\right\} \in \tau _u\\&\Rightarrow \left\{ \left[ \diamond _i T_{f_{N_i}(e)}(U)\right] _{e\in E} \right\} \in \tau _1, \,\,\,\, \left\{ \diamond _i \left[ I_{f_{N_i}(e)}(x)\right] ^c_{e\in E}\right\} \in \tau _2, \,\,\,\, \left\{ \diamond _i \left[ F_{f_{N_i}(e)}(U)\right] ^c_{e\in E}\right\} \in \tau _3; \end{aligned}$$

This ends the proposition.

It can be verified by example (2) in the subsection [3.2.1]. $\square$

3.5.1 Remark

Converse of the above proposition is not true as shown by the following example.

Let $U=\{x_{1},x_{2},x_{3}\}, \,\, E =\{e_{1}, e_{2}\}$ and $\tau _u = \{\phi _u, 1_u, N_1, N_2, N_3\}$ where $N_1, N_2, N_3$ being NSSs in $\textit{NSS}(U,E)$ are defined as following:

$$\begin{aligned} f_{N_1}(e_{1})& =\, {} \left\{ \langle x_{1},(0.8,0.4,0.3)\rangle , \langle x_{2},(0.6,0.5,0.7)\rangle , \langle x_{3},(0.5,0.4,0.2)\rangle \right\} ; \\ f_{N_1}(e_{2})& =\, {} \left\{ \langle x_{1},(0.5,0.6,0.4)\rangle , \langle x_{2},(0.3,0.4,0.6)\rangle , \langle x_{3},(0.1,0.5,0.6)\rangle \right\} ; \\ f_{N_2}(e_{1})& =\, {} \left\{ \langle x_{1},(0.7,0.6,0.4)\rangle , \langle x_{2},(0.5,0.7,0.7)\rangle , \langle x_{3},(0.4,0.6,0.7)\rangle \right\} ; \\ f_{N_2}(e_{2})& =\, {} \left\{ \langle x_{1},(0.4,0.7,0.6)\rangle , \langle x_{2},(0.2,0.8,0.7)\rangle , \langle x_{3},(0.1,0.7,0.6)\rangle \right\} ; \\ f_{N_3}(e_{1})& =\, {} \left\{ \langle x_{1},(0.9,0.5,0.6)\rangle , \langle x_{2},(0.6,0.6,0.9)\rangle , \langle x_{3},(0.6,0.5,0.8)\rangle \right\} ; \\ f_{N_3}(e_{2})& =\, {} \left\{ \langle x_{1},(0.6,0.6,0.8)\rangle , \langle x_{2},(0.5,0.7,0.8)\rangle , \langle x_{3},(0.7,0.6,1.0)\rangle \right\} ; \end{aligned}$$

The t-norm and s-norm are defined as $a*b = min\{a,b\}$ and $a\diamond b = max\{a,b\}$. Then,

$$\begin{aligned} \tau _1& =\, {} \left\{ \left\langle T_{f_{\phi _u}(e)}(U), T_{f_{1_u}(e)}(U), T_{f_{N_1}(e)}(U), T_{f_{N_2}(e)}(U), T_{f_{N_3}(e)}(U)\right\rangle _ {e\in E}\right\} \\ \tau _2& =\, {} \left\{ \left\langle I_{f_{\phi _u}(e)}(U), I_{f_{1_u}(e)}(U), I_{f_{N_1}(e)}(U), I_{f_{N_2}(e)}(U), I_{f_{N_3}(e)}(U)\right\rangle _ {e\in E}\right\} \\ \tau _3& =\, {} \left\{ \left\langle F_{f_{\phi _u}(e)}(U), F_{f_{1_u}(e)}(U), F_{f_{N_1}(e)}(U), F_{f_{N_2}(e)}(U), F_{f_{N_3}(e)}(U)\right\rangle _ {e\in E}\right\} \end{aligned}$$

are fuzzy soft topologies on (U, E). Elaborately,

$$\begin{aligned} \tau _1& =\, {} \left\{ \langle (0,0,0),(1,1,1),(0.8,0.6,0.5), (0.7,0.5,0.4),(0.9,0.6,0.6)\rangle _{e_1},\right. \\&\left. \langle (0,0,0),(1,1,1),(0.5,0.3,0.1), (0.4,0.2,0.1),(0.6,0.5,0.7)\rangle _{e_2}\right\} \end{aligned}$$

and so on.

Here, $\tau _u = \{\phi _u, 1_u, N_1, N_2, N_3\}$ is not a neutrosophic soft topology on (U, E), because $N_2\cap N_3 \notin \tau _u$, where $N_2\cap N_3 = N_4$, say, is given as following:

$$\begin{aligned} f_{N_4}(e_{1})& =\, {} \left\{ \langle x_{1},(0.7,0.6,0.6)\rangle , \langle x_{2},(0.5,0.7,0.9)\rangle , \langle x_{3},(0.4,0.6,0.8)\rangle \right\} ;\\ f_{N_4}(e_{2})& =\, {} \left\{ \langle x_{1},(0.4,0.7,0.8)\rangle , \langle x_{2},(0.2,0.8,0.8)\rangle , \langle x_{3},(0.1,0.7,1.0)\rangle \right\} ; \end{aligned}$$

3.5.2 Proposition

Let $(U, E,\tau _u)$ be a neutrosophic soft topological space defined over (U, E). Then $\tau _{1e} = \{[T_{f_M(e)}(U)] : M\in \tau _u\}$, $\tau _{2e} = \{[I_{f_M(e)}(U)]^c: M\in \tau _u\}$, $\tau _{3e} = \{[F_{f_M(e)}(U)]^c: M\in \tau _u\}$ for each $e\in E$, define fuzzy topologies on (U, E).

Proof

By proposition (3.5), $\tau _1= \{[T_{f_{N_i}(e)}(U)]_{e\in E}\}$, $\tau _2= \{[I_{f_{N_i}(e)}(U)]^c{_{e\in E}}\}$ and $\tau _3= \{[F_{f_{N_i}(e)}(U)]^c{_{e\in E}}\}$ are three fuzzy soft topologies on (U, E).

So for each $e\in E$, $\tau _{1e}, \tau _{2e}, \tau _{3e}$ are fuzzy topologies on (U, E). Let $\tau _{e} = \{\tau _{1e}, \tau _{2e}, \tau _{3e}\}$; It is called as fuzzy tritopology on (U, E). Thus corresponding to each parameter $e\in E$, we have a fuzzy tritopology $\tau _e$ on (U, E). Hence, a neutrosophic soft topology on (U, E) gives a parameterized family of fuzzy tritopologies on (U, E).

The reverse of that proposition may not be true. Following example shows the fact. $\square$

3.5.3 Example

We consider the example in remark [3.5.1]. Then,

$$\begin{aligned} \tau _{1e_1}& =\, {} \left\{ T_{f_{\phi _u}(e_1)}(U), T_{f_{1_u}(e_1)}(U), T_{f_{N_1}(e_1)}(U), T_{f_{N_2}(e_1)}(U), T_{f_{N_3}(e_1)}(U)\right\} \\ \tau _{2e_1}& =\, {} \left\{ I_{f_{\phi _u}(e_1)}(U), I_{f_{1_u}(e_1)}(U), I_{f_{N_1}(e_1)}(U), I_{f_{N_2}(e_1)}(U), I_{f_{N_3}(e_1)}(U)\right\} \\ \tau _{3e_1}& =\, {} \left\{ F_{f_{\phi _u}(e_1)}(U), F_{f_{1_u}(e_1)}(U), F_{f_{N_1}(e_1)}(U), F_{f_{N_2}(e_1)}(U), F_{f_{N_3}(e_1)}(U)\right\} \end{aligned}$$

are fuzzy topologies on (U, E), where,

$$\begin{aligned} \tau _{1e_1} = \left\{ (0,0,0),(1,1,1),(0.8,0.6,0.5),(0.7,0.5,0.4),(0.9,0.6,0.6)\right\} \hbox { and so on}. \end{aligned}$$

Here, $\{\tau _{1e_1},\tau _{2e_1}, \tau _{3e_1}\}$ is a fuzzy tritopology on (U, E). Also,

$$\begin{aligned} \tau _{1e_2}& =\, {} \left\{ T_{f_{\phi _N}(e_2)}(U), T_{f_{1_N}(e_2)}(U), T_{f_{N_1}(e_2)}(U), T_{f_{N_2}(e_2)}(U), T_{f_{N_3}(e_2)}(U)\right\} \\ \tau _{2e_2}& =\, {} \left\{ I_{f_{\phi _N}(e_2)}(U), I_{f_{1_N}(e_2)}(U), I_{f_{N_1}(e_2)}(U), I_{f_{N_2}(e_2)}(U), I_{f_{N_3}(e_2)}(U)\right\} \\ \tau _{3e_2}& =\, {} \left\{ F_{f_{\phi _N}(e_2)}(U), F_{f_{1_N}(e_2)}(U), F_{f_{N_1}(e_2)}(U), F_{f_{N_2}(e_2)}(U), F_{f_{N_3}(e_2)}(U)\right\} \end{aligned}$$

is another set of fuzzy topologies on (U, E) and consequently $\{\tau _{1e_2}, \tau _{2e_2}, \tau _{3e_2}\}$ is also a fuzzy tritopology on (U, E). But $\tau _u = \{\phi _u, 1_u, N_1, N_2, N_3\}$ is not a neutrosophic soft topology on (U, E).

3.6 Definition

Let $\tau _u$ be a neutrosophic soft topology on (U, E) and $N_1, N_2\in \textit{NSS}(U,E)$. Then $N_2$ is called a neighbourhood of $N_1$ if there exists an open neutrosophic soft set M (i.e., $M\in \tau _u$) such that $N_1\subset M\subset N_2$.

In the example (2) of subsection  [3.2.1], $N_3\subset N_2\subset N_1$ and so $N_1$ is a neighbourhood of $N_3$.

3.6.1 Theorem

An NSS $M\in \textit{NSS}(U,E)$ is an open NSS iff M is a neighbourhood of each NSS $N_1$ contained in M.

Proof

Let M be an open NSS and $N_1$ be any NSS contained in M. Since we have $N_1\subset M\subset M$, it follows that M is a neighbourhood of $N_1$.

Next, suppose that M is a neighbourhood of each NSS contained in M. Since $M\subset M$, there exists an open NSS $N_2$ such that $M\subset N_2\subset M$. Hence, $M = N_2$ and M is open. $\square$

3.6.2 Definition

Let $(U, E,\tau _u)$ be a neutrosophic soft topological space on (U, E) and $M\in \textit{NSS}(U,E)$. The family of all neighbourhoods of M is called the neighbourhood system or neighbourhood filter of M up to topology $\tau _u$ and is denoted by Nbd(M).

3.6.3 Theorem

Let Nbd(M) be the neighbourhood system of the NSS M. Then,

• (i) finite intersections of the members of Nbd(M) belongs to Nbd(M).

• (ii) each neutrosophic soft set containing a member of Nbd(M) belongs to Nbd(M).

Proof

(i) Let $N_1, N_2 \in Nbd(M)$. Then there exists $N_1', N_2' \in \tau _u$ such that $M\subset N_1'\subset N_1$ and $M\subset N_2'\subset N_2$. Since, $N_1' \cap N_2' \in \tau _u$, we have, $M \subset N_1' \cap N_2' \subset N_1 \cap N_2$. Thus, $N_1 \cap N_2 \in Nbd(M)$.

(ii) If $N_1\in Nbd(M)$ and $N_2$ be a neutrosophic soft set containing $N_1$, then there exists $N_1'\in \tau _u$ such that $M\subset N_1'\subset N_1\subset N_2$. This shows that $N_2\in Nbd(M)$. $\square$

3.7 Definition

Let $(U, E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M\in \textit{NSS}(U,E)$ be arbitrary. Then the interior of M is denoted by $M^o$ and is defined by

$$\begin{aligned} M^o = \cup \left\{ N_1: N_1 \, \text{ is }\,\text{ neutrosophic }\,\text{ soft }\,\text{ open }\,\text{ and } \, N_1\subset M\right\} \end{aligned}$$

i.e., it is the union of all open neutrosophic soft subsets of M.

3.7.1 Example

We consider the example (2) in subsection [3.2.1]. Let an arbitrary $M\in \textit{NSS}(U,E)$ be defined as following:

$$\begin{aligned} f_{M}(e_1)& =\, {} \{\langle x_{1},(0.9,0.4,0.5)\rangle , \langle x_{2},(0.5,0.6,0.6)\rangle , \langle x_{2},(0.7,0.6,0.5)\}; \\ f_{M}(e_2)& =\, {} \{\langle x_{1},(0.7,0.5,0.4)\rangle , \langle x_{2},(0.8,0.3,0.4)\rangle , \langle x_{2},(0.6,0.7,0.5)\}; \end{aligned}$$

Then $\phi _u, N_2, N_3\subset M$ and so, $M^o = \phi _u\cup N_2\cup N_3 = N_2$.

3.7.2 Theorem

Let $(U, E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M, P\in \textit{NSS}(U,E)$. Then,

• (i) $M^o \subset M$ and $M^o$ is the largest open set.

• (ii) $M\subset P \Rightarrow M^o\subset P^o$.

• (iii) $M^o$ is an open neutrosophic soft set i.e., $M^o\in \tau _u$.

• (iv) M is neutrosophic soft open set iff $M^o = M$.

• (v) $(M^o)^o = M^o$.

• (vi) $(\phi _u)^o = \phi _u$ and $1_u^o = 1_u$.

• (vii) $(M\cap P)^o = M^o\cap P^o$.

• (viii) $M^o\cup P^o \subset (M\cup P)^o$.

Proof

• (i) It is obvious from the definition.

• (ii) Here $M^o\subset M\subset P$. So $M^o\subset P$ and also $P^o\subset P$. But $P^o$ is the largest open set contained in P, hence $M^o\subset P^o$.

• (iii) By definition of $\tau _u$ and $M^o$, it is obvious.

• (iv) Here, $M^o\subset M$ and let M be neutrosophic soft open. Now,

  $$\begin{aligned} M\subset M \Rightarrow M\subset \cap \{P\in \tau _u: P\subset M\} = M^o \Rightarrow M\subset M^o \end{aligned}$$

  Hence, $M = M^o$

  Conversely, let $M = M^o$. Then, $M = M^o\in \tau _u \Rightarrow M$ is an open NSS.

• (v) If N is an open NSS then $N^o = N$. Clearly, $M^o$ is an open NSS. Replacing N by $M^o$, we have the result.

• (vi) Since, $\phi _u, 1_u\in \tau _u$ so they are open NSSs. Hence, the result follows from (iv).

• (vii) $M\cap P\subset M$ and $M\cap P\subset P\Rightarrow (M\cap P)^o\subset M^o$ and $(M\cap P)^o\subset P^o \Rightarrow (M\cap P)^o\subset M^o\cap P^o$

  Further, $M^o\subset M$ and $P^o\subset P$. Then $M^o\cap P^o\subset M\cap P$. But, $(M\cap P)^o \subset M\cap P$ and it is largest open NSS. So, $M^o\cap P^o \subset (M\cap P)^o$.

  Thus, $(M\cap P)^o = M^o\cap P^o$.

• (viii) $M\subset M\cup P$ and $P\subset M\cup P \Rightarrow M^o\subset (M\cup P)^o$ and $P^o\subset (M\cup P)^o \Rightarrow M^o\cup P^o\subset (M\cup P)^o$.$\square$

3.7.3 Definition

Let $(U, E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M\in \textit{NSS}(U,E)$. The the associated interior of M is an NSS over (U, E), denoted by $\{(T_{f_{M}(e)}^o, I_{f_{M}(e)}^o, F_{f_{M}(e)}^o )_{e\in E}\}$ and is defined as     $\{(T_{f_{M}(e)}^o, I_{f_{M}(e)}^o, F_{f_{M}(e)}^o )_{e\in E}\} = \{(\diamond _i T_{f_{N_i}(e)}, *_i I_{f_{N_i}(e)}, *_i F_{f_{N_i}(e)})_{e\in E}\}$, where $N_i\in \tau _u$ and $f_{N_i}(e)\subset f_{M}(e)$.

3.7.4 Proposition

Let $(U, E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M\in \textit{NSS}(U,E)$. Then, $M^o \subset \{(T_{f_{M}(e)}^o, I_{f_{M}(e)}^o, F_{f_{M}(e)}^o )_{e\in E}\}$   where $M^o = \{(f_M(e))_{e\in E}\}^o$.

Proof

For each $e\in E, \, (T_{f_{M}(e)}^o, I_{f_{M}(e)}^o, F_{f_{M}(e)}^o )$ is the largest open neutrosophic set contained in $f_M(e)$. Also $M^o \subset M$. Hence, $M^o \subset \{(T_{f_{M}(e)}^o, I_{f_{M}(e)}^o, F_{f_{M}(e)}^o )_{e\in E}\}$. $\square$

3.7.5 Example

We consider the example (2) of subsection [3.2.1]. Let $M\in \textit{NSS}(U,E)$ be defined as following:

$$\begin{aligned} f_{M}(e_{1})& =\, {} \left\{ \langle x_{1},(0.8,0.5,0.4)\rangle , \langle x_{2},(0.7,0.7,0.6)\rangle , \langle x_{3},(0.4,0.6,0.4)\rangle \right\} ; \\ f_{M}(e_{2})& =\, {} \left\{ \langle x_{1},(0.9,0.4,0.4)\rangle , \langle x_{2},(0.8,0.7,0.2)\rangle , \langle x_{3},(0.8,0.5,0.5)\rangle \right\} ; \end{aligned}$$

Then, $N_2\subset M$ and $N_3\subset M$. So, $M^o = N_2\cup N_3 = N_2$.

Further, $f_{N_2}(e_1), f_{N_3}(e_1)\subset f_{M}(e_1)$ and $f_{N_1}(e_2), f_{N_2}(e_2), f_{N_3}(e_2)\subset f_{M}(e_2)$.

The tabular representation of $\{(T_{f_{M}(e)}^o, I_{f_{M}(e)}^o, F_{f_{M}(e)}^o )_{e\in E}\}$ is as (Table 1):

Table 1 Tabular form of $\{(T_{f_{M}(e)}^o, I_{f_{M}(e)}^o, F_{f_{M}(e)}^o )_{e\in E}\}$

Hence, $M^o \ne \{(T_{f_{M}(e)}^o, I_{f_{M}(e)}^o, F_{f_{M}(e)}^o )_{e\in E}\}$

3.8 Definition

Let $(U, E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M\in \textit{NSS}(U,E)$ be arbitrary. Then the closure of M is denoted by $\overline{M}$ and is defined by:

$$\begin{aligned} \overline{M} = \cap \left\{ N_1 : N_1 \, \text{ is }\,\text{ neutrosophic }\,\text{ soft }\,\text{ closed }\,\text{ and } \, N_1\supset M\right\} \end{aligned}$$

i.e., it is the intersection of all closed neutrosophic soft supersets of M.

3.8.1 Example

Consider the example (2) in subsection [3.2.1]. Let an arbitrary $M\in \textit{NSS}(U,E)$ be defined as:

$$\begin{aligned} f_{M}(e_1)& =\, {} \left\{ \langle x_{1},(0.6,0.7,0.8)\rangle , \langle x_{2},(0.5,0.3,0.7)\rangle , \langle x_{3},(0.4,0.4,0.5)\rangle \right\} ; \\ f_{M}(e_2)& =\, {} \left\{ \langle x_{1},(0.4,0.5,0.7)\rangle , \langle x_{2},(0.3,0.4,0.8)\rangle , \langle x_{3},(0.6,0.4,0.6)\rangle \right\} ; \end{aligned}$$

Clearly, $\phi _u, 1_u, N^c_1, N^c_2, N^c_3, N^c_4$ are all closed NSSs over (U, E). They are given as:

$$\begin{aligned} f_{N^c_1}(e_1)& =\, {} \left\{ \langle x_{1},(0.4,0.5,1.0)\rangle , \langle x_{2},(0.6,0.4,0.6)\rangle , \langle x_{3},(0.4,0.4,0.5)\rangle \right\} ;\\ f_{N^c_1}(e_2)& =\, {} \left\{ \langle x_{1},(0.5,0.6,0.8)\rangle , \langle x_{2},(0.3,0.3,0.7)\rangle , \langle x_{3},(0.6,0.5,0.7)\rangle \right\} ; \\ f_{N^c_2}(e_1)& =\, {} \left\{ \langle x_{1},(0.6,0.5,0.8)\rangle , \langle x_{2},(0.6,0.3,0.5)\rangle , \langle x_{3},(0.5,0.3,0.4)\rangle \right\} ;\\ f_{N^c_2}(e_2)& =\, {} \left\{ \langle x_{1},(0.5,0.4,0.7)\rangle , \langle x_{2},(0.4,0.2,0.6)\rangle , \langle x_{3},(0.6,0.2,0.5)\rangle \right\} ; \\ f_{N^c_3}(e_1)& =\, {} \left\{ \langle x_{1},(0.7,0.4,0.6)\rangle , \langle x_{2},(0.8,0.2,0.4)\rangle , \langle x_{3},(0.6,0.2,0.3)\rangle \right\} ; \\ f_{N^c_3}(e_2)& =\, {} \left\{ \langle x_{1},(0.6,0.2,0.5)\rangle , \langle x_{2},(0.5,0.1,0.5)\rangle , \langle x_{3},(0.7,0.1,0.2)\rangle \right\} ; \end{aligned}$$

Then $1_u, N^c_2, N^c_3\supset M$ and so, $\overline{M} = 1_u\cap N^c_2\cap N^c_3 = N^c_2$.

3.8.2 Theorem

Let $(U, E, \tau _u)$ be a neutrosophic soft topological space over (U, E) and $M, P\in \textit{NSS}(U,E)$. Then,

• (i) $M\subset \overline{M}$ and $\overline{M}$ is the smallest closed set.

• (ii) $M\subset P \Rightarrow \overline{M}\subset \overline{P}$.

• (iii) $\overline{M}$ is closed neutrosophic soft set i.e., $\overline{M}\in \tau _{u}^c$.

• (iv) M is neutrosophic soft closed set iff $\overline{M} = M$.

• (v) $\overline{\overline{M}} = \overline{M}$.

• (vi) $\overline{\phi _u} = \phi _u$ and $\overline{1_u} = 1_u$.

• (vii) $\overline{M\cup P } = \overline{M}\cup \overline{P}$.

• (viii) $\overline{M\cap P } \subset \overline{M}\cap \overline{P}$.

Proof

• (i) It follows from definition directly.

• (ii) $M\subset \overline{M}$ and $P\subset \overline{P} \Rightarrow M\subset P\subset \overline{P} \Rightarrow M\subset \overline{P}$

  But $\overline{M}$ is the smallest closed set containing M i.e., $M\subset \overline{M}\subset \overline{P}$. Hence, $\overline{M}\subset \overline{P}$.

• (iii) It is obvious from the definition of $\tau _u$ and $\overline{M}$.

• (iv) Here, $M\subset \overline{M}$ and let M be closed. Then, $M\in \tau ^c_u$ and $M\subset M$.

  Now, $\overline{M} = \cap \left\{ P\in \tau ^c_u: P\supset M\right\} \subset \left\{ M\in \tau ^c_u: M\supset M\right\} = M$

  $\Rightarrow \overline{M}\subset M$ i.e., $M =\overline{M}$

  Next, let $M = \overline{M}$. Then $(\overline{M})^c \in \tau _u \Rightarrow M^c\in \tau _u \Rightarrow M^c$ is open $\Rightarrow M$ is closed.

• (v) If N is closed then $N = \overline{N}$. But $\overline{N}$ is closed by construction of $\overline{N}$. Replacing N by $\overline{M}$, we get $\overline{\overline{M} }= \overline{M}$.

• (vi) $\phi _u, 1_u$ are open as well as closed. So by (iv), the result follows.

• (vii) $M\subset M\cup P$ and $P\subset M\cup P \Rightarrow \overline{M}\subset \overline{M\cup P}$ and $\overline{P}\subset \overline{M\cup P}$

  $\Rightarrow \overline{M}\cup \overline{P}\subset \overline{M\cup P}$

  Also, $M\subset \overline{M}$ and $P\subset \overline{P} \Rightarrow M\cup P\subset \overline{M}\cup \overline{P}$. But we have, $M\cup P\subset \overline{M\cup P}\subset \overline{M}\cup \overline{P}$

  Thus, $\overline{M\cup P } = \overline{M}\cup \overline{P}$.

• (viii) $M\cup P\subset M$ and $M\cup P\subset P \Rightarrow \overline{M\cup P}\subset \overline{M}$ and $\overline{M\cup P}\subset \overline{P}$

  $\Rightarrow \overline{M\cup P}\subset \overline{M}\cup \overline{P}.$ $\square$

3.8.3 Definition

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M\in \textit{NSS}(U,E)$. The the associated closure of M is an NSS over (U, E), denoted by $\{(\overline{T_{f_{M}(e)}}, \overline{I_{f_{M}(e)}}, \overline{F_{f_{M}(e)}})_{e\in E}\}$ and is defined as     $\{(\overline{T_{f_{M}(e)}}, \overline{I_{f_{M}(e)}}, \overline{F_{f_{M}(e)}})_{e\in E}\} = \{(*_i F_{f_{N_i}(e)}, \diamond _i (1- I_{f_{N_i}(e)}), \diamond _i T_{f_{N_i}(e)})_{e\in E}\}$, where $N_i\in \tau _u$ and $f_{M}(e)\subset f_{N^c_i}(e)$.

3.8.4 Proposition

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M\in \textit{NSS}(U,E)$. Then, $\{(\overline{T_{f_{M}(e)}}, \overline{I_{f_{M}(e)}}, \overline{F_{f_{M}(e)}})_{e\in E}\}\subset \overline{M}$     where $\overline{M} = \overline{\{(f_M(e))_{e\in E}\}}$.

Proof

For each $e\in E, \, (\overline{T_{f_{M}(e)}}, \overline{I_{f_{M}(e)}}, \overline{F_{f_{M}(e)}})$ is the smallest closed neutrosophic set containing $f_M(e)$. Also $M \subset \overline{M}$. Hence, $\{(\overline{T_{f_{M}(e)}}, \overline{I_{f_{M}(e)}}, \overline{F_{f_{M}(e)}})_{e\in E}\}\subset \overline{M}$. $\square$

3.8.5 Example

Consider the example (2) of subsection [3.2.1]. Clearly $N^c_1, N^c_2, N^c_3$ are all neutrosophic soft closed. Define an arbitrary NSS $M\in \textit{NSS}(U,E)$ as:

$$\begin{aligned} f_{M}(e_{1})& =\, {} \left\{ \langle x_{1},(0.5,0.6,1.0)\rangle , \langle x_{2},(0.5,0.5,0.6)\rangle , \langle x_{3},(0.3,0.5,0.7)\rangle \right\} ; \\ f_{M}(e_{2})& =\, {} \left\{ \langle x_{1},(0.5,0.7,0.8)\rangle , \langle x_{2},(0.2,0.6,0.8)\rangle , \langle x_{3},(0.4,0.7,0.9)\rangle \right\} ; \end{aligned}$$

Then, $M\subset N^c_2$ and $M\subset N^c_3$. So, $\overline{M} = N^c_2\cap N^c_3 = N^c_2$ which is as following:

$$\begin{aligned} f_{\overline{M}}(e_{1})& =\, {} \left\{ \langle x_{1},(0.6,0.5,0.8)\rangle , \langle x_{2},(0.6,0.3,0.5)\rangle , \langle x_{3},(0.5,0.3,0.4)\rangle \right\} ; \\ f_{\overline{M}}(e_{2})& =\, {} \left\{ \langle x_{1},(0.5,0.4,0.7)\rangle , \langle x_{2},(0.4,0.2,0.6)\rangle , \langle x_{3},(0.6,0.2,0.5)\rangle \right\} ; \end{aligned}$$

Further, $f_{M}(e_1) \subset f_{N^c_2}(e_1), f_{N^c_3}(e_1)$ and $f_{M}(e_2) \subset f_{N^c_1}(e_2), f_{N^c_2}(e_2), f_{N^c_3}(e_2)$;

The tabular representation of $\{(\overline{T_{f_{M}(e)}}, \overline{I_{f_{M}(e)}}, \overline{F_{f_{M}(e)}})_{e\in E}\}$ is as (Table 2):

Table 2 Tabular form of $\{(\overline{T_{f_{M}(e)}}, \overline{I_{f_{M}(e)}}, \overline{F_{f_{M}(e)}})_{e\in E}\}$

Hence, $\overline{M} \ne \{(\overline{T_{f_{M}(e_1)}}, \overline{I_{f_{M}(e_1)}}, \overline{F_{f_{M}(e_1)}})_{e\in E}\}$.

3.8.6 Theorem

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M\in \textit{NSS}(U,E)$. Then,     (i) $(\overline{M})^c = (M^c)^o$       (ii)$(M^o)^c = \overline{(M^c)}$

Proof

• (i) $\overline{M} = \cap \left\{ Q\in \tau ^c_u: Q\supset M\right\}$

  $\Rightarrow (\overline{M})^c = \left( \cap \left\{ Q\in \tau ^c_u: Q\supset M\right\} \right) ^c = \cup \left\{ Q^c\in \tau _u: Q^c\subset M^c\right\} = (M^c)^o$

• (ii) $M^o = \cup \left\{ P\in \tau _u: P\subset M\right\}$

  $\Rightarrow (M^o)^c = \left( \cup \left\{ P\in \tau _u: P\subset M\right\} \right) ^c = \cap \left\{ P^c\in \tau ^c_u: P^c\supset M^c\right\} = \overline{(M^c)}$.

$\square$

3.9 Definition

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M \in \textit{NSS}(U,E)$. Then boundary of M is denoted by Bd(M) and is defined by $Bd(M) = \overline{M}\cap \overline{M^c}$.

3.9.1 Theorem

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $N \in \textit{NSS}(U,E)$. Then,

• (i) $N^o\cap Bd(N) = \phi _u.$

• (ii) $\overline{N} = N^o\cup Bd(N).$

• (iii) $Bd(N) = \phi _u$ iff N is closed and open.

• (iv) $Bd(N) = \overline{N} \cap (N^o)^c.$

Proof

• (i)
  $$\begin{aligned} N^o\cap Bd(N) = N^o\cap (\overline{N}\cap \overline{N^c}) = N^o\cap (\overline{N}\cap {(N^o)^c}) = N^o\cap {(N^o)^c}\cap \overline{N} = \phi _u \cap \overline{N} = \phi _u \end{aligned}$$
• (ii)
  $$\begin{aligned} N^o\cup Bd(N)& =\, {} N^o\cup (\overline{N}\cap \overline{N^c}) = N^o\cup (\overline{N}\cap {(N^o)^c}) = (N^o\cup \overline{N}) \cap \\ (N^o\cup (N^o)^c)& =\, {} (N^o\cup \overline{N})\cap 1_u = (N^o\cup \overline{N}) = \overline{N}, \hbox { as } N^o\subset N\subset \overline{N}. \end{aligned}$$
• (iii)
  $$\begin{aligned} Bd(N)& =\, {} \overline{N}\cup \overline{N^c} = \phi _u\\\Rightarrow & {} \overline{N}\cap (N^o)^c = \phi _u \Rightarrow \overline{N}\cap ((N^o)^c)^c \ne \phi _u \Rightarrow \overline{N}\cap N^o \ne \phi _u \\\Rightarrow & {} \overline{N}\subset N^o \hbox { i.e., } N\subset \overline{N}\subset {N^o} \Rightarrow N\subset N^o; \end{aligned}$$

  Also we have, $N^o\subset N$. Thus, $N = N^o$ i.e., N is open.

  Further, $\overline{N} \subset N^o \subset N \Rightarrow \overline{N}\subset N$; But we have, $N\subset \overline{N}$. Thus, $N = \overline{N}$ i.e., N is closed.

  Conversely, if N is closed and open then $N = N^o$ and $N = \overline{N}$.

  Now, $Bd(N) = \overline{N}\cap \overline{N^c} = \overline{N}\cap (N^o)^c = N\cap N^c = \phi _u$.

• (iv)
  $$\begin{aligned} Bd(N) = \overline{N}\cap \overline{N^c} = \overline{N}\cap (N^o)^c ; \end{aligned}$$

3.10 Definition

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M\in \textit{NSS}(U,E)$. Then,

• (i) an open neutrosophic soft set M is called regular if $M = (\overline{M})^o$ and

• (ii) a closed neutrosophic soft set M is called regular if $M = \overline{(M^o)}$.

3.10.1 Theorem

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and $M, P\in \textit{NSS}(U,E)$. Then,

• (i) if M be closed NSS then $M^o$ is a regular open NSS.

• (ii) if M be open NSS then $\overline{M}$ is a regular closed NSS.

• (iii) if M, P are regular open NSSs then $M\subset P$ iff $\overline{M}\subset \overline{P}$.

• (iv) if M, P are regular closed NSSs then $M\subset P$ iff $M^o\subset P^o$.

• (v) complement of regular closed (open) NSS is a regular open (closed) NSS.

Proof

• (i) Since, M is closed then $M = \overline{M}$. Also, $(M^o)^o = M^o$, as $M^o$ is open.

  Now we have, $M^o\subset M \Rightarrow \overline{M^o} \subset \overline{M} = M \Rightarrow \overline{M^o}\subset M \Rightarrow (\overline{M^o})^o\subset M^o$

  Further, $M^o\subset \overline{M^o} \Rightarrow (M^o)^o\subset (\overline{M^o})^o \Rightarrow M^o\subset (\overline{M^o})^o$;

  Hence, $M^o$ is regular.

• (ii) Since, M is open then $M = M^o$;

  Now, $(\overline{M})^o \subset \overline{M} \Rightarrow \overline{(\overline{M})^o} \subset \overline{\overline{M}} \Rightarrow \overline{(\overline{M})^o} \subset \overline{M}$

  Again, $M\subset \overline{M} \Rightarrow M^o\subset (\overline{M})^o \Rightarrow M\subset (\overline{M})^o \Rightarrow \overline{M} \subset \overline{(\overline{M})^o}$;

  This shows that $\overline{M} = \overline{(\overline{M})^o}$ i.e., $\overline{M}$ is regular closed NSS.

• (iii) Here, $M = (\overline{M})^o$ and $P = (\overline{P})^o$

  If $M\subset P$, then $\overline{M}\subset \overline{P}$.

  Next, $\overline{M}\subset \overline{P} \Rightarrow (\overline{M})^o\subset (\overline{P})^o \Rightarrow M\subset N$.

• (iv) Here, $M = \overline{M^o}$ and $P = \overline{P^o}$;

  Now, $M\subset P \Rightarrow \overline{M^o} \subset \overline{N^o} \Rightarrow M^o\subset N^o$;

  Next, $M^o\subset N^o \Rightarrow \overline{M^o}\subset \overline{N^o} \Rightarrow M\subset N$;

• (v) Let M be a regular closed NSS. We shall show $M^c = (\overline{M^c})^o$

  Here, $M = \overline{M^o}$. Now $(\overline{M^c})^o = ((M^o)^c)^o = (\overline{M^o})^c = M^c$. Next, let M be regular open NSS i.e., $M = (\overline{M})^o$. . Then $\overline{(M^c)^o} = \overline{(\overline{M})^c} = ((\overline{M})^o)^c = M^c$ i.e., $M^c$ is regular closed NSS.$\square$

4 Base for neutrosophic soft topology

4.1 Definition

• 1. A neutrosophic soft point in an NSS N is defined as an element $(e,f_N(e))$ of N, for $e\in E$ and is denoted by $e_N$, if $f_N(e)\notin \phi _u$ and $f_N(e')\in \phi _u, \forall e'\in E-\{e\}$.

• 2. The complement of a neutrosophic soft point $e_N$ is another neutrosophic soft point $e^c_N$ such that $f^c_N(e) = (f_N(e))^c$.

• 3. A neutrosophic soft point $e_N\in M, M$ being an NSS if for the element $e\in E, f_N(e)\le f_M(e)$.

4.1.1 Example

Let $U =\{x_1, x_2, x_3\}$ and $E = \{e_1, e_2\}$. Then,

$$\begin{aligned} e_{1N} = \left\{ \langle x_1,(0.6,0.4,0.8)\rangle , \langle x_2,(0.8,0.3,0.5)\rangle , \langle x_3,(0.3,0.7,0.6)\rangle \right\} \end{aligned}$$

is a neutrosophic soft point whose complement is

$$\begin{aligned} e^c_{1N} = \left\{ \langle x_1,(0.8,0.6,0.6)\rangle , \langle x_2,(0.5,0.7,0.8)\rangle , \langle x_3,(0.6,0.3,0.3)\rangle \right\} . \end{aligned}$$

For another NSS M defined on same (U, E), let,

$$\begin{aligned} f_M(e_1) = \left\{ \langle x_1,(0.7,0.4,0.7)\rangle , \langle x_2,(0.8,0.2,0.4)\rangle , \langle x_3,(0.5,0.6,0.5)\rangle \right\} . \end{aligned}$$

Then, $f_N(e_1) \le f_M(e_1)$   i.e.,   $e_{1N} \in M$.

4.2 Definition

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E). Then a collection ${\ss }_u\subset \tau _u$ is a base for $\tau _u$ if arbitrary $M\in \tau _u$ can be expressed as the union of the members of ${\ss }_u$ i.e., if $M = \cup _i B_i$ for $B_i\in {\ss }_u$. Members of ${\ss }_u$ are called basic open sets.

4.2.1 Example

In the example (1) in [3.2.1],   ${\ss }_u = \{\phi _u, N_1, N_2, N_3\}$ is a base for neutrosophic soft topological space $(U,E,\tau _u)$ over (U, E). Because,

$$\begin{aligned} \phi _u = \phi _u\cup \phi _u, 1_u = N_1\cup N_2, N_1 = N_1\cup N_3, N_2 = N_2\cup N_2, N_3 = N_3\cup N_3, N_4 = N_2\cup N_3; \end{aligned}$$

4.2.2 Theorem

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and ${\ss }_u\subset \tau _u$. Then the followings are equivalent.

• (i) ${\ss }_u$ is a basis of $\tau _u$.

• (ii) For every $M\in \tau _u$ and $e_M \in M$ there exists $B_i\in {\ss }_u$ such that $e_M\in B_k \subset M$.

Proof

(i) $\rightarrow$ (ii)     Let ${\ss }_u$ be a basis of $\tau _N$ and $M\in \tau _u$. Then, $M = \cup _i B_i$ for $B_i\in {\ss }_u$. Now, $e_M\in M \Rightarrow e_M\in \cup _i B_i \Rightarrow e_M\in B_k$, for some k.

Thus, $e_M\in B_k\subset \cup _i B_i = M$.

(ii) $\rightarrow$ (i)     Here M is open NSS and $e_M\in B_i \subset M$ for $B_i\in {\ss }_u$.

Then, $M = \cup _i B_i \Rightarrow {\ss }_u$ is a basis for $\tau _u$. $\square$

4.2.3 Theorem

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) and ${\ss }_u\subset \textit{NSS}(U,E)$. Then ${\ss }_u$ is a basis for topology $\tau _u$ iff followings hold for arbitrary $N\in \tau _u$.

• (i) for every $e_N\in N$ there exists $B_i\in {\ss }_u$ such that $e_N\in B_i$.

• (ii) for every $B_1, B_2\in {\ss }_u$ and $e_N\in B_1\cap B_2$, there exists $B_3\in {\ss }_u$ such that $e_N\in B_3\subset B_1\cap B_2$.

Proof

First suppose that ${\ss }_u$ is a basis for topology $\tau _u$ on (U, E). Then $N = \cup _iB_i$ for $B_i\in {\ss }_N$. Now $e_N\in N \Rightarrow e_N\in B_i$ for some i. Thus condition (i) is satisfied.

Next suppose that $B_1, B_2\in {\ss }_u$. Then $B_1, B_2$ are basic open NSS and $B_1\cap B_2$ is also an open NSS. So for $e_N\in B_1\cap B_2$, there exists another $B_3\in {\ss }_u$ such that $e_N\in B_3\subset B_1\cap B_2$.

Conversely, suppose that the given conditions hold. We are to prove that ${\ss }_u$ is a basis for topology $\tau _u$ on (U, E).

Let $\tau _u$ be a family of union of sets of ${\ss }_u$. Since by condition (i), $N = \cup _i B_i$ for $B_i\in {\ss }_u$ then $N\in \tau _u$. Now if $N_1, N_2\in \tau _u$ and $e_N\in N_1\cap N_2$ there exists $B_1, B_2\in {\ss }_u$ such that $e_N\in B_1\subset N_1$ and $e_N\in B_2\subset N_2$. This implies $e_N\in B_1\cap B_2 \subset N_1\cap N_2$.

By condition (ii), there exists $B_3\in {\ss }_u$ such that $e_N\in B_3\subset B_1\cap B_2 \subset N_1\cap N_2$. This shows that $N_1\cap N_2$ can be expressed as the union of the members in ${\ss }_u$ i.e., $N_1\cap N_2\in \tau _u$. Finally, $N_1\cap N_2 = \phi _u\in \tau _u$ if $N_1, N_2$ are disjoint. Hence, ${\ss }_u$ is a base for the topology $\tau _u$ on (U, E). $\square$

4.2.4 Theorem

If ${\ss }_u$ is a base for a topology $\tau _u$, then $\tau _u$ is the smallest topology containing ${\ss }_u$.

Proof

Let $\tau _{u^1}$ be another topology containing ${\ss }_u$ and $M\in \tau _u$. Then for $e_M\in M$, there exists $B_{e_M} \in {\ss }_u$ such that $e_M \in B_{e_M}\subset M$ i.e., $M = \cup _{e_M} B_{e_M}$ for $e_M\in M$. Now since $\tau _{u^1}$ contains ${\ss }_u$ and $B_{e_M} \in {\ss }_u$ then $B_{e_M} \in \tau _{u^1}$. Also $\tau _{u^1}$ is a topology, so $\cup _{e_M} B_{e_M} \in \tau _{u^1}$ i.e., $M\in \tau _{u^1}$. This shows that $\tau _u\subset \tau _{u^1}$. $\square$

4.2.5 Theorem

Let N be an NSS over (U, E). Suppose $\tau _{u^1}$ and $\tau _{u^2}$ be two topologies on (U, E) generated by the bases ${\ss }_{u^1}$ and ${\ss }_{u^2}$, respectively. Then $\tau _{u^1}\subset \tau _{u^2}$ iff for each $e_N\in N$ and for each $B_1\subset {\ss }_{u^1}$ containing $e_N$, there exists $B_2\in {\ss }_{u^2}$ such that $e_N\in B_2\subset B_1$.

Proof

First suppose, $\tau _{u^1}\subset \tau _{u^2}$ and $e_N \in N, B_1\in {\ss }_{u^1}$ such that $e_N\in B_1$. Since ${\ss }_{u^1}$ is a basis for neutrosophic soft topology $\tau _{u^1}$ on (U, E), then ${\ss }_{u^1}\subset \tau _{u^1} \Rightarrow e_N\in B_1\in {\ss }_{u^1} \subset \tau _{u^1} \subset \tau _{u^2}$ i.e., $e_N\in B_1\in \tau _{u^2}$. Since, ${\ss }_{u^2}$ is the base for the topology $\tau _{u^2}$, so for $B_2\in {\ss }_{u^2}$ we have, $e_N\in B_2\subset B_1$.

Conversely suppose that the hypothesis holds. we show that $\tau _{u^1}\subset \tau _{u^2}$.

Let $M\in \tau _{u^1}$. Since ${\ss }_{u^1}$ is a basis for the topology $\tau _{u^1}$, then for $e_N\in M$ there exists $B_1 \in {\ss }_{u^1}$ such that $e_N\in B_1\subset M$. Now by hypothesis, there exists $B_2 \in {\ss }_{u^2}$ such that $B_2\subset B_1 \Rightarrow B_2\subset B_1 \subset M \Rightarrow B_2\subset M \Rightarrow M\in \tau _{u^2}$. This shows that $\tau _{u^1}\subset \tau _{u^2}$. $\square$

5 Subspace topology

5.1 Definition

Let $(U,E,\tau _u)$ be a neutrosophic soft topological space over (U, E) where $\tau _u$ is a topology on (U, E) and $M\in \textit{NSS}(U,E)$ an arbitrary NSS. Suppose $\tau _M = \{M\cap N_i: N_i\in \tau _u\}$. Then $\tau _M$ forms also a topology on (U, E). Thus $(U,E,\tau _M)$ is a neutrosophic soft subspace topology of $(U,E,\tau _u)$.

5.1.1 Example

Let us consider the example (2) in   subsection [3.2.1]. We define $M\in \textit{NSS}(U,E)$ as following:

$$\begin{aligned} f_{M}(e_{1})& =\, {} \left\{ \langle x_{1},(0.4,0.6,0.8)\rangle , \langle x_{2},(0.7,0.3,0.2)\rangle , \langle x_{3},(0.5,0.5,0.7)\rangle \right\} ; \\ f_{M}(e_{2})& =\, {} \left\{ \langle x_{1},(0.6,0.3,0.5)\rangle , \langle x_{2},(0.4,0.7,0.6)\rangle , \langle x_{3},(0.8,0.3,0.5)\rangle \right\} ; \end{aligned}$$

We denote   $M\cap \phi _u = \phi _M, M\cap 1_u = 1_M, M\cap N_1 = M_1, M\cap N_2 = M_2, M\cap N_3 = M_3$; Then $M_1, M_2, M_3$ are given as following:

$$\begin{aligned} f_{M_1}(e_{1})=\left\{ \langle x_{1},(0.4,0.6,0.8)\rangle , \langle x_{2},(0.6,0.6,0.6)\rangle , \langle x_{3},(0.5,0.6,0.7)\rangle \right\} ; \\ f_{M_1}(e_{2})=\left\{ \langle x_{1},(0.6,0.4,0.5)\rangle , \langle x_{2},(0.4,0.7,0.6)\rangle , \langle x_{3},(0.7,0.5,0.6)\rangle \right\} ; \\ f_{M_2}(e_{1})=\left\{ \langle x_{1},(0.4,0.6,0.8)\rangle , \langle x_{2},(0.5,0.7,0.6)\rangle , \langle x_{3},(0.4,0.7,0.7)\rangle \right\} ; \\ f_{M_2}(e_{2})=\left\{ \langle x_{1},(0.6,0.6,0.5)\rangle , \langle x_{2},(0.4,0.8,0.6)\rangle , \langle x_{3},(0.5,0.8,0.6)\rangle \right\} ; \\ f_{M_3}(e_{1})=\left\{ \langle x_{1},(0.4,0.6,0.8)\rangle , \langle x_{2},(0.4,0.8,0.8)\rangle , \langle x_{3},(0.3,0.8,0.7)\rangle \right\} ; \\ f_{M_3}(e_{2})=\left\{ \langle x_{1},(0.5,0.8,0.6)\rangle , \langle x_{2},(0.4,0.9,0.6)\rangle , \langle x_{3},(0.2,0.9,0.7)\rangle \right\} ; \end{aligned}$$

Here $M_1\cap M_2=M_2, M_1\cap M_3=M_3, M_2\cap M_3=M_3$   and   $M_1\cup M_2=M_2, M_1\cup M_3=M_3, M_2\cup M_3=M_3$. Then $\tau _M = \{\phi _M, 1_M, M_1, M_2, M_3\}$ is neutrosophic soft subspace topology on (U, E).

5.2 Definition

Let M, N be two NSSs over (U, E). Then $M-N$ may be defined as:

$$\begin{aligned} M-N = \, \left\{ \langle x, T_{f_{M}(e)(x)}*F_{f_{N}(e)(x)}, I_{f_{M}(e)(x)}\diamond (1-I_{f_{N}(e)}(x)), F_{f_{M}(e)}(x)\diamond T_{f_{N}(e)}(x)\rangle \right\} \end{aligned}$$

for all $x\in U, \, e\in E$.

5.3 Theorem

Let $(U, E, \tau _u)$ be a neutrosophic soft topological space over (U, E) and $M, N\in \textit{NSS}(U,E)$. Then,

• (i) If ${\ss }_u$ is a base of $\tau _u$ then ${\ss }_M = \{B\cap M: B\in {\ss }_u \}$ is a base for the topology $\tau _M$.

• (ii) If Q is closed NSS in M and M is closed NSS in N, then Q is closed in N.

• (iii) Let $Q\subset M$. If $\overline{Q}$ is the closure of Q then $\overline{Q}\cap M$ is the closure of Q in M.

Proof

• (i) Since ${\ss }_u$ is a base for $\tau _u$ so for arbitrary $N_K\in \tau _u$, we have $N_K = \cup _{B\in {\ss }_u} B$. This implies $N_K\cap M = (\cup _{B\in {\ss }_u} B)\cap M = \cup _{B\in {\ss }_u} (B\cap M)$ for $N_K\cap M \in \tau _M$. Since arbitrary member of $\tau _M$ can be expressed as the union of members of ${\ss }_M$, hence the theorem is completed.

• (ii) To prove this part, we first show that if Q is closed in M then there exists a closed set $V\subset N$ i.e., $V\notin \tau _u$ such that $Q = V\cap M$.

  Let Q be closed in M. Then $Q^c$ is open in M i.e., $Q^c$ can be put as $Q^c = P\cap M$ for $P\in \tau _u \Rightarrow (Q^c)^c = M\cap (P\cap M)^c = P^c\cap M$. Here $P^c\notin \tau _u$ i.e., $P^c$ is closed NSS. So $P^c$ here acts as $V\subset N$.

  Conversely, suppose that $Q = V\cap M$ where $M\subset N$ and V is closed in N. Clearly, $V^c\in \tau _u$ so that $V^c\cap M\in \tau _M$. Now, $V^c\cap M = (N-V)\cap M = (N\cap M)-(V\cap M) = M-Q$. This implies $M-Q$ is open in M i.e., Q is closed in M.

  We now prove the main theorem.

  Since Q is closed NSS in M, then $Q = V\cap M$, for V being a closed NSS in N. By hypothesis, M is closed in N also. Thus Q is closed in N.

• (iii) $\overline{Q} = \cap \{Q_i: Q_i \, \text{ is }\,\text{ closed }\, \text{ and } \, Q_i\supset Q\}$ is the closure of Q and so $\overline{Q}$ is closed.

  Now, $\overline{Q}\cap M = \cap \{ Q_i: Q_i \, \text{ is }\,\text{ closed }\, \text{ and } \, Q_i\supset Q\}\cap M = \cap (Q_i\cap M)$. Since each $Q_i$ is closed then each $Q_i\cap M$ is closed in M by theorem (ii). Now, $Q\subset Q_i$ and $Q\subset M$. So $Q\cap M\subset Q_i\cap M \Rightarrow Q\subset Q_i\cap M$.

  Thus, $\overline{Q}\cap M = \cap \{(Q_i\cap M): (Q_i\cap M)\, \text{ is }\,\text{ closed }\, \text{ and } \, (Q_i\cap M)\supset Q\}$ Hence $\overline{Q}\cap M$ is a closure of Q in M.$\square$

5.4 Theorem

Let $(U, E,\tau _M)$ be a subspace topology of a topological space $(U, E,\tau _u)$ over (U, E). If M is open NSS in $(U, E,\tau _u)$, then an NSS $M_1\subset M$ is open in $(U, E, \tau _M)$ iff $M_1$ is open in $(U, E,\tau _u)$. The interior of a neutrosophic soft subset P of the open NSS M in $(U, E,\tau _M)$ is the same as the interior of P in $(U, E,\tau _u)$.

Proof

First suppose that M is open NSS in $(U, E, \tau _u)$ such that a neutrosophic soft subset $M_1$ of M is open in $(U, E, \tau _M)$. Then $M_1\in \tau _M$ and so $M_1 = N_1\cap M$ for $N_1\in \tau _u$. But $M_1$ is open NSS in $(U, E,\tau _u)$ as $N_1, M$ both are open NSSs in $(U, E, \tau _u)$.

Conversely, assume that $M_1$ is open NSS in $(U, E, \tau _u)$ when M is open NSS in $(U, E,\tau _u)$ and $M_1\subset M$. Then $M_1\in \tau _u$. But $M_1\cap M = M_1$ and so $M_1$ is open NSS in $(U, E,\tau _M)$. Hence the first part is proved. Now,

$$\begin{aligned} P^o \, \text{ in } \, (U, E,\tau _M)& =\, {} \cup \left\{ M_1: M_1\subset P \, \text{ and } \, M_1\in \tau _M \right\} \\& =\, {} \cup \left\{ (N_1\cap M): (N_1\cap M)\subset P \, \text{ and } \, N_1\in \tau _u \right\} \\& =\, {} \cup \left\{ (N_1\cap M): (N_1\cap M)\subset P \, \text{ and } \, (N_1\cap M)\in \tau _u\right\} \\ & \quad\left[ \text{ as }\, M \, \text{ is }\,\text{ open }\, \text{ NSS } \, \text{ in }\, (U, E, \tau _u)\right] \\& =\, {} \cup \left\{ M_1: M_1\subset P \, \text{ and } \, M_1\in \tau _u\right\} \\& =\, {} P^o \, \text{ in } \, (U, E,\tau _u) \end{aligned}$$

Thus the theorem is proved. $\square$

5.5 Theorem

Let $(U, E,\tau _Q)$ be a subspace topology of a topological space $(U, E,\tau _u)$ over (U, E). If Q is closed NSS in $(U, E, \tau _u)$, then an NSS $Q_1\subset Q$ is closed in $(U, E,\tau _Q)$ iff $Q_1$ is closed in $(U, E,\tau _u)$. The closure of a neutrosophic soft subset S of the closed NSS Q in $(U, E,\tau _Q)$ is the same as the closure of S in $(U, E,\tau _u)$.

Proof

First suppose that Q is closed NSS in $(U, E, \tau _u)$ such that a neutrosophic soft subset $Q_1$ of Q is closed in $(U, E, \tau _Q)$. Since $Q_1$ is closed in $(U, E, \tau _Q)$ and so $Q_1 = N_2\cap Q$ for $N_2$ being closed in $(U, E,\tau _u)$. But $Q_1$ is closed NSS in $(U, E,\tau _u)$ as $N_2, Q$ both are closed NSSs in $(U, E,\tau _u)$.

Conversely, assume that $Q_1$ is closed NSS in $(U, E,\tau _u)$ when Q is closed NSS in $(U, E,\tau _u)$ and $Q_1\subset Q$. Then $Q_1\cap Q = Q_1$ and so $Q_1$ is closed NSS in $(U, E,\tau _Q)$. Hence the first part is proved. Now,

$$\begin{aligned} \overline{S} \, \text{ in } \, (U, E,\tau _Q)& =\, {} \cap \left\{ Q_1: Q_1\supset S \, \text{ and } \, Q^c_1\in \tau _Q \right\} \\& =\, {} \cap \left\{ (N_2\cap Q): (N_2\cap Q)\supset S \, \text{ and } \, N^c_2\in \tau _u \right\} \\& =\, {} \cap \left\{ (N_2\cap Q): (N_2\cap Q)\supset S \, \text{ and } \, (N_2\cap Q)^c\in \tau _u\right\} \\&\left[ \text{ as }\, Q \, \text{ is }\,\text{ closed }\, \text{ NSS }\, \text{ in }\, (U, E,\tau _u)\right] \\& =\, {} \cap \left\{ Q_1: Q_1\supset S \, \text{ and } \, Q^c_1\in \tau _u \right\} \\& =\, {} \overline{S} \, \text{ in } \, (U, E,\tau _u) \end{aligned}$$

Thus the theorem is proved. $\square$

6 Separation axioms

6.1 Definition

$\mathbf{T_0 :}$   A neutrosophic soft topological space $(U, E,\tau _u)$ over (U, E) is said to be a neutrosophic soft $T_0$ space if for every pair of disjoint neutrosophic soft points $e_{N_1}$ and $e_{N_2}$ there exists a neutrosophic soft open set containing one but not the other.

A discrete neutrosophic soft topological space is a neutrosophic soft $T_0$-space since every neutrosophic soft point is a neutrosophic soft open set in the discrete space.

$\mathbf{T_1 :}$   A neutrosophic soft topological space $(U, E,\tau _u)$ over (U, E) is said to be a neutrosophic soft $T_1$ space if for every pair of disjoint neutrosophic soft points $e_K$ and $e_S$, there exists neutrosophic soft open sets M and P such that $e_K\in M, e_K\notin P$ and $e_S\in P, e_S\notin M$.

Let $U=\{h_{1},h_{2}\}, \,\, E =\{e_{1}, e_{2}\}$ and $\tau _u = \{\phi _u, 1_u, M, P\}$ where $M, P\in \textit{NSS}(U,E)$ are defined as following:

$$\begin{aligned} f_{M}(e_{1})& =\, {} \left\{ \langle h_{1},(1,0,1)\rangle , \langle h_{2},(0,0,1)\rangle \right\} ; \\ f_{M}(e_{2})& =\, {} \left\{ \langle h_{1},(0,1,0)\rangle , \langle h_{2},(1,0,0)\rangle \right\} ; \\ f_{P}(e_{1})& =\, {} \left\{ \langle h_{1},(0,1,0)\rangle ,\langle h_{2},(1,1,0)\rangle \right\} ; \\ f_{P}(e_{2})& =\, {} \left\{ \langle h_{1},(1,0,1)\rangle ,\langle h_{2},(0,1,1)\rangle \right\} ; \end{aligned}$$

Then $\tau _u$ is a neutrosophic soft topology on (U, E) with respect to the t-norm and s-norm defined as $a*b = max\{a+b-1,0\}$ and $a\diamond b = min\{a+b,1\}$. Here $e_{1M}\in M, e_{1M}\notin P$ and $e_{2P}\in P, e_{2P}\notin M$ though $e_{1M}, e_{2P}$ are distinct.

$\mathbf{T_2 :}$   Let $(U, E,\tau _u)$ be a neutrosophic soft topological space over (U, E). For two distinct neutrosophic soft points $e_K, e_S$ if there exists disjoint neutrosophic soft open sets M, P such that $e_K\in M$ and $e_S\in P$ then $(U, E,\tau _u)$ is called $T_2$ space or Hausdorff space.

Let $U=\{h_{1},h_{2}\}, \,\, E =\{e\}$ and $\tau _u = \{\phi _u, 1_u, M, P\}$ where $M, P\in \textit{NSS}(U,E)$ are defined as following:

$$\begin{aligned} f_{M}(e)=\left\{ \langle h_{1},(1,0,1)\rangle , \langle h_{2},(0,0,1)\rangle \right\} ; \\ f_{P}(e)=\left\{ \langle h_{1},(0,1,0)\rangle ,\langle h_{2},(1,1,0)\rangle \right\} ; \end{aligned}$$

Then $\tau _u$ is a neutrosophic soft topology on (U, E) with respect to the t-norm and s-norm defined as $a*b = max\{a+b-1,0\}$ and $a\diamond b = min\{a+b,1\}$. Here $e_{M}\in M$ and $e_{P}\in P$ where $e_{M}\ne e_{P}$ and $M\cap P = \phi _u$.

Regular:  A neutrosophic soft topological space $(U, E,\tau _u)$ over (U, E) is said to be a neutrosophic soft regular space if for every neutrosophic soft point $e_K$ and neutrosophic soft closed set M not containing $e_K$, there exists disjoint neutrosophic soft open sets $P_1, P_2$ such that $e_K\in P_1$ and $M\subset P_2$.

Let $U=\{h_{1},h_{2}\}, \,\, E =\{e\}$ and $\tau _u = \{\phi _u, 1_u, P_1, P_2\}$ where $P_1, P_2\in \textit{NSS}(U,E)$ are defined as following:

$$\begin{aligned} f_{P_1}(e)& =\, {} \left\{ \langle h_{1},(0,1,0)\rangle , \langle h_{2},(0,1,1)\rangle \right\} ; \\ f_{P_2}(e)& =\, {} \left\{ \langle h_{1},(1,0,1)\rangle , \langle h_{2},(1,0,0)\rangle \right\} ; \end{aligned}$$

Then $\tau _u$ is a neutrosophic soft topology on (U, E) with respect to the t-norm and s-norm defined as $a*b = max\{a+b-1,0\}$ and $a\diamond b = min\{a+b,1\}$ where $P_1\cap P_2 = \phi _u$. Here, $e_{P_1}\in P_1$ and $e_{P_1}\notin M \subset P_2$ where $M = P^c_2$ is a closed NSS.

$\mathbf{T_3:}$   A neutrosophic soft topological space $(U, E,\tau _u)$ over (U, E) is said to be a neutrosophic soft $T_3$-space if it is neutrosophic soft regular and neutrosophic soft $T_1$-space.

Normal:   A neutrosophic soft topological space $(U, E,\tau _u)$ over (U, E) is said to be a neutrosophic soft normal space if for every pair of disjoint neutrosophic soft closed sets $M_1, M_2$ there exists disjoint neutrosophic soft open sets $P_1, P_2$ such that $M_1 \subseteq P_1$ and $M_2\subseteq P_2$.

Let $U=\{x_{1},x_{2}\}, \,\, E =\{e\}$ and $\tau _u = \{\phi _u, 1_u, P_1, P_2\}$ where $P_1, P_2\in \textit{NSS}(U,E)$ are defined as following:

$$\begin{aligned} f_{P_1}(e)& =\, {} \left\{ \langle x_{1},(1,1,0)\rangle , \langle x_{2},(0,1,1)\rangle \right\} ; \\ f_{P_2}(e)& =\, {} \left\{ \langle x_{1},(0,0,1)\rangle , \langle x_{2},(1,0,0)\rangle \right\} ; \end{aligned}$$

Then $\tau _u$ is a neutrosophic soft topology on (U, E) with respect to the t-norm and s-norm, $a*b = max\{a+b-1,0\}$ and $a\diamond b = min\{a+b,1\}$. Here $P_1, P_2$ are disjoint neutrosophic soft open sets. Then $P^c_1, P^c_2$ are disjoint neutrosophic soft closed sets in $(U, E,\tau _u)$ with $P^c_1 = P_2, P^c_2 =P_1$

$\mathbf{T_4:}$   A neutrosophic soft topological space $(U, E,\tau _u)$ over (U, E) is said to be a neutrosophic soft $T_4$-space if it is neutrosophic soft normal and neutrosophic soft $T_1$-space.

6.2 Proposition

Let $(U, E,\tau _M)$ be a neutrosophic soft subspace topology of a neutrosophic soft topological space $(U, E,\tau _u)$ over (U, E). Then,

• (i) if $(U, E,\tau _u)$ is a neutrosophic soft $T_0$-space then $(U, E,\tau _M)$ is also so.

• (ii) if $(U, E,\tau _u)$ is a neutrosophic soft $T_1$-space then $(U, E,\tau _M)$ is also so.

• (iii) if $(U, E,\tau _u)$ is a neutrosophic soft $T_2$-space then $(U, E,\tau _M)$ is also so.

• (iv) if $(U, E,\tau _u)$ is a neutrosophic soft regular space then $(U, E,\tau _M)$ is also so.

• (v) if $(U, E,\tau _u)$ is a neutrosophic soft $T_3$-space then $(U, E,\tau _M)$ is also so.

Proof

(i) Let $e_K, e_S \in M$ with $e_K\ne e_S$. Since $(U, E,\tau _u)$ is a neutrosophic soft $T_0$-space then $N_1, N_2\in \tau _u$ such that $e_K\in N_1, e_S\notin N_1$ or $e_S\in N_2, e_K\notin N_2$. Hence, $e_K\in M\cap N_1, e_S \notin M\cap N_1$ or $e_S\in M\cap N_2, e_K\notin M\cap N_2$. Thus $(U, E,\tau _M)$ is a neutrosophic soft $T_0$-space.

The others can be proved in the similar manner. $\square$

6.3 Proposition

Neutrosophic soft $T_4$- space $\Rightarrow$ neutrosophic soft $T_3$- space $\Rightarrow$ neutrosophic soft $T_2$- space $\Rightarrow$ neutrosophic soft $T_1$- space $\Rightarrow$ neutrosophic soft $T_0$- space.

Proof

We here give the proof of last $(\Rightarrow )$. The proof of others are in similar way.

Let $(U, E,\tau _u)$ be a neutrosophic soft topological space over (U, E). We consider two distinct neutrosophic soft points $e_k, e_S$. Since $(U, E,\tau _u)$ is a neutrosophic soft $T_1$-space then there exists $M, P\in \tau _u$ such that $e_K\in M, e_K\notin P$ and $e_S\notin M, e_S\in P$. This shows $e_K\in M, e_S\notin M$ or $e_K\notin P, e_S\in P$. Hence, $(U, E,\tau _u)$ is a $T_0$-space over (U, E). $\square$

6.4 Theorem

A neutrosophic soft topological space $(U, E,\tau _u)$ is a neutrosophic soft $T_2$-space iff for distinct neutrosophic soft points $e_K, e_S$ there exists a neutrosophic soft open set M containing $e_K$ but not $e_S$ such that $e_S\notin \overline{M}$.

Proof

Let $e_K, e_S$ be two distinct neutrosophic soft points in a neutrosophic soft $T_2$-space $(U, E,\tau _u)$. Then there exists disjoint neutrosophic soft open sets $N_1, N_2$ such that $e_K\in N_1, e_S\in N_2$. Since $e_K\ne e_S$ and $N_1, N_2$ are disjoint NSSs, then $e_S\notin N_1$. It implies $e_S\notin \overline{N_1}$ as $N_1\subset \overline{ N_1}$. By similar argument $e_K\notin N_2$ and so $e_K\notin \overline{N_2}$.

Next suppose, for distinct neutrosophic soft points $e_K, e_S$ there exists a neutrosophic soft open set M containing $e_K$ but not $e_S$ such that $e_S\notin \overline{M}$. Then $e_S\in (\overline{M})^c$ i.e., M and $(\overline{M})^c$ are disjoint neutrosophic soft open set containing $e_K$ and $e_S$, respectively. $\square$

6.5 Theorem

A neutrosophic soft topological space $(U, E,\tau _u)$ in which every neutrosophic soft point is neutrosophic soft closed, is neutrosophic soft regular space iff for a neutrosophic soft open set M containing a neutrosophic soft point $e_K$, there exists a neutrosophic soft open set P containing $e_S$, such that $\overline{P}\subset M$.

Proof

We take a neutrosophic soft open set M containing $e_K$ in a regular neutrosophic soft topological space $(U, E,\tau _u)$. Then $M^c$ is neutrosophic soft closed and $e_K\notin M^c$. By hypothesis, there exists disjoint neutrosophic soft open sets $N_1, N_2$ such that $e_K\in N_1$ and $M^c\subset N_2$. Now since $N_1\cap N_2 = \phi _u$, so $N_1\subset N^c_2 \Rightarrow \overline{N_1}\subset \overline{N^c_2} = N^c_2$ as $N^c_2$ is closed. But $(M^c)^c\supset N^c_2 \Rightarrow M\supset N^c_2$. Thus $\overline{N_1}\subset M$.

Conversely, assume that the hypothesis hold. Take a neutrosophic soft closed set Q not containing a neutrosophic soft point $e_K$. Then $e_K\in Q^c$ and $Q^c$ is a neutrosophic soft open set. This implies there exists a neutrosophic soft open set P containing $e_K$ such that $\overline{P}\subset Q^c \Rightarrow Q\subset (\overline{P})^c \Rightarrow (\overline{P})^c$ is a neutrosophic soft open set containing Q and $P\cap (\overline{P})^c = \phi _u$ $[\text{ as } \, P\subset \overline{P}\, ]$. $\square$

6.6 Theorem

A neutrosophic soft topological space $(U, E,\tau _u)$ is neutrosophic soft normal space iff for any neutrosophic soft closed set Q and neutrosophic soft open set P containing Q, there exists a neutrosophic soft open set M such that $Q\subset M$ and $\overline{M}\subset P$.

Proof

Let $(U, E,\tau _u)$ be a neutrosophic soft normal space and P be a neutrosophic soft open set containing Q where Q be a neutrosophic soft closed set i.e., $P^c, Q$ be two disjoint neutrosophic soft closed sets. Then there exists disjoint neutrosophic soft open sets $N_1, N_2$ such that $Q\subset N_1$ and $P^c\subset N_2$. Now $N_1\subset N^c_2 \Rightarrow \overline{N_1}\subset \overline{N^c_2}= N^c_2$. Also, $P^c \subset N_2 \Rightarrow N^c_2 \subset P \Rightarrow \overline{N_1}\subset P$.

Conversely, let $Q_1, Q_2$ be two disjoint pair of neutrosophic soft closed sets. Then $Q_1\subset Q^c_2$. By hypothesis there exists a neutrosophic soft open set M such that $Q_1\subset M$ and $\overline{M}\subset Q^c_2 \Rightarrow Q_2\subset (\overline{M})^c \Rightarrow M$ and $(\overline{M})^c$ are disjoint neutrosophic soft open sets such that $Q_1\subset M$ and $Q_2\subset (\overline{M})^c$. $\square$

7 Conclusion

In the present paper, the topological structure on neutrosophic soft set has been introduced. We propose some properties of neutrosophic soft interior and neutrosophic soft closure. We can say that a neutrosophic soft topological space gives a parameterized family of fuzzy tritopologies on the initial universe but the reverse is not true. Besides, here we also have defined the base for neutrosophic soft topological space, subspace on neutrosophic soft set, separation axioms with suitable examples. Several related properties and structural characteristics in each case have been investigated. This concept will bring a new opportunity in future research and development of NSS theory.