Separation Axioms of Interval-Valued Fuzzy Soft Topology via Quasi-Neighborhood Structure

Abstract

In this study, we present the concept of the interval-valued fuzzy soft point and then introduce the notions of its neighborhood and quasi-neighborhood in interval-valued fuzzy soft topological spaces. Separation axioms in an interval-valued fuzzy soft topology, so-called q-$T_i$ for $i=0,1,2,3,4$, are introduced, and some of their basic properties are also studied.

1. Introduction

In 1999, Molodtsov [1] proposed a new mathematical approach known as soft set theory for dealing with uncertainties and vagueness. Traditional tools such as fuzzy sets [2] and rough sets [3] cannot clearly define objects. Soft set theory is different from traditional tools for dealing with uncertainties. A soft set was defined by a collection of approximate descriptions of an object based on parameters by a given set-valued map. Maji et al. [4] initiated the research on both fuzzy set and soft set hybrid structures called fuzzy soft sets and presented a concept that was subsequently discussed by many researchers. Different extensions of the classical fuzzy soft sets were introduced, such as generalized fuzzy soft sets [5], intuitionist fuzzy soft sets [6,7], vague soft sets [8], interval-valued fuzzy soft sets [9], and interval-valued intuitive fuzzy soft sets [10]. In particular, to alleviate some disadvantages of fuzzy soft sets, interval-valued fuzzy soft sets were introduced where no objective procedure was available to select the crisp membership degree of elements in fuzzy soft sets. Tanya and Kandemir [11] started topological studies of fuzzy soft sets. They used the classical concept of topology to construct a topological space over a fuzzy soft set and named it the fuzzy soft topology. They also studied some fundamental topological properties for the fuzzy soft topology, such as interior, closure, and base. Later, Simsekler and Yuksel [12] studied the fuzzy soft topological space in the case of Tanay and Kandemir [11]. However, they established the concept of the fuzzy soft topology over a fuzzy soft set with a set of fixed parameters and considered some topological concepts for fuzzy soft topological spaces such as the base, subbase, neighborhood, and Q-neighborhood. Roy and Samanta [13] noted a new concept of the fuzzy soft topology. They suggested the notion of the fuzzy soft topology over an ordinary set by adding fuzzy soft subsets of it, where everywhere, the parameter set is supposed to be fixed. Then, in [14], they continued to study the fuzzy soft topology and established a fuzzy soft point definition and various neighborhood structures. Atmaca and Zorlutuna [15] considered the concept of soft quasi-coincidence for fuzzy soft sets. By applying this new concept, they also studied the basic topological notions such as interior and closure for fuzzy soft sets. The concept of the product fuzzy soft topology and the boundary fuzzy soft topology was introduced by Zahedi et al. [16,17], and they studied some of their properties. They also suggested a new definition for the fuzzy soft point and then different neighborhood structures. Separation axioms of the fuzzy topological space and fuzzy soft topological space were studied by many authors, see [18,19,20,21,22,23,24,25,26,27]. The aim of this work is to develop interval-valued fuzzy soft separation axioms. We start with preliminaries and then give the definition of the interval-valued fuzzy soft point as a generalization of the interval-valued fuzzy point and fuzzy soft point in order to create different neighborhood structures in the interval-valued fuzzy soft topological space in Section 3 and Section 4. Finally, in Section 5, the notion of separation axioms q-$T_i,i=0,1,2,3,4$ in the interval-valued fuzzy soft topology is introduced, and some of their basic properties are also studied.

2. Preliminaries

Throughout this paper, X is the set of objects and E is the set of parameters. The set of all subsets of X is denoted by $P\left(X\right)$ and $A\subset E$, showing a subset of E.

Definition 1

([1]). A pair $(f,A)$ is called a soft set over X, if f is a mapping given by $f:A\to P\left(X\right)$. For any parameter $e\in A,f\left(e\right)\subset X$ may be considered as the set e-approximate elements of the soft set $(f,A)$. In other words, the soft set is not a kind of set, but a parameterized family of subsets of the set X.

Before introducing the notion of the interval-valued fuzzy soft sets, we give the concept of the interval-valued fuzzy set.

Definition 2

([28]). An interval-valued fuzzy $\left(IVF\right)$ set over X is defined by the membership function $f:X\to int\left(\right[0,1\left]\right)$, where $int\left(\right[0,1\left]\right)$ denotes the set of all closed subintervals of $[0,1]$. Suppose that $x\in X$. Then, $f\left(x\right)=[f^{-}\left(x\right),f^{+}\left(x\right)]$ is called the degree of membership of the element $x\in X$, where $f^{-}\left(x\right)$ and $f^{+}$ are the lower and upper degrees of the membership of x and $0<f^{-}\left(x\right)<f^{+}\left(x\right)<1$.

Yang et al. [9] suggested the concept of interval-valued fuzzy soft set by combining the interval-valued fuzzy set and soft set as below.

Definition 3

([9]). An interval-valued fuzzy soft $\left(IVFS\right)$ set over X denoted by $f_E$ or $(f,E)$ is defined by the mapping $f:E\to \mathcal{IVF}\left(X\right)$, where $\mathcal{IVF}\left(X\right)$ is the set of all interval-valued fuzzy sets over X. For any $e\in E,f\left(e\right)$ can be written as an interval-valued fuzzy set such that $f\left(e\right)=\{\langle x,[f_e^{-}\left(x\right),f_e^{+}\left(x\right)]\rangle :x\in X\}$ where $f_e^{-}\left(x\right)and{f}_e^{+}\left(x\right)$ are the lower and upper degrees of the membership of x with respect to e, where $0\le f_e^{-}\left(x\right)\le f_e^{+}\left(x\right)\le 1$.

Note that $\mathcal{IVFS}(X,E)$ shows the set of all $IVFS$ sets over X.

Definition 4

([9]). Let $f_A$ and $g_B$ be two $IVFS$ sets overX. We say that:

1. 

$f_A$ is an interval-valued fuzzy soft subset of $g_B$, denoted by $f_A\tilde{\le }{g}_B$, if and only if:

(i) 

$A\le B$,

(ii) 

For all $e\in A,f_e^{-}\left(x\right)\le g_e^{-}\left(x\right)$ and $f_e^{+}\left(x\right)\le g_e^{+}\left(x\right),\forall x\in X$.

2. 

$f_A=g_B$ if and only if $f_A\tilde{\le }{g}_B$ and $g_A\tilde{\le }{f}_B$.

3. 

The union of two $IVFS$ sets $f_A$ and $g_B$, denoted by $f_A\tilde{\vee }{g}_B$, is the $IVFS$ set $(f\vee g,C)$, where $C=A\cup B$, and for all $e\in C$, we have:

$${(f\vee g)}_e\left(x\right)=\left\{\begin{array}{ll}[f_e^{-}\left(x\right),f_e^{+}\left(x\right)],& e\in A-B\\ [g_e^{-}\left(x\right),g_e^{+}\left(x\right)],& e\in B-A\\ [max(f_e^{-}\left(x\right),g_e^{-}\left(x\right),max(f_e^{+}\left(x\right),g_e^{+}\left(x\right)]& e\in A\cap B,\end{array}\right.$$

for all $x\in X$.

4. 

The intersection of two $IVFS$ sets $f_A$ and $g_B$, denoted by $f_A\tilde{\wedge }{g}_B$, is the $IVFS$ set $(f\wedge g,C)$, where $C=A\cap B$, and for all $e\in C$, we have ${(f\wedge g)}_e\left(x\right)=[min{f}_e^{-}\left(x\right),g_e^{-}\left(x\right),min{f}_e^{+}\left(x\right),g_e^{+}\left(x\right)]$ for all $x\in X$.

5. 

The complement of the $IVFS$ set $f_A$ is denoted by $f_A^c\left(x\right)$ where for all $e\in A$, we have $f_e^c\left(x\right)=[1-f_e^{+}\left(x\right),1-f_e^{-}\left(x\right)]$.

Definition 5

([9]). Let $f_E$ be an $IVFS$ set. Then:

1. 

$f_E$ is called the null interval-valued fuzzy soft set, denoted by ${\varnothing }_E$, if $f_e^{-}\left(x\right)=f_e^{+}\left(x\right)=0$, for all $x\in X,e\in E$.

2. 

$f_E$ is called the absolute interval-valued fuzzy soft set, denoted by $X_E$, if $f_e^{-}\left(x\right)=f_e^{+}\left(x\right)=1$, for all $x\in X,e\in E$.

Motivated by the definition of the soft mapping, discussed in [29], we define the concept of the $IVFS$ mapping as the following:

Definition 6.

Let $f_A$ be an $IVFS$ set over $X_1$ and $g_B$ be an $IVFS$ set over $X_2$, where $A\subseteq E_1$ and $B\subseteq E_2$. Let ${\Phi }_u:X_1\to X_2$ and ${\Phi }_p:E_1\to E_2$ be two mappings. Then:

1. 

The map $\Phi :\mathcal{IVFS}(X_1,E_1)\to \mathcal{IVFS}(X_2,E_2)$ is called an $IVFS$ map from $X_1$ to $X_2$, and for any $y\in X_2$ and $\epsilon \in B\subseteq E_2$, the lower image and the upper image of $f_A$ under Φ is the $IVFS\Phi \left(f_A\right)$ over $X_2$, respectively, defined as below:

$$[\Phi \left(f^{-}\right)]\left(\epsilon \right)\left(y\right)=\left\{\begin{array}{ll}{sup}_{x\in {\Phi }_{u^{-1}}\left(y\right)}\left[\underset{e\in {\Phi }_{p^{-1}\cap A}}{sup}{f}^{-}\left(e\right)\right]\left(x\right),& \mathrm{if}{\Phi }_p^{-1}\left(\epsilon \right)\cap A\ne \varphi \mathrm{and}{\Phi }_u^{-1}\left(y\right)\ne \varphi \\ 0,& otherwise,\end{array}\right.$$

$$[\Phi \left(f^{+}\right)]\left(\epsilon \right)\left(y\right)=\left\{\begin{array}{ll}{sup}_{x\in {\Phi }_{u^{-1}}\left(y\right)}\left[\underset{e\in {\Phi }_{p^{-1}\cap A}}{sup}{f}^{+}\left(e\right)\right]\left(x\right),& \mathrm{if}{\Phi }_p^{-1}\left(\epsilon \right)\cap A\ne \varphi \mathrm{and}{\Phi }_u^{-1}\left(y\right)\ne \varphi \\ 0,& otherwise.\end{array}\right.$$

2. 

Let $\Phi :\mathcal{IVFS}(X_1,E_1)\to \mathcal{IVFS}(X_2,E_2)$ be an $IVFS$ map from $X_1$ to $X_2$. The lower inverse image and the upper inverse image of $IVFS{g}_B$ under Φ denoted by ${\Phi }^{-1}\left(g_B\right)$ is an $IVFS$ over $X_1$, respectively, such that for all $x\in X_1$ and $e\in E_1$, it is defined as below:

$$\left[{\Phi }^{-1}\left(g^{-}\right)\right]\left(e\right)\left(x\right)=\left\{\begin{array}{ll}{g}_{{\Phi }_{p\left(e\right)}}^{-}{\Phi }_u\left(x\right),& \mathrm{if}{\Phi }_p\left(e\right)\in B\\ 0,& otherwise,\end{array}\right.$$

$$\left[{\Phi }^{-1}\left(g^{+}\right)\right]\left(e\right)\left(x\right)=\left\{\begin{array}{ll}{g}_{{\Phi }_{p\left(e\right)}}^{+}{\Phi }_u\left(x\right),& if{\Phi }_p\left(e\right)\in B\\ 0& otherwise.\end{array}\right.$$

Proposition 1.

Let $\Phi :\mathcal{IVFS}(X,E)\to \mathcal{IVFS}(Y,F)$ be an $IVFS$ mapping between X and X, and let ${\left\{f_{iA}\right\}}_{i\in J}\subset \mathcal{IVFS}(X,E)$ and ${\left\{g_{iB}\right\}}_{i\in J}\subset \mathcal{IVFS}(Y,F)$ be two families of $IVFS$ sets over X and Y, respectively, where $A\subseteq E$ and $B\subseteq F$, then the following properties hold.

1. 

${[\Phi \left(f_{jA}\right)]}^c\tilde{\le }\Phi {\left(f_{jA}\right)}^c$ for each $j\in J$.

2. 

${\left[{\Phi }^{-1}\left(g_{jB}\right)\right]}^c={\Phi }^{-1}{\left(g_{jB}\right)}^c$ for each $j\in J$.

3. 

If $g_{iB}\tilde{\le }{g}_{jB}$, then ${\Phi }^{-1}\left(g_{iB}\right)\tilde{\le }{\Phi }^{-1}\left(g_{jB}\right)$ for each $i,j\in J$.

4. 

If $f_{iA}\tilde{\le }{f}_{jA}$, then $\Phi \left(f_{iA}\right)\tilde{\le }\Phi \left(f_{jA}\right)$ for each $i,j\in J$.

5. 

$\Phi \left[{\tilde{\vee }}_{j\in J}{f}_{jA}\right]={\tilde{\vee }}_{j\in J}\Phi \left(f_{jA}\right)$ and ${\Phi }^{-1}\left[{\tilde{\vee }}_{j\in J}{g}_{jB}\right]={\tilde{\vee }}_{j\in J}{\Phi }^{-1}\left(g_{jB}\right)$.

6. 

$\Phi \left[{\tilde{\wedge }}_{j\in J}{f}_{jA}\right]={\tilde{\wedge }}_{j\in J}\Phi \left(f_{jA}\right)$ and ${\Phi }^{-1}\left[{\tilde{\wedge }}_{j\in J}{g}_{jB}\right]={\tilde{\wedge }}_{j\in J}{\Phi }^{-1}\left(g_{jB}\right)$.

Proof. 

We only prove Part (5). The other parts follow a similar technique. For any $k\in F,y\in Y$, and $a\in A$, then:

$$\begin{array}{lll}\Phi \left[{\tilde{\vee }}_{j\in J}{f}_{jA}\right]\left(k\right)\left(y\right)& =& {\displaystyle \underset{x\in {\Phi }_u^{-}\left(y\right)}{sup}}({\displaystyle \underset{z\in {\Phi }_p^{-1}\left(k\right)}{sup}}\left({\tilde{\vee }}_{j\in J}\right)f_{jA})\left(z\right)\left(x\right)\\ & =& {\displaystyle \underset{x\in {\Phi }_u^{-1}\left(y\right)}{sup}}({\displaystyle \underset{z\in {\Phi }_p^{-1}\left(k\right)}{sup}}({\displaystyle \underset{j\in J}{max}}([f_{ja}^{-},f_{ja}^{+}])))\left(k\right)\left(y\right)\hfill \\ & =& {\displaystyle \underset{x\in {\Phi }_u^{-1}\left(y\right)}{sup}}({\displaystyle \underset{j\in J}{max}}({\displaystyle \underset{z\in {\Phi }_p^{-1}\left(k\right)}{sup}}[f_{ja}^{-}\left(k\right),f_{ja}^{+}\left(k\right)]))\left(y\right)\hfill \\ & =& {\displaystyle \underset{j\in J}{max}}({\displaystyle \underset{x\in {\Phi }_u^{-1}\left(y\right)}{sup}}({\displaystyle \underset{z\in {\Phi }_p^{-1}\left(k\right)}{sup}}[f_{ja}^{-}\left(k\right))\left(y\right),f_{ja}^{+}\left(k\right)]))\left(y\right)]\hfill \\ & =& {\displaystyle \underset{j\in J}{max}}({\displaystyle \underset{x\in {\Phi }_u^{-}\left(y\right)}{sup}}({\displaystyle \underset{z\in {\Phi }_p^{-1}\left(k\right)}{sup}}{f}_{jA}\left(k\right)\left(y\right)))\hfill \\ & =& {\displaystyle \underset{j\in J}{max}}\Phi \left(f_{jA}\right)\left(k\right)\left(y\right)\hfill \\ & =& {\tilde{\vee }}_{j\in J}\Phi \left(f_{jA}\right)\left(k\right)\left(y\right).\hfill \end{array}$$

Now, we prove that ${\Phi }^{-1}\left[{\tilde{\vee }}_{j\in J}{g}_{jB}\right]={\tilde{\vee }}_{j\in J}{\Phi }^{-1}\left(g_{jB}\right)$. For any $e\in E,x\in X$ and $b\in B$:

$$\begin{array}{lll}{\Phi }^{-1}\left[{\tilde{\vee }}_{j\in J}{g}_{jB}\right]\left(e\right)\left(x\right)& =& \left({\tilde{\vee }}_{j\in J}\right)g_{jB}\left({\Phi }_p\left(e\right)\right)\left({\Phi }_u\left(x\right)\right)\hfill \\ & =& [{\displaystyle \underset{j\in J}{max}}{g}_{jb}^{-},{\displaystyle \underset{j\in J}{max}}{g}_{jb}^{+}]\left({\Phi }_p\left(e\right)\right)\left({\Phi }_u\left(x\right)\right)\hfill \\ & =& [[{\displaystyle \underset{j\in J}{max}}{g}_{jb}^{-}\left({\Phi }_p\left(e\right)\right)\left({\Phi }_u\left(x\right)\right),{\displaystyle \underset{j\in J}{max}}{g}_{jb}^{+}\left({\Phi }_p\left(e\right)\right)\left({\Phi }_u\left(x\right)\right)]\hfill \\ & =& [{\displaystyle \underset{j\in J}{max}}{\Phi }_u^{-1}\left(g_{jb}^{-}\right)\left(e\right)\left(x\right),{\displaystyle \underset{j\in J}{max}}{\Phi }_u^{-1}\left(g_{jb}^{+}\right)\left(e\right)\left(x\right)]\hfill \\ & =& {\displaystyle \underset{j\in J}{max}}[{\Phi }_u^{-1}\left(g_{jb}^{-}\right)\left(e\right)\left(x\right),{\Phi }_u^{-1}\left(g_{jb}^{+}\right)\left(e\right)\left(x\right)]\hfill \\ & =& {\displaystyle \underset{j\in J}{max}}{\Phi }_u^{-1}\left(g_{jB}\right)\left(e\right)\left(x\right)\hfill \\ & =& {\tilde{\vee }}_{j\in J}{\Phi }_u^{-1}\left(g_{jB}\right)\left(e\right)\left(x\right).\hfill \end{array}$$

□

3. Interval-Valued Fuzzy Soft Topological Spaces

The interval-valued fuzzy topology $IVFT$ was discussed by Mondal and Samanta [30]. In this section, we recall their definition and then present different neighborhood structures in the interval-valued fuzzy soft topology $\left(IVFST\right)$.

Definition 7.

Let X be a non-empty set, and let τ be a collection of interval valued fuzzy soft sets over X with the following properties:

(i) 

${\varnothing }_E$, $X_E$ belong to τ,

(ii) 

If $f_{1E},f_{2E}$ are $IVFS$ sets belong to τ, then $f_{1E}\tilde{\wedge }{f}_{2E}$ belong to τ,

(iii) 

If the collection of $IVFS$ sets $\left\{f_{jE}\right|j\in J\}$ where J is an index set, belonging to τ, then ${\tilde{\vee }}_{j\in J}{f}_{jE}$ belong to τ.

Then, τ is called the interval-valued fuzzy soft topology over X, and the triplet $(X,E,\tau )$ is called the interval-valued fuzzy soft topological space $\left(IVFST\right)$.

As the ordinary topologies, the indiscrete $IVFST$ over X contains only ${\varnothing }_E$ and $X_E$, while the discrete $IVFST$ over X contains all $IVFS$ sets. Every member of τ is called an interval-valued fuzzy soft open set ($IVFS$-open) in X. The complement of an $IVFS$-open set is called an $IVFS$-closed set.

Remark 1.

If $f_e^{-}\left(x\right)=f_e^{+}\left(x\right)=a\in [0,1]$, then we put $[f_e^{-}\left(x\right),f_e^{+}\left(x\right)]=[a,a]=a$.

Example 1.

Let $X=[0,1]$ and E be any subset of X. Consider the $IVFS$ set $f_E$ over X by the mapping:

$$f:E\to \mathcal{IVF}\left(\right[0,1\left]\right)$$

such that for any $e\in E,x\in X$:

$${\tilde{f}}_e\left(x\right)=\left\{\begin{array}{cc}1& 0\le x\le e\\ 0& e<x\le 1.\end{array}\right.$$

Then, the collection $\tau =\{{\Phi }_E,X_E,f_E\}$ is an $IVFST$ over X.

1. 

Clearly $X_E,{\varnothing }_E\in \tau$.

2. 

Let ${\left\{f_{jE}\right\}}_{j\in J}$ be a sub-family of τ where for any $j\in J$ if $x\in X$ such that for all $e\in E$:

$$f_{je}\left(x\right)=\left\{\begin{array}{cc}1& 0\le x\le e\\ 0& e<x\le 1.\end{array}\right.$$

Since:

$${\vee }_j{f}_{je}\left(x\right)=\left\{\begin{array}{cc}1& 0\le x\le e\\ 0& e<x\le 1,\end{array}\right.$$

then ${\tilde{\vee }}_j{f}_{jE}\in \tau$.

3. 

Let $f_E,g_E\in \tau$, where:

$$f_e\left(x\right)=\left\{\begin{array}{cc}1& 0\le x\le e\\ 0& e<x\le 1,\end{array}\right.$$

and:

$$g_e\left(x\right)=\left\{\begin{array}{cc}1& 0\le x\le e\\ 0& e<x\le 1.\end{array}\right.$$

Since:

$$f_e\left(x\right)\wedge g_e\left(x\right)=\left\{\begin{array}{cc}1& 0\le x\le e\\ 0& e<x\le 1.\end{array}\right.$$

Thus, $f_E\wedge g_E\in \tau$.

Example 2

([23]). Let $\mathbb{R}$ be the set of all real numbers with the usual topology ${\tau }_u$ where ${\tau }_u=\langle \{(a,b),a,b\in \mathbb{R}\}\rangle$ and E is a parameter set. Let $U=(a,b)\subset \mathbb{R}$ be an open interval in $\mathbb{R}$; we define $IVFS$ ${\tilde{U}}_E$ over $\mathbb{R}$ by the mapping:

$$\tilde{U}:E\to {\left(Int[0,1]\right)}^{\mathbb{R}}$$

such that for all $x\in \mathbb{R}$:

$${\tilde{U}}_e\left(x\right)=\left\{\begin{array}{cc}1& x\in (a,b)\\ 0& x\notin (a,b).\end{array}\right.$$

Then, the family $\{{\tilde{U}}_E:(a,b)\subset \mathbb{R},\forall a,b\in \mathbb{R}\}$ generates an $IVFS$ over $\mathbb{R}$, and we denote it by ${\tau }_u^{\left(IVFS\right)}:$

1. 

Clearly, ${\mathbb{R}}_E,{\varnothing }_E\in {\tau }_u^{\left(IVFS\right)}$ where for all $e\in E$, $k\in \mathbb{R},{\mathbb{R}}_E\left(e\right)\left(k\right)=[1,1]$, and ${\varnothing }_e\left(k\right)=0$

2. 

Let ${\left\{{\tilde{U}}_{jE}\right\}}_{j\in J}$ be a sub-family of ${\tau }_u^{\left(IVFS\right)}$ where for any $j\in J$ if $x\in (a_j,b_j)$ and interval $(a_j,b_j)$ in $\mathbb{R}$ such that for all $e\in E$:

$${\tilde{U}}_{je}\left(x\right)=\left\{\begin{array}{cc}1& x\in (a_j,b_j)\\ 0& x\notin (a_j,b_j).\end{array}\right.$$

Since ${\tilde{\vee }}_j{\tilde{U}}_{jE}=(\widetilde{{\cup }_j{U}_j},E)$ where ${\cup }_j{U}_{jE}\in {\tau }_u$, then ${\tilde{\vee }}_j{\tilde{U}}_{jE}\in {\tau }_u^{\left(IVFS\right)}$

3. 

Let ${\tilde{U}}_E,{\tilde{V}}_E\in {\tau }_u^{\left(IVFS\right)}$, then ${\tilde{U}}_E\tilde{\wedge }{\tilde{V}}_E\in {\tau }_u^{\left(IVFS\right)}$ since ${\tilde{U}}_E\tilde{\wedge }{\tilde{V}}_E=(\widetilde{U\cap V},E)$ where $U\cap V\in {\tau }_u$.

Definition 8.

Let interval $[{\lambda }_e^{-},{\lambda }_e^{+}]\subseteq [0,1]$ for all $e\in E$. Then, ${\tilde{x}}_E$ is called an interval-valued fuzzy soft point ($IVFS$ point) with support $x\in X$ and e lower value ${\lambda }_e^{-}$ and e upper value ${\lambda }_e^{+}$, if for each $y\in X$:

$$\tilde{x}\left(e\right)\left(y\right)=\left\{\begin{array}{cc}[{\lambda }_e^{-},{\lambda }_e^{+}]& y=x\\ 0& otherwise.\end{array}\right.$$

Example 3.

Let $X=[0,1]$ and E be any subset of X. Consider $IVFS$ point ${\tilde{x}}_E$ with support x, lower value zero, and upper value $0.3$, we define $IVFS$ point ${\tilde{x}}_E$ by:

$$\tilde{x}\left(e\right)\left(c\right)=\left\{\begin{array}{cc}[0,0.3]& c=x\\ 0& otherwise,\end{array}\right.$$

for any $e\in E$ and $c\in X$.

Definition 9.

The $IVFS$ point ${\tilde{x}}_E$ belongs to $IVFS$ set $f_E$, denoted by ${\tilde{x}}_E\tilde{\in }{f}_E$, whenever for all $e\in E$, we have ${\lambda }_e^{-}\le f_e^{-}\left(x\right)$ and ${\lambda }_e^{+}\le f_e^{+}\left(x\right)$.

Theorem 1.

Let $f_E$ be an $IVFS$ set, then $f_E$ is the union of all its $IVFS$ points,

i.e., $f_E={\tilde{\vee }}_{{\tilde{x}}_E\tilde{\in }{f}_E}{\tilde{x}}_E$.

Proof. 

Let $x\in X$ be a fixed point, $y\in X$ and $e\in E$. Take all ${\tilde{x}}_E\tilde{\in }{f}_E$ with different e lower and e upper values ${\lambda }_{je}^{-},{\lambda }_{je}^{+}$ where $j\in J$. Then, there exists ${\lambda }_{je}^{-}=f_e^{-},{\lambda }_{je}^{+}=f_e^{+}$ where:

$$\begin{array}{lll}{\tilde{\vee }}_{{\tilde{x}}_E\in f_E}{\tilde{x}}_e\left(y\right)& =& [sup{\tilde{x}}_e^{-}\left(y\right),sup{\tilde{x}}_e^{+}\left(y\right)]\hfill \\ & =& [{\displaystyle \underset{{\lambda }_{je}^{-}\tilde{\le }{f}^{-}\left(x\right)}{sup}}{\lambda }_{je}^{-},{\displaystyle \underset{{\lambda }_{je}^{+}\tilde{\le }{f}^{+}\left(x\right)}{sup}}{\lambda }_{je}^{+}]\hfill \\ & =& [f_e^{-}\left(x\right),f_e^{+}\left(x\right)].\hfill \end{array}$$

□

Proposition 2.

Let ${\left\{f_{jE}\right\}}_{j\in J}$ be a family of $IVFS$ sets over X, where J is an index set and ${\tilde{x}}_E$ is an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$. If $\tilde{x}\tilde{\in }{\tilde{\wedge }}_{j\in J}\left\{f_{jE}\right\}$, then ${\tilde{x}}_E\tilde{\in }\left\{f_{jE}\right\}$ for each $j\in J$.

Proof. 

Let ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$, and let $\tilde{x}\tilde{\in }{\tilde{\wedge }}_{j\in J}\left\{f_{jE}\right\}$. Then, ${\lambda }_e^{-}\le {\wedge }_{j\in J}\left\{f_{je}^{-}\right\}\left(x\right)\le \left\{f_{je}^{-}\right\}\left(x\right)$ for each $e\in E$, $x\in X$ and ${\lambda }_e^{+}\le {\wedge }_{j\in J}\left\{f_{je}^{+}\right\}\left(x\right)\le \left\{f_{je}^{+}\right\}\left(x\right)$ for each $e\in E$, $x\in X$. Thus, $[{\lambda }_e^{-},{\lambda }_e^{+}]\le [\left\{f_{je}^{-}\right\}\left(x\right),\left\{f_{je}^{+}\right\}\left(x\right)]$, for each $e\in E$, $x\in X$. Hence, ${\tilde{x}}_E\tilde{\in }{\left\{f_{jE}\right\}}_{j\in J}$. □

Remark 2.

If ${\tilde{x}}_E\tilde{\in }{f}_E\tilde{\vee }{g}_E$ does not imply ${\tilde{x}}_E\tilde{\in }{f}_E$ or ${\tilde{x}}_E\tilde{\in }{g}_E$.

This is shown in the following example.

Example 4.

Let τ be an $IVFST$ over X, where $\tau =\{{\varnothing }_E,X_E,f_E,g_E,f_E\tilde{\wedge }{g}_E\}$, and ${\tilde{x}}_E$ be the absolute $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$. If $f_E$ and $g_E$ are two $IVFS$ sets in X defined as below:

$$f:E\to \mathcal{IVF}\left(\right[0,1\left]\right)$$

and:

$$g:E\to \mathcal{IVF}\left(\right[0,1\left]\right)$$

such that for any $e\in E,x\in X$:

$$f_e\left(x\right)=\left\{\begin{array}{cc}[1,0.5]& 0\le x\le e\\ 0& e<x\le 1\end{array}\right.$$

and:

$$g_e\left(x\right)=\left\{\begin{array}{cc}[0.2,1]& 0\le x\le e\\ 0& e<x\le 1.\end{array}\right.$$

Since:

$$f_e\left(x\right)\vee g_e\left(x\right)=\left\{\begin{array}{cc}1& if0\le x\le e\\ 0& ife<x\le 1,\end{array}\right.$$

then ${\tilde{x}}_E\tilde{\in }{f}_E\tilde{\vee }{g}_E$, but ${\tilde{x}}_E\tilde{\notin }{f}_E$ and ${\tilde{x}}_E\tilde{\notin }{g}_E$.

Theorem 2.

Let ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$ and $f_E$ and $g_E$ be $IVFS$ sets. If ${\tilde{x}}_E\tilde{\in }{f}_E\tilde{\vee }{g}_E$, then there exists $IVFS$ point ${\tilde{x}}_{1E}\tilde{\in }{f}_E$ and $IVFS$ point ${\tilde{x}}_{2E}\tilde{\in }{g}_E$ such that ${\tilde{x}}_E={\tilde{x}}_{1E}\tilde{\vee }{\tilde{x}}_{2E}$.

Proof. 

Let ${\tilde{x}}_E\tilde{\in }{f}_E\tilde{\vee }{g}_E$. Then, ${\lambda }_e^{-}\le f_e^{-}\left(x\right)\vee g_e^{-}\left(x\right)$ and ${\lambda }_e^{+}\le f_e^{+}\left(x\right)\vee g_e^{+}\left(x\right)$, for each $e\in E$, $x\in X$. Let us choose

$E_1=\{e\in E|{\lambda }_e^{-}\le f_e^{-}\left(x\right),{\lambda }_e^{+}\le f_e^{+}\left(x\right):x\in X\}$,

$E_2=\{e\in E|{\lambda }_e^{-}\le g_E^{-}\left(x\right),{\lambda }_e^{+}\le g_E^{+}\left(x\right):x\in X\}$

and:

$${\tilde{x}}_1\left(e\right)\left(y\right)=\left\{\begin{array}{ll}[{\lambda }_e^{-},{\lambda }_e^{+}]& ify=x_1,e\in E_1\hfill \\ 0,& otherwise,\end{array}\right.$$

$${\tilde{x}}_2\left(e\right)\left(y\right)=\left\{\begin{array}{ll}[{\lambda }_e^{-},{\lambda }_e^{+}],& ify=x_2,e\in E_2\hfill \\ 0,& otherwise.\end{array}\right.$$

Since $x_{1e}^{-}\le f_{1e}^{-}\left(x\right)$ and $x_{1e}^{+}\le f_{1e}^{+}\left(x\right)$ for each $e\in E_1,x\in X$, that implies ${\tilde{x}}_{1E}\tilde{\in }{f}_{1E}$ and also $x_{2e}^{-}\le f_{2e}^{-}\left(x\right)$, and $x_{2e}^{+}\le f_{2e}^{+}\left(x\right)$ for each $e\in E_2,x\in X$, that implies ${\tilde{x}}_{2E}\tilde{\in }{f}_{2E}$. Consequently, $E_1\tilde{\vee }{E}_2=E$ and ${\tilde{x}}_E={\tilde{x}}_{1E}\tilde{\vee }{\tilde{x}}_{2E}$. □

Definition 10.

Let $(X,E,\tau )$ be an $IVFST$ space and ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$. The $IVFS$ set $g_E$ is called the interval-valued fuzzy soft neighborhood $\left(IVFSN\right)$ of $IVFS$ point ${\tilde{x}}_E$, if there exists the $IVFS$-open set $f_E$ in X such that ${\tilde{x}}_E\tilde{\in }{f}_E\widetilde{<}{g}_E$. Therefore, the $IVFS$-open set $f_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$ if $\forall e\in E,x\in X$ such that ${\lambda }_e^{-}<f_e^{-}\left(x\right)$ and ${\lambda }_e^{+}<f_e^{+}\left(x\right)$.

Definition 11.

Let $(X,E,\tau )$ be an $IVFST$ space and ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$ and ${{\tilde{x}}_E}^{☆}$ be an $IVFS$ point with support $x^{☆}$, e lower value ${\epsilon }_e^{-}$, and e upper value ${\epsilon }_e^{+}$. ${{\tilde{x}}_E}^{☆}$ is said to be compatible with ${\lambda }_e^{-},{\lambda }_e^{+}$, if ${{\tilde{x}}_E}^{☆}$ provides that $0\le {\epsilon }_e^{-}\le {\lambda }_e^{-}$ and $0\le {\epsilon }_e^{+}\le {\lambda }_e^{+}$ for each $e\in E$.

Proposition 3.

1. 

If $f_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$ and $f_E\tilde{\le }{h}_E$, then $h_E$ is also an $IVFSN$ of ${\tilde{x}}_E$.

2. 

If $f_E$ and $g_E$ are two $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$, then $f_E\tilde{\wedge }{g}_E$ is also the $IVFSN$ of ${\tilde{x}}_E$.

3. 

If $f_E$ is an $IVFSN$ of the $IVFS$ point ${{\tilde{x}}_E}^{☆}$ with support $x^{☆}$, e lower value ${\lambda }_e^{-}-{\epsilon }_e^{-}$, and e upper value ${\lambda }_e^{+}-{\epsilon }_e^{+}$, for all ${\epsilon }_e^{-}$ compatible with ${\lambda }_e^{-}$ and ${\epsilon }_e^{+}$ compatible with ${\lambda }_e^{+}$, then $f_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$.

4. 

If $f_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_{1E}$ and $g_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_{2E}$, then $f_E\tilde{\vee }{g}_E$ is also an $IVFSN$ of ${\tilde{x}}_{1E}$ and ${\tilde{x}}_{2E}$.

5. 

If $f_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$, then there exists $IVFSN$$g_E$ of ${\tilde{x}}_E$ such that $g_E\tilde{\le }{f}_E$ and $g_E$ is $IVFSN$ of $IVFS$ point $\tilde{y}$ with support y, e lower value ${\gamma }_e^{-}$, and e upper value ${\gamma }_e^{+}$, for all ${\tilde{y}}_E\tilde{\in }{g}_E$.

Proof. 

1.

Let $f_E$ be an $IVFSN$ of the $IVFS$ point $\tilde{x}$. Then, there exists the $IVFS$-open set $g_E$ in X such that ${\tilde{x}}_E\tilde{\in }{g}_E\tilde{\le }{f}_E$. Since $f_E\tilde{\le }{h}_E$, ${\tilde{x}}_E\tilde{\in }{g}_E\tilde{\le }{f}_E\tilde{\le }{h}_E$. Thus, $h_E$ is an $IVFSN$ of ${\tilde{x}}_E$.

2.

Let $f_E$ and $g_E$ be two $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$. Then, there exists two $IVFS$-open sets $h_E$, $k_E$ in X such that ${\tilde{x}}_E\tilde{\in }{h}_E\tilde{\le }{f}_E$ and ${\tilde{x}}_E\tilde{\in }{k}_E\tilde{\le }{g}_E$. Thus, ${\tilde{x}}_E\tilde{\in }{h}_E\tilde{\wedge }{k}_E\tilde{\le }{f}_E\tilde{\wedge }{g}_E$. Since $h_E\tilde{\wedge }{k}_E$ is an $IVFS$-open set, $g_E\tilde{\wedge }{f}_E$ is an $IVFSN$ of ${\tilde{x}}_E$.

3.

Let $f_E$ be an $IVFSN$ of the $IVFS$ point ${{\tilde{x}}_E}^{☆}$ with support $x^{☆}$, e lower value ${\lambda }_e^{-}-{\epsilon }_e^{-}$, and e upper value ${\lambda }_e^{+}-{\epsilon }_e^{+}$, for all ${\epsilon }_e^{-}$ compatible with ${\lambda }_e^{-}$ and ${\epsilon }_e^{+}$ compatible with ${\lambda }_e^{+}$. Then, there exists $IVFS$-open set $g_E^{x^{☆}}$ such that ${{\tilde{x}}_E}^{☆}\tilde{\in }{g}_E^{x^{☆}}\tilde{\le }{f}_E$. Let $g_E={\tilde{\vee }}_{x^{☆}}{g}_E^{x^{☆}}$, then $g_E$ is $IVFS$-open in X and $g_E\tilde{\le }{f}_E$. By Theorem 1 and since for all $e\in E$, $\tilde{\vee }{{\tilde{x}}_E}^{☆}={\tilde{x}}_E\tilde{\le }{\tilde{\vee }}_{x^{☆}}{g}_E^{x^{☆}}=g_E\tilde{\le }{f}_E$. Hence, ${\tilde{x}}_E\tilde{\in }{g}_E\tilde{\le }{f}_E$, i.e., $f_E$ is an $IVFSN$ of ${\tilde{x}}_E$.

4.

Let $f_E$ be an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_{1E}$ with support $x_1$, e lower value ${\lambda }_{1e}^{-}$, and e upper value ${\lambda }_{1e}^{+}$ and $g_E$ be an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_{2E}$ with support $x_2$, e lower value ${\lambda }_{2e}^{-}$, and e upper value ${\lambda }_{2e}^{+}$. Then, there exists $IVFS$-open sets $h_{1E},h_{2E}$ such that ${\tilde{x}}_{1E}\tilde{\in }{h}_{1E}\tilde{\le }{f}_E$ and ${\tilde{x}}_{2E}\tilde{\in }{h}_{2E}\tilde{\le }{f}_E$, respectively. Since ${\tilde{x}}_{1E}\tilde{\in }{h}_{1E}$, ${\lambda }_{1e}^{-}\le h_{1e}^{-}\left(x\right),{\lambda }_{1e}^{+}\le h_{1e}^{+}\left(x\right)$ for each $e\in E$ and $x\in X$. Since ${\tilde{x}}_{2E}\tilde{\in }{h}_{2E}$, ${\lambda }_{2e}^{-}\le h_{2e}^{-}\left(x\right),{\lambda }_{2e}^{+}\le h_{2e}^{+}\left(x\right)$ for each $e\in E$ and $x\in X$. Thus, we have:

$$max\{[{\lambda }_{1e}^{-},{\lambda }_{1e}^{+}],[{\lambda }_{2e}^{-},{\lambda }_{2e}^{+}]\}\le max\{[h_{1e}^{-}\left(x\right),h_{1e}^{+}\left(x\right)],[h_{2e}^{-}\left(x\right),h_{2e}^{+}\left(x\right)]\}$$

for each $e\in E$, $x\in X$. Therefore, ${\tilde{x}}_{1E}\tilde{\vee }{\tilde{x}}_{2E}\tilde{\in }{h}_{1E}\tilde{\vee }{h}_{2E}$, $h_{1E}\tilde{\vee }{h}_{2E}\in \tau$, and $h_{1E}\tilde{\vee }{h}_{2E}\tilde{\le }{f}_E\tilde{\vee }{g}_E$. Consequently, $f_E\tilde{\vee }{g}_E$ is an $IVFSN$ of $x_{1E}\tilde{\vee }{x}_{2E}$.

5.

Let $f_E$ be an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$, with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$. Then, there exists $IVFS$-open set $g_E$ such that ${\tilde{x}}_E\tilde{\in }{g}_E\tilde{\le }{f}_E$. Since $g_E$ is an $IVFS$-open set, $g_E$ is a neighborhood of its points, i.e., $g_E$ is an $IVFSN$ of $IVFS$ point ${\tilde{y}}_E$ with support y, e lower value ${\gamma }_e^{-}$, and e upper value ${\gamma }_e^{+}$, for all $e\in E$. Furthermore, $g_E$ is an $IVFSN$ of $IVFS$ point ${\tilde{x}}_E$ since ${\tilde{x}}_E\tilde{\in }{g}_E$. Therefore, there exists $g_E$ that is an $IVFSN$ of ${\tilde{x}}_E$ such that $g_E\tilde{\le }{f}_E$ and $g_E$ is an $IVFSN$ of ${\tilde{y}}_E$; since $f_E$ is an $IVFSN$ of ${\tilde{x}}_E$.

□

Definition 12.

Let $(X,E,\tau )$ be an $IVFST$ space and $f_E$ be an $IVFS$ set. The $IVFS$-closure of $f_E$ denoted by $Cl{f}_E$ is the intersection of all $IVFS$-closed super sets of $f_E$. Clearly, $Cl{f}_E$ is the smallest $IVFS$-closed set over X that contains $f_E$.

Example 5

([23]). Consider $IVFST$ ${\tau }_u^{IVFS}$ over $\mathbb{R}$ as introduced in Example 2, and if ${\tilde{H}}_E$ is an $IVFS$ over $\mathbb{R}$ related of the open interval $H=(a,b)\subset \mathbb{R}$ by mapping:

$$\tilde{H}:E\to {\left(Int[0,1]\right)}^{\mathbb{R}}$$

$${\tilde{H}}_e\left(x\right)=\left\{\begin{array}{cc}1& x\in (a,b)\\ 0& x\notin (a,b),\end{array}\right.$$

where $e\in E$ and $x\in \mathbb{R}$, then the closure of ${\tilde{H}}_E$ is defined as:

$$Cl\tilde{H}:E\to {\left(Int[0,1]\right)}^{\mathbb{R}}$$

$${\tilde{H}}_e\left(x\right)=\left\{\begin{array}{cc}1& x\in [a,b]\\ 0& x\notin [a,b].\end{array}\right.$$

Remark 3.

By replacing ${\tilde{x}}_E$ for $f_E$, the $IVFS$-closure of ${\tilde{x}}_E$ denoted by $Cl{\tilde{x}}_E$ is the intersection of all $IVFS$-closed super sets of ${\tilde{x}}_E$.

Proposition 4.

Let $(X,E,\tau )$ be an $IVFST$ space and $f_E$ and $g_E$ be two $IVFSS$ over X. Then:

1. 

$Cl{\varnothing }_E={\varnothing }_E$ and $Cl{\tilde{X}}_E={\tilde{X}}_E$,

2. 

$f_E\tilde{\le }Cl{f}_E$, and $Cl{f}_E$ is the smallest $IVFS$-closed set containing $IVFS{f}_E$,

3. 

$Cl\left(Cl{f}_E\right)=Cl{f}_E$,

4. 

if $f_E\tilde{\le }{g}_E$, then $\left(Cl{f}_E\right)\tilde{\le }Cl{g}_E$.

5. 

$f_E$ is an $IVFS$-closed set if and only if $f_E=Cl{f}_E$,

6. 

$Cl\left(f_E\tilde{\vee }{g}_E\right)=Cl{f}_E\tilde{\vee }Cl{g}_E$,

7. 

$Cl\left(f_E\tilde{\wedge }{g}_E\right)\tilde{\le }Cl{f}_E\tilde{\wedge }Cl{g}_E$.

Proof. 

We only prove Part (6). A similar technique is used to show the other parts.

Since $f_E\tilde{\le }{f}_E\tilde{\vee }{g}_E$ and $g_E\tilde{\le }{f}_E\tilde{\vee }{g}_E$, by Part (4), we have $Cl{f}_E\tilde{\le }Cl\left(f_E\tilde{\vee }{g}_E\right)$ and $Cl{g}_{}\tilde{\le }Cl\left(f_E\tilde{\vee }{g}_E\right)$. Then, $Cl{f}_E\tilde{\vee }Cl{g}_E\tilde{\le }Cl\left(f_E\tilde{\vee }{g}_E\right)$.

Conversely, we have $f_E\tilde{\le }Cl{f}_E$ and $g_E\tilde{\le }Cl{g}_E$, by Part (2). Then, $f_E\tilde{\vee }{g}_E\tilde{\le }Cl{f}_E\tilde{\vee }Cl{g}_E$ where $Cl{f}_E\tilde{\vee }Cl{g}_E$ is an $IVFS$-closed set. Thus, $Cl\left(f_E\tilde{\vee }{g}_E\right)\tilde{\le }Cl{f}_E\tilde{\vee }Cl{g}_E$.

Therefore, $Cl\left(f_E\tilde{\vee }{g}_E\right)=Cl{f}_E\tilde{\vee }Cl{g}_E$. □

Definition 13.

Let $(X_1,E_1,{\tau }_1)$ and $(X_2,E_2,{\tau }_2)$ be two $IVFSTS$ and:

$$\begin{array}{c}\hfill \Phi :(X_1,E_1,{\tau }_1)\to (X_2,E_2,{\tau }_2)\end{array}$$

be an $IVFS$ map. Then, Φ is called an:

1. 

interval-valued fuzzy soft continuous $\left(IVFSC\right)$ map if and only if for each $g_{E_2}\in {\tau }_2$, we have ${\Phi }^{-1}\left(g_{E_2}\right)\in {\tau }_1$,

2. 

interval-valued fuzzy soft open $\left(IVFSO\right)$ map if and only if for each $f_E\in {\tau }_1$, we have $\Phi \left(f_{E_1}\right)\in {\tau }_2$.

Theorem 3.

Let $(X_1,E_1,{\tau }_1)$ and $(X_2,E_2,{\tau }_2)$ be two $IVFST$ and Φ be an $IVFS$ mapping from $X_1$ to $X_2$, then the following statements are equivalent:

1. 

Φ is $IVFC$,

2. 

For each $IVFS$ point ${\tilde{x}}_E$ on $X_1$, the inverse of every neighborhood of $\Phi \left({\tilde{x}}_E\right)$ under Φ is a neighborhood of ${\tilde{x}}_E$,

3. 

For each $IVFS$ point ${\tilde{x}}_E$ on $X_1$ and each neighborhood $g_E$ of $\Phi \left({\tilde{x}}_E\right)$, there exists a neighborhood $f_E$ of ${\tilde{x}}_E$ such that $\Phi \left(f_E\right)\tilde{\le }{g}_E$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ Let $g_E$ be an $IVFSN$ of $\Phi \left({\tilde{x}}_E\right)$ in ${\tau }_2$. Then, there exists an $IVFS$-open set $f_E$ in ${\tau }_2$ such that $\Phi \left({\tilde{x}}_E\right)\tilde{\in }{f}_E\tilde{\le }{g}_E$. Since $\Phi$ is $IVFSC$, ${\Phi }^{-1}\left(f_E\right)$ is an $IVFS$-open in ${\tau }_1$, and we have ${\tilde{x}}_E\tilde{\in }{\Phi }^{-1}\left(f_E\right)\tilde{\le }{\Phi }^{-1}\left(g_E\right)$.

$\left(2\right)\Rightarrow \left(3\right)$ Let $g_E$ be an $IVFSN$ of $\Phi \left({\tilde{x}}_E\right)$. By the hypothesis, ${\Phi }^{-1}\left(g_E\right)$ is an $IVFSN$ of ${\tilde{x}}_E$. Consider $f_E={\Phi }^{-1}\left(g_E\right)$ to be an $IVFSN$ of ${\tilde{x}}_E$. Then, we have $\Phi \left(f_E\right)=\Phi \left({\Phi }^{-1}\left(g_E\right)\right)\tilde{\le }{g}_E$.

$\left(3\right)\Rightarrow \left(1\right)$ Let $g_E$ be an $IVFS$-open set in ${\tau }_2$. We must show that ${\Phi }^{-1}\left(g_E\right)$ is an $IVFS$-open set in ${\tau }_1$. Now, let ${\tilde{x}}_E\tilde{\in }{\Phi }^{-1}\left(g_E\right)$. Then, $\Phi \left({\tilde{x}}_E\right)\tilde{\in }{g}_E$. Since $g_E$ is an $IVFS$-open set in ${\tau }_2$, we get that $g_E$ is an $IVFSN$$\Phi \left({\tilde{x}}_E\right)$ in ${\tau }_2$. By the hypothesis, there exists $IVFS$-open set $f_E$ that is an $IVFSN$ of ${\tilde{x}}_E$ such that $\Phi \left(f_E\right)\tilde{\le }{g}_E$. Thus, $f_E\tilde{\le }{\Phi }^{-1}[\Phi \left(f_E\right)]\tilde{\le }{\Phi }^{-1}\left(g_E\right)$ for $f_E$ is an $IVFSN$ of ${\tilde{x}}_E$. From here, $f_E\tilde{\le }{\Phi }^{-1}\left(g_E\right)$, as $f_E$ is an $IVFSN$ of ${\tilde{x}}_E$. Hence, ${\Phi }^{-1}\left(g_E\right)\tilde{\in }{\tau }_1$. □

4. Quasi-Coincident Neighborhood Structure of Interval-Valued Fuzzy Soft Topological Spaces

In this section, we present the quasi-coincident neighborhood structure in the interval-valued fuzzy soft topology $\left(IVFST\right)$ and its properties.

Definition 14.

The $IVFS$ point ${\tilde{x}}_E$ is called soft quasi-coincident with $IVFS$$f_E$, denoted by ${\tilde{x}}_E\tilde{q}{f}_E$, if there exists $e\in E$ such that ${\lambda }_e^{-}+f_e^{-}\left(x\right)>1$ and ${\lambda }_e^{+}+f_e^{+}\left(x\right)>1$. If $f_E$ is not soft quasi-coincident with $f_E$, we write $f_E\neg \tilde{q}{g}_E$.

Definition 15.

The $IVFS$ set $f_E$ is called soft quasi-coincident with $IVFS$$g_E$, denoted by $f_E\tilde{q}{g}_E$, if there exists $e\in E$ such that $f_e^{-}\left(x\right)+g_e^{-}\left(x\right)>1$ and $f_e^{+}\left(x\right)+g_e^{+}\left(x\right)>1$.

Proposition 5.

Let ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$ and $f_E,g_E$ two $IVFS$ sets. Then:

(i) 

$f_E\tilde{\le }{g}_E\iff f_E\neg \tilde{q}{g}_E^c$,

(ii) 

${\tilde{x}}_E\tilde{\in }{f}_E\iff {\tilde{x}}_E\neg \tilde{q}{f}_E^c$.

Proof. 

We just prove Part (i). A similar technique is used to show Part (ii). For two $IVFS$ sets $f_E,g_E$, we have:

$$\begin{array}{lll}{f}_E\tilde{\le }{g}_E& \iff & \forall e\in E:[f_e^{-}\left(x\right),f_e^{+}\left(x\right)]\le [g_e^{-}\left(x\right),g_e^{+}\left(x\right)],\forall x\in X\hfill \\ & \iff & \forall e\in E:f_e^{-}\left(x\right)\le g_e^{-}\left(x\right)and{f}_e^{+}\left(x\right)\le g_e^{+}\left(x\right),\forall x\in X\hfill \\ & \iff & \forall e\in E:f_e^{-}\left(x\right)+1-g_e^{-}\left(x\right)\le 1and{f}_e^{+}\left(x\right)+1-g_e^{+}\left(x\right)\le 1,\forall x\in X\hfill \\ & \iff & \forall e\in E:f_e^{-}\left(x\right)+{g^{-}}_e^c\left(x\right)\le 1and{f}_e^{+}\left(x\right)+{g^{+}}_e^c\left(x\right)\le 1,\forall x\in X\hfill \\ & \iff & f_E\neg \tilde{q}{g}_E^c.\hfill \end{array}$$

□

Proposition 6.

Let $\{f_{jE}:j\in J\}$ be a family of $IVFS$ sets over X and ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$. If ${\tilde{x}}_E\tilde{q}\left(\tilde{\wedge }{f}_{jE}\right)$, then ${\tilde{x}}_E\tilde{q}{f}_{jE}$ for each $j\in J$.

Proof. 

Let ${\tilde{x}}_E\tilde{q}\left(\tilde{\wedge }{f}_{jE}\right)$. Then, ${\lambda }_e^{-}\tilde{q}\left({\tilde{\wedge }}_j{f}_{je}^{-}\right)\left(x\right)$, ${\lambda }_e^{+}\tilde{q}\left({\tilde{\wedge }}_j{f}_{je}^{+}\right)\left(x\right)$ for $e\in E$, and $x\in X$. This implies that ${\lambda }_e^{-}>1-{\wedge }_j\left(f_{je}^{-}\right)\left(x\right)$ and ${\lambda }_e^{+}>1-{\wedge }_j\left(f_{je}^{+}\right)\left(x\right)$, $x\in X$. Since ${\wedge }_j{f}_{je}^{-}\left(x\right)\le f_{je}^{-}\left(x\right)$ and ${\wedge }_j{f}_{je}^{+}\left(x\right)\le f_{je}^{+}\left(x\right)$, then ${\lambda }_e^{-}>1-{\wedge }_j\left(f_{je}^{-}\right)\left(x\right)>1-f_{je}^{-}\left(x\right)$ for each $e\in E,x\in X$ and ${\lambda }_e^{+}>1-{\wedge }_j\left(f_{je}^{+}\right)\left(x\right)>1-f_{je}^{+}\left(x\right)$ for each $e\in E,x\in X$. Hence, ${\lambda }_e^{-}>1-f_{je}^{-}\left(x\right)$ and ${\lambda }_e^{+}>1-f_{je}^{+}\left(x\right)$. Therefore, $[{\lambda }_e^{-},{\lambda }_e^{+}]>[1,1]-[f_{je}^{-}\left(x\right),f_{je}^{+}\left(x\right)]$ implies that ${\tilde{x}}_E>1-f_{jE}^{-}$ and ${\tilde{x}}_E\tilde{q}{f}_{jE}$ for each $j\in J$. □

Remark 4.

${\tilde{x}}_E\tilde{q}(f_E\vee g_E)$ does not imply ${\tilde{x}}_E\tilde{q}{f}_E$ or ${\tilde{x}}_E\tilde{q}{g}_E$. This is shown in the following example.

Example 6.

Let us consider Example 4 where ${\tilde{x}}_E\tilde{q}\left(f_E\tilde{\vee }{g}_E\right)$, but ${\tilde{x}}_E\neg \tilde{q}{f}_E$ and ${\tilde{x}}_E\neg \tilde{q}{g}_E$.

Theorem 4.

Let ${\tilde{x}}_E$ be an $IVFS$ point ${\tilde{x}}_E$ with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$ and $f_E,g_E$ be $IVFS$ sets over X. If ${\tilde{x}}_E\tilde{q}(f_E\vee g_E)$, then there exists ${\tilde{x}}_{1E}\tilde{q}{f}_E$ and ${\tilde{x}}_{2E}\tilde{q}{g}_E$ such that ${\tilde{x}}_E={\tilde{x}}_{1E}\tilde{\vee }{\tilde{x}}_{2E}$.

The proof is very similar to the proof of Theorem 2.

Definition 16.

Let $(X,E,\tau )$ be an $IVFSTS$ and ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower values ${\lambda }_e^{-},$ and e upper values ${\lambda }_e^{+}$. The $IVFS$ set $g_E$ is called a quasi-soft neighborhood $\left(QIVFSN\right)$ of $IVFS$ point ${\tilde{x}}_E$ if there exists the $IVFS$-open set $f_E$ in X such that ${\tilde{x}}_E\tilde{q}{f}_E\tilde{\le }{g}_E$. Thus, the $IVFS$-open set $f_E$ is a $QIVFSN$ of the $IVFS$ point ${\tilde{x}}_E$ if and only if $\exists e\in E,x\in X$ such that ${\lambda }_e^{-}+f_e^{-}\left(x\right)>1$ and ${\lambda }_e^{+}+f_e^{+}\left(x\right)>1$.

Remark 5.

A quasi-coincident soft neighborhood of an $IVFS$ point generally does not contain the point itself. This is shown by the following:

Example 7.

Let $X=[0,1]$ and E be any subset of X. Consider two $IVFS$ sets $f_E,g_E$ over X by the mapping $f:E\to \mathcal{IVF}\left(\right[0,1\left]\right)$ and $f:E\to \mathcal{IVF}\left(\right[0,1\left]\right)$ such that for any $e\in E,x\in X$:

$${\tilde{f}}_e\left(x\right)=\left\{\begin{array}{cc}[0.4,0.5]& 0\le x\le e\\ 0& e<x\le 1,\end{array}\right.$$

and:

$${\tilde{g}}_e\left(x\right)=\left\{\begin{array}{cc}[0.6,0.7]& 0\le x\le e\\ 0& e<x\le 1,\end{array}\right.$$

and ${\tilde{x}}_E$ be any $IVFS$ point defined by:

$${\tilde{x}}_e\left(c\right)=\left\{\begin{array}{cc}[0.4,0.5]& c=x\\ 0& c\ne x.\end{array}\right.$$

Let $\tau =\{{\varnothing }_E,X_E,f_E,g_E\}$. Then clearly, τ is an $IVFST$ over X. Since $f_E\tilde{\le }{g}_E$ and $\tilde{x}\tilde{q}{f}_E$, thus $g_E$ is a $QIVFSN$ of ${\tilde{x}}_E$. However, ${\tilde{x}}_E\notin g_E$.

Proposition 7.

(1) 

If $f_E\tilde{\le }{g}_E$ and $f_E$ is a $QINVSN$ of ${\tilde{x}}_E$, then $g_E$ is also a $QINVSN$ of ${\tilde{x}}_E$,

(2) 

If $f_E,g_E$ are $QINVSN$ of ${\tilde{x}}_E$, then $f_E\tilde{\wedge }{g}_E$ is also a $QINVSN$ of ${\tilde{x}}_E$.

(3) 

If $f_E$ is a $QINVSN$ of ${\tilde{x}}_{1E}$ and $g_E$ is a $QINVSN$ of ${\tilde{x}}_{2E}$, then $f_E\tilde{\vee }{g}_E$ is also a $QINVSN$ of ${\tilde{x}}_{1E}\tilde{\vee }{\tilde{x}}_{2E}$.

(4) 

If $f_E$ is a $QINVSN$ of ${\tilde{x}}_E$, then there exists $g_E$ that is a $QINVSN$ of ${\tilde{x}}_E$, such that $g_E\tilde{\le }{f}_E$, and $g_E$ is a $QINVSN$ of $y_E$, $\forall y_E\tilde{q}{g}_E$.

Proof. 

(1) and (2) are straightforward.

(3)

Let $f_E$ be a $QINVSN$ of ${\tilde{x}}_{1E}$ and $g_E$ be a $QINVSN$ of ${\tilde{x}}_{2E}$. Then, there exists an $IVFS$-open set $h_{1E}$ in X such that ${\tilde{x}}_{1E}\tilde{q}{h}_{1E}\tilde{\le }{f}_E$ and $g_E$ is a $QINVSN$ of ${\tilde{x}}_{2E}$. Thus, there exists an $IVFS$-open set $h_{2E}$ in X such that ${\tilde{x}}_{2E}\tilde{q}{h}_{2E}\tilde{\le }{g}_E$. Since ${\tilde{x}}_{1E}\tilde{q}{h}_{1E}$ for each $e\in E,x\in X,{\lambda }_{1e}^{-}+h_{1e}^{-}>1$, ${\lambda }_{1e}^{+}+h_{1e}^{+}>1$, this implies that ${\lambda }_{1e}^{-}>1-h_{1e}^{-},{\lambda }_{1e}^{+}>1-h_{1e}^{+}$ for each $e\in E$. Since ${\tilde{x}}_{2E}\tilde{q}{h}_{2E}$, for each $e\in E,{\lambda }_{2e}^{-}+h_{2e}^{-}>1$, ${\lambda }_{2e}^{+}+h_{2e}^{+}>1$, this implies that ${\lambda }_{2e}^{-}>1-h_{2e}^{-},{\lambda }_{2e}^{+}>1-h_{2e}^{+}$ for each $e\in E,x\in X$. From here,

$$max({\lambda }_{1e}^{-},{\lambda }_{2e}^{-})>max(1-h_{1e}^{-}\left(x\right)),(1-h_{2e}^{-}\left(x\right)),max({\lambda }_{1e}^{+},{\lambda }_{2e}^{+})>max(1-h_{1e}^{+}\left(x\right)),(1-h_{2e}^{+}\left(x\right)).$$

Therefore, ${\tilde{x}}_{1E}\tilde{\vee }{\tilde{x}}_{2E}\tilde{q}\left(h_{1E}\tilde{\vee }{h}_{2E}\right)\tilde{\le }{f}_E\tilde{\vee }{g}_E$. Consequently, $f_E\tilde{\vee }{g}_E$ is a $QINVSN$ of ${\tilde{x}}_{1E}\tilde{\vee }{\tilde{x}}_{2E}$.

(4)

Let $f_E$ be a $QINVSN$ of ${\tilde{x}}_E$. Then, there exists $g_E$ that is a $QINVSN$ of ${\tilde{x}}_E$ such that ${\tilde{x}}_E\tilde{q}{g}_E\tilde{\le }{f}_E$. Consider the $g_E=h_E$. Indeed, since ${\tilde{x}}_E\tilde{q}{h}_E$ and $h_E$ is an $IVFS$-open set, $h_E$ is a $QINVSN$ of ${\tilde{x}}_E$. Thus, we obtain $h_E$ that is a $QINVSN$ of ${\tilde{y}}_E$.

□

Theorem 5.

In $IVFST(X,E,\tau )$, the $IVFS$ point ${\tilde{x}}_E$ belongs to $Cl{f}_E$ if and only if each $QIVFS$ of ${\tilde{x}}_E$ is soft quasi-coincident with $f_E$.

Proof. 

Let $IVFS$ point ${\tilde{x}}_E$ with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$ belong to $Cl{f}_E,i.e,{\tilde{x}}_E\tilde{\in }Cl{f}_E$. For any $IVFS$-closed $g_E$ containing $f_E$, ${\tilde{x}}_E\tilde{\in }{g}_E$, which implies that ${\lambda }_e^{-}\le g_e^{-}\left(x\right)$ and ${\lambda }_e^{+}\le g_e^{+}\left(x\right)$, for all $x\in X,e\in E$. Consider $h_E$ to be an $QIVFN$ of the $IVFS$ point ${\tilde{x}}_E$ and $h_E\neg \tilde{q}{f}_E$. Then, for any $e\in E$ and $x\in X$, $h_e^{-}\left(x\right)+f_e^{-}\left(x\right)\le 1,h_e^{+}\left(x\right)+f_e^{+}\left(x\right)\le 1$, and so, $f_E\tilde{\le }{h}_E^c$. Since $h_E$ is a $QIVFSN$ of the $IVFS$ point ${\tilde{x}}_E$, by ${\tilde{x}}_E$, it does not belong to $h_E^c$. Therefore, we have that ${\tilde{x}}_E$ does not belong to $Cl{f}_E$. This is a contradiction.

Conversely, let any $QIVFSN$ of the $IVFS$ point ${\tilde{x}}_E$ be soft quasi-coincident with $f_E$. Consider that ${\tilde{x}}_E$ doe not belong to $Cl{f}_E,i.e,{\tilde{x}}_E\notin Cl{f}_E$. Then, there exists an $IVFS$-closed set $g_E$, which contains $f_E$ such that ${\tilde{x}}_E$ does not belong to $g_E$. We have ${\tilde{x}}_E\tilde{q}{g}_E^c$. Then, $g_E^c$ is an $QIVFSN$ of the $IVFS$ point ${\tilde{x}}_E$ and $f_E\neg \tilde{q}{g}_E^c$. This is a contradiction with the hypothesis. □

5. IVFS Quasi-Separation Axioms

In this section, we develop the separation axioms to $IVFST$, so-called $IVFSQ$ separation axioms ($IVFSq$-$T_i$ axioms) for $i=0,1,2,3,4$, and consider some of their properties.

Definition 17.

Let $(X,E,\tau )$ be an $IVFST$ space. Let ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be $IVFS$ points over X, where:

$$\tilde{x}\left(e\right)\left(z\right)=\left\{\begin{array}{cc}[{\lambda }_e^{-},{\lambda }_e^{+}]& z=x\\ 0& otherwise\end{array}\right.$$

and:

$$\tilde{y}\left(e\right)\left(z\right)=\left\{\begin{array}{cc}[{\gamma }_e^{-},{\gamma }_e^{+}]& z=y\\ 0& otherwise,\end{array}\right.$$

then ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are said to be distinct if and only if ${\tilde{x}}_E\tilde{\wedge }{\tilde{y}}_E={\varnothing }_E$, which means $x\ne y$.

Definition 18.

Let $(X,E,\tau )$ be an $IVFST$ space. The $IVFS$ point ${\tilde{x}}_E$ is called a crisp $IVFS$ point $x_E^{[1,1]}$, if ${\lambda }_e^{-}={\lambda }_e^{+}=1$ for all $e\in E$.

Definition 19.

Let $(X,E,\tau )$ be an $IVFST$ space and ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be two $IVFS$ points. If there exists $IVFS$ open sets $f_E$ and $g_E$ such that:

(a) 

when ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two distinct $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\lambda }_e^{+}$ and ${\gamma }_e^{-},{\gamma }_e^{+}$, respectively, and $f_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$ and ${\tilde{y}}_E\neg \tilde{q}{f}_E$ or $g_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{y}}_E$ and ${\tilde{x}}_E\neg \tilde{q}{g}_E$,

(b) 

when ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two $IVFS$ points with the same supports $x=y$, e value ${\lambda }_e^{-}<{\gamma }_e^{-}$, and e value ${\lambda }_e^{+}<{\gamma }_e^{+}$ and $f_E$ is a $QIVFSN$ of the $IVFS$ point ${\tilde{y}}_E$ such that ${\tilde{x}}_E\neg \tilde{q}{f}_E$,

then $(X,E,\tau )$ is an interval-valued fuzzy soft quasi-$T_0$ space ($IVFS$q-$T_0$ space).

Example 8.

Consider the $IVFS$ set defined in Example $3$ and ${\tilde{x}}_E,{\tilde{y}}_E$ to be any two distinct $IVFS$ points in X defined by:

$$\tilde{x}\left(e\right)\left(z\right)=\left\{\begin{array}{cc}1& z=x\\ 0& z\ne x\end{array}\right.$$

and:

$$\tilde{y}\left(e\right)\left(z\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}z=y\hfill \\ 1\hfill & \mathrm{if}z\ne y.\hfill \end{array}\right.$$

Then, $f_E$ is an $IVFSN$ of ${\tilde{x}}_E$ and ${\tilde{y}}_E\neg \tilde{q}{f}_E$. Thus, X is an $IVFS$q-$T_0$ space.

Theorem 6.

$(X,E,\tau )$ is an $IVFS$q-$T_0$ space if and only if for every two $IVFS$ points ${\tilde{x}}_E$, $\widetilde{y_E}$ and ${\tilde{x}}_E\notin Cl{\tilde{y}}_E$ or ${\tilde{y}}_E\notin Cl{\tilde{x}}_E$.

Proof. 

Let $(X,E,\tau )$ be an $IVFS$q-$T_0$ space and ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be two $IVFS$ points in X.

First consider that ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two distinct $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\gamma }_e^{-}$ and ${\lambda }_e^{+},{\gamma }_e^{+}$, respectively. Then, a crisp $IVFS$ point ${\tilde{x}}_E^{[1,1]}$ has an $IVFSN$$f_E$ such that ${\tilde{y}}_E\neg \tilde{q}{f}_E$ or a crisp $IVFS$ point ${\tilde{y}}_E^{[1,1]}$ has an $IVFSN$$g_E$ such that ${\tilde{x}}_E\neg \tilde{q}{f}_E$. Consider that the crisp $IVFS$ point ${\tilde{x}}_E^{[1,1]}$ has an $IVFSN$$f_E$ such that ${\tilde{y}}_E\neg \tilde{q}{f}_E$. Moreover, $f_E$ is an $QINFSN$ of ${\tilde{x}}_E$ and ${\tilde{y}}_E\neg \tilde{q}{f}_E$. Hence, $\widetilde{x_E}\notin Cl\widetilde{y_E}$. Next, we consider the case ${\tilde{x}}_E$ and ${\tilde{y}}_E$ to be two $IVFS$ points with the same supports $x=y$, e lower value ${\lambda }_e^{-}<{\gamma }_e^{-}$, and e upper value ${\lambda }_e^{+}<{\gamma }_e^{+}$. Then, ${\tilde{y}}_E$ has a $QIVFSN$ that is not quasi-coincident with ${\tilde{x}}_E$, and so, by Theorem 5, $\widetilde{x_E}\notin Cl\widetilde{y_E}$.

Conversely, let ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be two $IVFS$ points in X. Consider without loss of generality that $\widetilde{x_E}\notin Cl\widetilde{y_E}$. First, consider that ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two distinct $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\gamma }_e^{-}$ and ${\lambda }_e^{+},{\gamma }_e^{-}$, respectively, since ${\tilde{x}}_E\notin Cl{\tilde{y}}_E$ for any $e\in E$, $f_e^{-}\left(y\right)=f_e^{+}\left(y\right)=0$ and $f_e^{-}\left(x\right)=f_e^{+}\left(x\right)=1$. Then, $Cl{\left({\tilde{y}}_E\right)}^c$ is an $IVFSN$ of ${\tilde{x}}_E$ such that $Cl{\left({\tilde{y}}_E\right)}^c\neg \tilde{q}{\tilde{y}}_E$. Next, let ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be two $IVFS$ points with the same supports $x=y$, and we must have e lower value ${\lambda }_e^{-}>{\gamma }_e^{-}$ and e upper value ${\lambda }_e^{+}>{\gamma }_e^{+}$, then ${\tilde{x}}_E$ has a $QIVFSN$ that is not quasi-coincident with ${\tilde{y}}_E$. □

Definition 20.

Let $(X,E,\tau )$ be an $IVFST$ and ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be two $IVFS$ points, if there exists $IVFS$ open sets $f_E$ and $g_E$ such that:

(a) 

when ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two distinct $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\gamma }_e^{-}$ and ${\lambda }_e^{+},{\gamma }_e^{+}$, respectively, $f_E$ is an $IVFSN$ of $IVFS$ points ${\tilde{x}}_E$ and ${\tilde{y}}_E\neg \tilde{q}{f}_E$, and $g_E$ is an $IVFSN$ of $IVFS$ points ${\tilde{y}}_E$ and ${\tilde{x}}_E\neg \tilde{q}{g}_E$,

(b) 

when ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two $IVFS$ points with the same supports $x=y$, e value ${\lambda }_e^{-}<{\gamma }_e^{-}$, and e value ${\lambda }_e^{+}<{\gamma }_e^{+}$, $f_E$ is an $QIVFSN$ of the $IVFS$ point ${\tilde{y}}_E$ such that ${\tilde{x}}_E\neg \tilde{q}{f}_E$,

then $(X,E,\tau )$ is an interval-valued fuzzy soft quasi-$T_1$ space ($IVFS$q-$T_1$ space).

Theorem 7.

$(X,E,\tau )$ is an $IVFS$q-$T_1$ space if and only if any $IVFS$ point ${\tilde{x}}_E$ in X is an $IVFS$-closed set.

Proof. 

Suppose that each $IVFS$ point ${\tilde{x}}_E$ in X is an $IVFS$-closed set, i.e., $g_E={\tilde{x}}_E^c$. Then, $g_E$ is an $IVFS$-open set. Let $x_E$ and $y_E$ be two $IVFS$ points as follows: First, consider that ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two distinct $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\gamma }_e^{-}$ and ${\lambda }_e^{+},{\gamma }_e^{+}$, respectively. Then, $g_E$ is an $IVFS$-open set such that $g_E$ is an $IVFSN$ of $IVFS$ point ${\tilde{y}}_E$ and ${\tilde{x}}_E\neg \tilde{q}{g}_E$. Similarly, $f_E={\tilde{y}}_E^c$ is an $IVFS$-open set and $f_E$ is an $IVFSN$ of the $IVFS$ points ${\tilde{x}}_E$ and ${\tilde{y}}_E\neg \tilde{q}{f}_E$. Next, we consider the case ${\tilde{x}}_E$ and ${\tilde{y}}_E$ to be two $IVFS$ points with the same supports $x=y$, e value ${\lambda }_e^{-}<{\gamma }_e^{-}$, and e value ${\lambda }_e^{+}<{\gamma }_e^{+}$. Then, ${\tilde{y}}_E$ has a $QIVFSN$$g_E$, which is not quasi-coincident with ${\tilde{x}}_E$. Thus, X is an $IVFS$q-$T_1$ space.

Conversely, Let $(X,E,\tau )$ be an $IVFS$q-$T_1$ space. Suppose that any $IVFS$ point ${\tilde{x}}_E$ is not an $IVFS$-closet set in X, i.e., $f_E\doteq {\tilde{x}}_E^c$. Then, ${\tilde{f}}_E\ne Cl{\tilde{f}}_E$, and there exists ${\tilde{y}}_E\tilde{\in }Cl{\tilde{f}}_E$ such that ${\tilde{x}}_E\ne {\tilde{y}}_E$.

First, consider that ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two distinct $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\gamma }_e^{-}$ and ${\lambda }_e^{+},{\gamma }_e^{+}$, respectively. Suppose that e lower value ${\lambda }_e^{-}\le 0.5$ and e upper value ${\lambda }_e^{+}\le 0.5$. Since ${\tilde{y}}_E\tilde{\in }Cl{f}_E$, by Theorem $4.1$, any $f_E$ is a $QIVFSN$ of ${\tilde{y}}_E$ and ${\tilde{x}}_E\tilde{q}{f}_E$. Then, there exists $IVFS$-open set $h_E$ such that $\tilde{y}\tilde{q}{h}_E\tilde{\le }{f}_E$. Hence, $h_e^{-}\left(y\right)+{\gamma }_e^{-}>1$. Next, let ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be two $IVFS$ points with the same supports $x=y$, e value ${\lambda }_e^{-}<{\gamma }_e^{-}$, and e value ${\lambda }_e^{+}<{\gamma }_e^{+}$. Since $y_E\tilde{\in }Cl{x}_E$, by Theorem 5, each $f_E$ is a $QIVFSN$ of $IVFS$ points ${\tilde{y}}_E$, ${\tilde{x}}_E\tilde{q}{f}_E$. This is a contradiction. □

Definition 21.

Let $(X,E,\tau )$ be an $IVFST$ and ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be two $IVFS$ points, if there exists $IVFS$ open sets $f_E$ and $g_E$ such that:

(a) 

when ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two distinct $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\gamma }_e^{-}$ and ${\lambda }_e^{+},{\gamma }_e^{+}$, respectively, $f_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{x}}_E$ and $g_E$ is an $IVFSN$ of the $IVFS$ point ${\tilde{y}}_E$, such that $f_E\neg \tilde{q}{g}_E$,

(b) 

when ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two $IVFS$ points with the same supports $x=y$, e value ${\lambda }_e^{-}<{\gamma }_e^{-}$, and e value ${\lambda }_e^{+}<{\gamma }_e^{+}$, $f_E$ is an $IVFSN$ of $IVFS$ point ${\tilde{x}}_E$ and $g_E$ is a $QIVFSN$ of $IVFS$ point ${\tilde{y}}_E$,

then $(X,E,\tau )$ is an interval-valued fuzzy soft quasi-$T_2$ space ($IVFS$ q-$T_2$ space).

Example 9.

Suppose that $X=[0,1]$ and E are any proper $(E\subset X)$. Consider $IVFS$ sets $f_E$ and $g_E$ over X defined as below: $f:E\to \mathcal{IVF}\left(\right[0,1\left]\right)$ and $g:E\to \mathcal{IVF}\left(\right[0,1\left]\right)$, such that for any $e\in E,x\in X$:

$$f\left(e\right)\left(x\right)=\left\{\begin{array}{cc}1& 0\le x\le e\\ 0& e<x\le 1\end{array}\right.$$

and:

$$g\left(e\right)\left(x\right)=\left\{\begin{array}{cc}0\hfill & 0\le x\le e\hfill \\ 1\hfill & e\le x\le 1.\hfill \end{array}\right.$$

Let $\tau =\{{\varnothing }_E,X_E,f_E,g_E\}$. Then clearly, τ is an $IVFST$ over X. Therefore, for any two absolute distinct $IVFS$ points ${\tilde{x}}_E,{\tilde{y}}_E$ in X defined by:

$$\tilde{x}\left(e\right)\left(z\right)=\left\{\begin{array}{cc}1& z=x\\ 0& z\ne x\end{array}\right.$$

and:

$$\tilde{y}\left(e\right)\left(z\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}z=y\hfill \\ 1\hfill & \mathrm{if}z\ne y.\hfill \end{array}\right.$$

Then, $f_E$ is an $IVFSN$ of ${\tilde{x}}_E$, and $g_E$ is an $IVFSN$ of ${\tilde{y}}_E$, such that $f_E\neg \tilde{q}{g}_E$. Then, X is an $IVFS$ q-$T_2$ space.

Theorem 8.

$IVFST(X,E,\tau )$ is an $IVFS$q-$T_2$ space if and only if for any $x\in X$, we have ${\tilde{x}}_E={\displaystyle \tilde{\bigwedge }}\{Cl{f}_E:f_E\in IVFSNof{\tilde{x}}_E\}$.

Proof. 

Let $(X,E,\tau )$ be a crisp $IVFS$q-$T_2$ space and ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\gamma }_e^{+}$. Let $y_E$ be a crisp $IVFS$ point with support y, e lower value ${\gamma }_e^{-}$, and e upper value ${\lambda }_e^{+}$. If ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\gamma }_e^{-}$ and ${\lambda }_e^{+},{\gamma }_e^{+}$, respectively, then there exist two $IVFS$-open sets $f_E$ and $g_E$ containing $IVFS$ points ${\tilde{y}}_E$ and ${\tilde{x}}_E$, respectively, such that $f_E\neg \tilde{q}{g}_E$. Then, $g_E$ is an $IVFSN$ of $IVFS$ point ${\tilde{x}}_E$ and $f_E$ is a $QIVFSN$ of ${\tilde{y}}_E$ such that $f_E\neg \tilde{q}{g}_E$. Hence, ${\tilde{y}}_E\notin Cl{g}_E$. If ${\tilde{x}}_E$ and ${\tilde{y}}_E$ are two $IVFS$ points with the same supports $x=y$, then ${\gamma }_e^{-}>{\lambda }_e^{-}$ and ${\gamma }_e^{-}>{\lambda }_e^{+}$. Thus, there are $QIVFSN$$f_E$ of $IVFS$ points ${\tilde{y}}_E$ and $IVFSN$$g_E$ such that $f_E\neg \tilde{q}{g}_E$. Hence, ${\tilde{y}}_E\notin Cl{g}_E$.

Conversely, let ${\tilde{x}}_E$ and ${\tilde{y}}_E$ be two distinct $IVFS$ points with different supports x and y, e lower values, and e upper values ${\lambda }_e^{-},{\lambda }_e^{+}$ and ${\gamma }_e^{-},{\gamma }_e^{+}$, respectively. Since:

$${\tilde{x}}_E=\tilde{\bigwedge }\{Cl{f}_E:f_E\in IVFSNof{\tilde{x}}_E\},\mathrm{and}\tilde{\bigwedge }\{Cl\left([f_e^{-},f_e^{+}]\right)\left(y\right):f_E\in IVFSNof{\tilde{x}}_E\}=0.$$

Thus, ${\tilde{y}}_E\neg \tilde{q}{\displaystyle \tilde{\bigwedge }}\{Cl{f}_E:f_E\in IVFSNof{\tilde{x}}_E\}$. Therefore, there exists $f_E$ that is an $IVFSN$ of $\tilde{x}$ and ${\tilde{y}}_E\neg \tilde{q}Cl{f}_E$. Take two $\tau$-$IVFS$-open sets $f_E$ and ${\left(Cl{f}_E\right)}^c$. Therefore, $f_E$ is an $IVFSN$ of $IVFS$ point ${\tilde{x}}_E$, ${\left(Cl{f}_E\right)}^c$ an $IVFSN$ of $IVFS$ point ${\tilde{y}}_E$, and $f_E\neg \tilde{q}{\left(Cl{f}_E\right)}^c$. □

Definition 22.

Let $(X,E,\tau )$ be an $IVFST$. If for any $IVFS$ point ${\tilde{x}}_E$ with support x, e lower values ${\lambda }_e^{-}$, and e upper values ${\lambda }_e^{+}$ and any $IVFS$-closed set $f_E$ in X such that ${\tilde{x}}_E\neg \tilde{q}{f}_E$, there exists two $IVFS$-open sets $h_E$ and $k_E$ such that ${\tilde{x}}_E\tilde{\in }{h}_E$ and $f_E\tilde{\le }{k}_E,h_E\neg \tilde{q}{k}_E$, then $(X,E,\tau )$ is called an interval-valued fuzzy soft quasi regular space ($IVFS$ q-regular space).

$(X,E,\tau )$ is called an interval-valued fuzzy soft quasi-$T_3$ space, if it is an $IVFS$ q-regular space and an $IVFS$ q-$T_1$ space.

Theorem 9.

$IVFST(X,E,\tau )$ is an $IVFS$ q-$T_3$ space if and only if for any $IVFSN$$g_E$ of $IVFS$ point ${\tilde{x}}_E$ there exists an $IVFS$-open set $f_E$ in X such that ${\tilde{x}}_E\tilde{\in }{f}_E\tilde{\le }cl{f}_E\tilde{\le }{g}_E$.

Proof. 

Let $g_E$ be an $IVFS$ set in X and ${\tilde{x}}_E$ be an $IVFS$ point with support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$ such that ${\tilde{x}}_E\tilde{\in }{g}_E$. Then, clearly, $g_E^c$ is an $IVFS$-closed set. Since X is an $IVFS$ q-$T_3$ space, there exist two $IVFS$-open sets $f_E,h_E$ such that ${\tilde{x}}_E\tilde{\in }{f}_E,g_E^c\tilde{\le }{h}_E,h_E$ and $f_E\neg \tilde{q}{h}_E$. Thus, $f_E^c\tilde{\le }{h}_E^c$. Therefore, $Cl{f}_E\tilde{\le }{h}_E^c$ implies $Cl{f}_E\tilde{\le }{g}_E$. Hence, ${\tilde{x}}_E\tilde{\in }{f}_E\tilde{\le }Cl{f}_E\tilde{\le }{g}_E$.

Conversely, let ${\tilde{x}}_E$ be an $IVFS$ point with different support x, e lower value ${\lambda }_e^{-}$, and e upper value ${\lambda }_e^{+}$, and let $g_E$ be an $IVFS$-closed set such that ${\tilde{x}}_E\neg \tilde{q}{g}_E$. Then, $g_E^c$ is an $IVFS$-open set containing the $IVFS$ point ${\tilde{x}}_E$, i.e., ${\tilde{x}}_E\tilde{\in }{g}_E^c$. Thus, there exists an $IVFS$-open set $f_E$ containing ${\tilde{x}}_E$ such that ${\tilde{x}}_E\tilde{\in }{f}_E\tilde{\le }Cl{f}_E\tilde{\le }{g}_E$ $g_E\tilde{\le }{\left(Cl{f}_E\right)}^c$. Therefore, clearly, ${\left(Cl{f}_E\right)}^c$ is an $IVFS$-open set containing $g_E$ and $f_E\neg \tilde{q}{\left(Cl{f}_E\right)}^c$. Hence, X is an $IVFS$ q-$T_3$ space. □

Definition 23.

Let $(X,E,\tau )$ be an $IVFST$. If for any two $IVFS$-closed sets $f_E$ and $g_E$ such that $f_E\neg \tilde{q}{g}_E$, there exists two $IVFS$-open sets $h_E$ and $k_E$ such that $f_E\tilde{\le }{h}_E$ and $g_E\tilde{\le }{k}_E$, then $(X,E,\tau )$ is called an interval-valued fuzzy soft quasi-normal space ($IVFS$ q-normal space).

$(X,E,\tau )$ is called an interval-valued fuzzy soft quasi $T_4$ space if it is an $IVFS$ q-normal space and an $IVFS$q-$T_1$ space.

Theorem 10.

$IVFST(X,E,\tau )$ is an $IVFS$ q-$T_4$ space if and only if for any $IVFS$-closed set $f_E$ and $IVFS$-open set containing $f_E$, there exists an $IVFS$-open set $h_E$ in X such that $f_E\tilde{\le }{h}_E\tilde{\le }cl{h}_E\tilde{\le }{g}_E$.

Proof. 

Let $f_E$ be an $IVFS$-closed set in X and $g_E$ be an $IVFS$-open set in X containing $f_E$, i.e., $f_E\tilde{\le }{g}_E$. Then, $g_E^c$ is an $IVFS$-closed set such that $f_E\neg \tilde{q}{g}_E^c$.

Since X is an $IVFS$ q-$T_4$ space, there exist two $IVFS$-open sets $h_E,k_E$ such that $f_E\tilde{\le }{h}_E,g_E^c\tilde{\le }{k}_E$, and $h_E\neg \tilde{q}{k}_E$. Thus, $h_E\tilde{\le }{k}_E^c$, but $Cl{h}_E\tilde{\le }Cl{k}_E^c=k_E$. Furthermore, $g_E^c\tilde{\le }{k}_E$ implies $k^c\tilde{\le }{g}_E$. That is an $IVFS$-closed set over X. Therefore, $Cl{h}_E\tilde{\le }{k}_E^c$. Hence, we have $f_E\tilde{\le }{h}_E\tilde{\le }Cl{h}_E\tilde{\le }{g}_E$.

Conversely, let ${\tilde{f}}_E$ and $g_E$ be any $IVFS$-closed sets such that $f_E\neg \tilde{q}{g}_E$. Then, $f_E\tilde{\le }{g}_E^c$. Thus, there exists an $IVFS$-open set $h_E$ such that $f_E\tilde{\le }{h}_E\tilde{\le }Cl{h}_E\tilde{\le }{g}_E$. Therefore, there are two $IVFS$-open sets $h_E$ and ${\left(Cl{h}_E\right)}^c$ such that $f_E\tilde{\le }{h}_E$, $g_E\tilde{\le }{\left(Cl{h}_E\right)}^c$. This shows that X is an $IVFS$ q-$T_4$ space. □

Theorem 11.

If $\Phi :(X_1,E_1,{\tau }_1)\to (X_2,E_2,{\tau }_2)$ is an $IVFSC$ and $IVFSO$ map where ${\Phi }_u{X}_1\to X_2$ and ${\Phi }_p{E}_1\to E_2$ are two ordinary bijections, then $X_1$ is an $IVFS$q-$T_i$ space if and only if $X_2$ is an $IVFS$q-$T_i$ space for $i=0,1,2,3,4$.

Proof. 

We just prove when $i=2$. The other parts are similar.

Suppose that we have two $IVFS$ points ${\tilde{k}}_{E_2}$ and ${\tilde{s}}_{E_2}$ with different supports k and s, e lowers value, and e upper values ${\lambda }_e^{-},{\lambda }_e^{+}$ and ${\gamma }_e^{-},{\gamma }_e^{+}$, respectively, for any $e\in E_2$. Then, the inverse lower and upper image of $IVFS$ point ${\tilde{k}}_{E_2}$ under the $IVFSO$ map $\Phi$ is an $IVFS$ point in $X_1$ with different support ${\Phi }^{-1}\left(k\right)$ as below:

$${\Phi }^{-1}\left({\tilde{k}}^{-}\right)\left(e\right)\left(x\right)={\tilde{k}}^{-}\left({\Phi }_p\left(e\right)\right)\left({\Phi }_u\left(x\right)\right)and{\Phi }^{-1}\left({\tilde{k}}^{+}\right)\left(e\right)\left(x\right)={\tilde{k}}^{+}\left({\Phi }_p\left(e\right)\right)\left({\Phi }_u\left(x\right)\right).$$

Furthermore, the inverse lower and upper image of $IVFS$ point ${\tilde{s}}_{E_2}$ under the $IVFSO$ map $\Phi$ is an $IVFS$ point in $X_1$ with different support ${\Phi }^{-1}\left(s\right)$ as below:

$${\Phi }^{-1}\left({\tilde{s}}^{-}\right)\left(e\right)\left(x\right)={\tilde{s}}^{-}\left({\Phi }_p\left(e\right)\right)\left({\Phi }_u\left(x\right)\right)and{\Phi }^{-1}\left({\tilde{s}}^{+}\right)\left(e\right)\left(x\right)={\tilde{s}}^{+}\left({\Phi }_p\left(e\right)\right)\left({\Phi }_u\left(x\right)\right).$$

Since $(X_1,E_1,{\tau }_1)$ is an $IVFS$q-$T_2$ space, there exist two $IVFS$-open sets $f_E$ and $g_E$ in $X_1$ such that ${\Phi }^{-1}\left({\tilde{k}}_{E_2}\right)\tilde{\in }{f}_E$, ${\Phi }^{-1}\left({\tilde{s}}_{E_2}\right)\tilde{\in }{g}_E$, and $f_E\neg \tilde{q}{g}_E$. Thus, ${\tilde{k}}_{E_2}\tilde{\in }{f}_E$ and ${\tilde{s}}_{E_2}\tilde{\in }{g}_E$, while $\Phi \left(f_E\right)\neg \tilde{q}\Phi \left(g_E\right)$. Therefore, $(X_2,E_2,{\tau }_2)$ is an $IVFS$q-$T_2$ space.

Conversely, suppose that we have two $IVFS$ points ${\tilde{x}}_E$ and ${\tilde{y}}_E$ with different supports $x,y\in X_1$, e lower value, and e upper value ${\lambda }_e^{-},{\lambda }_e^{+}$ and ${\gamma }_e^{-},{\gamma }_e^{+}$, respectively. Then, the lower and upper image of an $IVFS$ point ${\tilde{x}}_E$ under the $IVFSC$ map $\Phi$ is an $IVFS$ point in $X_2$ with different support ${\Phi }_u\left(x\right)$ as below:

$$\begin{array}{lll}\Phi \left({\tilde{x}}^{-}\right)\left(\epsilon \right)\left(k\right)& =& {\displaystyle \underset{z\in {\Phi }^{-1}\left(k\right)}{sup}}\left[{\displaystyle \underset{e\in {\Phi }_p^{-1}\left(\epsilon \right)}{sup}}\left({\tilde{x}}^{-}\right)\left(e\right)\right]\left(z\right)\hfill \\ & =& \left\{\begin{array}{cc}{\lambda }_e^{-}& ifk={\Phi }_u\left(x\right)\\ 0& otherwise,\end{array}\right.\hfill \end{array}$$

and:

$$\begin{array}{lll}\Phi \left({\tilde{x}}^{+}\right)\left(\epsilon \right)\left(k\right)& =& {\displaystyle \underset{z\in {\Phi }^{-1}\left(k\right)}{sup}}\left[{\displaystyle \underset{e\in {\Phi }_p^{-1}\left(\epsilon \right)}{sup}}\left({\tilde{x}}^{+}\right)\left(e\right)\right]\left(z\right)\hfill \\ & =& \left\{\begin{array}{cc}{\lambda }_e^{+}& ifk={\Phi }_u\left(x\right)\\ 0& otherwise\end{array}\right.\hfill \end{array}$$

and the lower and upper image of an $IVFS$ point ${\tilde{y}}_E$ under the $IVFSC$ map $\Phi$ is an $IVFS$ point in $X_2$ with different support ${\Phi }_u\left(y\right)$ as below:

$$\begin{array}{lll}\Phi \left({\tilde{y}}^{-}\right)\left(\epsilon \right)\left(k\right)& =& {\displaystyle \underset{z\in {\Phi }^{-1}\left(k\right)}{sup}}\left[{\displaystyle \underset{e\in {\Phi }_p^{-1}\left(\epsilon \right)}{sup}}\left({\tilde{y}}^{-}\right)\left(e\right)\right]\left(z\right)\hfill \\ & =& \left\{\begin{array}{cc}{\gamma }_e^{-}& ifk={\Phi }_u\left(y\right)\\ 0& otherwise\end{array}\right.\hfill \end{array}$$

and:

$$\begin{array}{lll}\Phi \left({\tilde{y}}^{+}\right)\left(\epsilon \right)\left(k\right)& =& {\displaystyle \underset{z\in {\Phi }^{-1}\left(k\right)}{sup}}\left[{\displaystyle \underset{e\in {\Phi }_p^{-1}\left(\epsilon \right)}{sup}}\left({\tilde{y}}^{+}\right)\left(e\right)\right]\left(z\right)\hfill \\ & =& \left\{\begin{array}{cc}{\gamma }_e^{+}& ifk={\Phi }_u\left(y\right)\\ 0& otherwise.\end{array}\right.\hfill \end{array}$$

Since $(X_2,E_2,{\tau }_2)$ is an $IVFS$q-$T_2$ space, there exist two $IVFS$-open sets $f_{E_2}$ and $g_{E_2}$ in $X_2$ such that $\Phi \left(\tilde{x}\right)\tilde{\in }{f}_{E_2}$, $\Phi \left(\tilde{y}\right)\tilde{\in }{g}_{E_2}$, and $f_{E_2}\neg \tilde{q}{g}_{E_2}$. Clearly, ${\tilde{x}}_E\tilde{\in }{\Phi }^{-1}\left(f_{E_2}\right),{\tilde{y}}_E\tilde{\in }{\Phi }^{-1}\left(g_{E_2}\right)$ and ${\Phi }^{-1}\left(f_{E_2}\right)\neg \tilde{q}{\Phi }^{-1}\left(g_{E_2}\right)$. Then, $(X_1,E_1,{\tau }_1)$ is an $IVFS$q-$T_2$ space. □

6. Conclusions

The aim of this study was to develop the interval-valued fuzzy soft separation axioms in order to build a framework that will provide a method for object ranking. Thus, in this paper, we introduced a new definition of the interval-valued fuzzy soft point and then considered some of its properties, and different types of neighborhoods of the $IVFS$ point were studied in interval-valued fuzzy soft topological spaces. The separation axioms of interval-valued fuzzy soft topological spaces were presented, and furthermore, the basic properties were also studied.