**2. Preliminaries**

In this section, we recall some definitions and properties of interval-valued fuzzy sets (*IVF*) and interval-valued fuzzy soft sets (*IVFS*). Note that, throughout this paper, *X* and *E* denote the sets of objects and parameters, respectively. I*<sup>X</sup>* and [I] *X*, where I = [0, 1] and [I] = {[*<sup>a</sup>*, *b*], *a* ≤ *b*, *a*, *b* ∈ I} denote, respectively, the set of all fuzzy subsets and the set of all interval-valued fuzzy subsets of *X*.

**Definition 1.** *Ref. [2] A pair* (*f* , *<sup>X</sup>*)*, is called an IVF subset of X if f is a mapping given by f* : *X* → [I] *such that for any x* ∈ *X*, *f*(*x*)=[ *f* <sup>−</sup>(*x*), *f* +(*x*)] *is a closed subinterval of* [0, 1] *where f* <sup>−</sup>(*x*) *and f* +(*x*) *are referred to as the lower and upper degrees of membership x to f and* 0 ≤ *f* <sup>−</sup>(*x*) ≤ *f* +(*x*) ≤ 1*.*

In 1999, Molodtsov [5] defined the concept of soft sets (SS) for the first time as a pair of (*f* , *E*) or *fE* such that *E* is a parameter set and *f* is the mapping *f* : *E* → 2*<sup>X</sup>* where for any *e* ∈ *E*, *f*(*e*) is a subset of *X*. By combining the concepts of soft sets and interval-valued fuzzy sets, a new hybrid tool was defined as the following.

**Definition 2.** *Ref. [25] A pair* (*f* , *E*) *is called an IVFS set over X if the mapping f is given by f* : *E* → [I] *X where for any e* ∈ *E and x* ∈ *X, f*(*e*)(*x*)=[ *f* <sup>−</sup>(*e*)(*x*), *f* +(*e*)(*x*)]*.*

Consider two *IVFSs fE*, *gE* over the common universe *X*. The union of *fE* and *gE*, denoted by *fE*∨˜ *gE*, is the *IVFSs* (*f* ∨˜ *g*)*<sup>E</sup>*, where ∀*e* ∈ *E* and any *x* ∈ *X*, we have (*f* ∨˜ *g*)(*e*)(*x*)=[max{ *f* − *e* (*x*), *g*<sup>−</sup> *e* (*x*)}, max{ *f* + *e* (*x*), *g*<sup>+</sup> *e* (*x*)}]. The intersection of *fE* and *gE*, denoted by *fE*∧˜ *gE*, is the *IVFSs* (*f* ∧˜ *g*)*<sup>E</sup>*, where ∀*e* ∈ *E* and ∀*x* ∈ *X*, we have (*f* ∧˜ *g*)(*e*)(*x*)=[min{ *f* − *e* (*x*), *g*<sup>−</sup> *e* (*x*)}, min{ *f* + *e* (*x*), *g*<sup>+</sup> *e* (*x*)}]. The complement of *fE* is denoted by *f c E* and is defined by *f c* : *E* → [I] *X* where ∀*e* ∈ *E* and any *x* ∈ *X*, *f <sup>c</sup>*(*e*)(*x*) = [1 − *f* + *e* (*x*), 1 − *f* − *e* (*x*)]. The null *IVFSs*, denoted by ∅*E*, is defined as an *IVFSs* over *X* such that *f* − *e* (*x*) = *f* + *e* (*x*) = 0 for all *e* ∈ *E* and any *x* ∈ *X*. The absolute *IVFSs*, denoted by *XE*, is defined as an *IVFSs* over *X* where *f* − *e* (*x*) = *f* + *e* (*x*) = 1, ∀*e* ∈ *E* and any *x* ∈ *X*.

Using the matrix form of interval-valued fuzzy relations, authors in [39] represented a finite IVFSs *fE* as the following *n* × *m* matrix

$$f\_{\mathbb{E}} = \left[ [f\_{ij}^-, f\_{ij}^+] \right]\_{n \times m} = \begin{bmatrix} [f\_{\varepsilon\_1}^-(\mathbf{x}\_1), f\_{\varepsilon\_1}^+(\mathbf{x}\_1)] & \dots & [f\_{\varepsilon\_1}^-(\mathbf{x}\_m), f\_{\varepsilon\_1}^+(\mathbf{x}\_m)] \\ \vdots & \dots & \vdots \\ [f\_{\varepsilon\_n}^-(\mathbf{x}\_1), f\_{\varepsilon\_1}^+(\mathbf{x}\_1)] & \dots & [f\_{\varepsilon\_n}^-(\mathbf{x}\_m), f\_{\varepsilon\_1}^+(\mathbf{x}\_m)] \end{bmatrix}\_{n \times m}$$

where | *E* |= *n*, | *X* |= *m* and *f* −*ij*= *f* − *ei*(*xj*), *f* +*ij* = *f* + *ei*(*xj*) for *i* = 1, . . . , *n* and *j* = 1, . . . , *m*.

Accordingly, the concepts of union, intersection, complement, etc., can be represented in a matrix format in the finite case.

**Definition 3.** *Ref. [46] A triplet* (*<sup>X</sup>*, *E*, *τ*) *is called an interval-valued fuzzy soft topological space (IVFST) if τ is a collection of interval-valued fuzzy soft subsets of X containing absolute and null IVFSs and closed under arbitrary union and finite intersection.*
