Multi-criteria group decision making with Pythagorean fuzzy soft topology

1 Introduction

To tackle real world problems, the techniques usually employed in classical mathematics are not always beneficial due to uncertainties and vagueness present in these problems. There are numerous techniques, like theory of probability and the interval mathematics which can be thought of as mathematical models for coping with uncertainties and vagueness. Unluckily, all these models have their own drawbacks and limitations. Zadeh [53] initiated the idea of fuzzy sets as an extension of the traditional crisp set. A fuzzy set is a momentous mathematical model to typify an assembling of bits and pieces possessing incomprehensible boundary. Pursuing the path of Zadeh, Atanassov [5–7] proposed the idea of intuitionistic fuzzy sets as an extension of fuzzy set by introducing the concepts of membership (denoted by μ (x)) and non-membership grades (denoted by ν (x)) along with the restriction that sum of these two grades must not exceed unity. Atanassov [8] portrayed geometrical elucidation of the elements of intuitionistic fuzzy entities. Molodtsov [29] originated the notion of a new kind of sets conventionally known as soft sets, as another model for sorting out uncertainties.

Fuzzy sets, soft sets and their hybrid structures are strong mathematical models for solving many real world problems. Indeed, the real power of these models is in their ability to handle and manipulate verbally-stated information based on perceptions rather than equations. Mathematical modeling on real world problems often involve multi-attribute, multi-agent, multi-polar information, uncertainties and vagueness. Recently, the researchers have introduced different mathematical models including fuzzy soft set (FS-set) introduced by Çağman et al. [12] and Maji et al. [27], fuzzy parameterized soft set (FP-soft set) introduced by Çağman et al. [13], fuzzy parameterized fuzzy soft set (FPFSS) introduced by Çağman et al. [14], and Zorlutuna and Atmaca [61]. Maji et al. [28] familiarized the concept of intuitionistic fuzzy soft set (IFS-set).

Topological structure on these sets have many applications towards decision making problems, medical diagnosis, artificial intelligence, pattern recognition, forecasting and image processing etc. Soft topology was introduced by Bashir and Sabir [10], Çağman et al. [11], Hazra et al. [23], Roy and Samanta [45], Shabir and Naz [46], and Varol et al. [47]. Aygunoglu et al. [9] introduced fuzzy soft topological spaces. Zorlutuna and Atmaca [61] introduced fuzzy parameterized fuzzy soft topology (FPFS-topology). Osmanoglu and Tokat [32], and Li and Cui [25] introduced intuitionistic fuzzy soft topology (IFS-topology). Riaz et al. [37–39] introduced N-soft topology and soft rough topology with applications to multi-criteria group decision making (MCGDM). Yu et al. [51] presented consensus reaching for MAGDM with multi-granular hesitant fuzzy linguistic term sets. Zhang et al. [60] presented managing multigranular unbalanced hesitant fuzzy linguistic information in multiattribute large-scale group decision making based on linguistic distribution approach. Yu et al. [52] studied extended TODIM for multi-criteria group decision making based on unbalanced hesitant fuzzy linguistic term sets.

Yager [48–50] introduced Pythagorean fuzzy set as an extension of Atanassov’s intuitionistic fuzzy set and presented Pythagorean membership grades with applications to multi-criteria decision making (MCDM). The eminent characteristics of PFSs is that the sum of squares of membership degree and nonmembership degree to each alternative is less than or equal to 1. Obviously, PFSs have more capability than IFSs to model the vagueness in practical MCDM problems.

Fuzzy sets, Pythagorean fuzzy sets, soft sets, rough sets, neutrosophic sets and their hybrid structures have been studied by many researchers including Akram et al. [1–3], Eraslan and Karaaslan [15], Feng et al. [16, 17], Garg [19, 20], Guleria and Bajaj [18], Hashmi et al. [21], Hashmi and Riaz [22], Kumar and Garg [24], Naeem et al. [30, 31], Peng and Yang [33], Peng and Selvachandran [34], Peng et al. [36], Riaz et al. [40–44], Zhang and Xu [57], Zhan et al. [54–56], Zhang and Zhan [58], and Zhang et al. [59].

The goal of this paper is to introduce PFS-topology on PFSS. PFS-topology is the extension of IFS-topology. In order to solve the real world problems, IFS-topology cannot deal with the situation if the sum of membership degree and non-membership degree of the parameter is larger than 1. It makes the MCGDM limited, and affects the optimum decision. PFSS and PFS-topology provide a large number of applications to MCGDM for such real world problems.

Rest of the paper is sorted out as pursues. In Section 2, we recall some primary notions of fuzzy set, soft set, IFS, PFS and PFSS. In Section 3, we introduce PFS-topology and its properties with the help of examples. In Section 4, we construct Pythagorean fuzzy soft separation axioms, Pythagorean fuzzy soft regular and normal spaces along with related results. In Section 5, we present an application of PFSSs using choice value method based on MCGDM which yields the optimum results for investment in the stock exchange using hypothetical data. In Section 6, we render an application of TOPSIS in medical diagnosis where PFS-topology is used. Finally, in Section 7, we have ended with conclusion and some future directions.

2 Preliminaries

In this section, we concisely review some primary concepts related to different kinds of sets including fuzzy set, soft set, intuitionistic fuzzy set, Pythagorean fuzzy set and Pythagorean fuzzy soft set that would be employed in the remnant part of the paper.

Definition 2.1. [53] Let X be a classical set and μ _A  : X → [0, 1] be the membership function. Then a fuzzy set A can be written in the form A = ( X , μ A ) = { ( ρ , μ A ( ρ ) ) : ρ ∈ X } The value of the mapping μ _A at ρ ∈ X i.e. μ _A  (ρ) denotes the degree of membership of ρ to the fuzzy set A. The aggregate of all fuzzy sets in X is designated as F ( X ) .

Definition 2.2. [5–7] An intuitionistic fuzzy set (IFS) A over the universe X comprises ordered triplets ( ρ , μ A ( ρ ) , ν A ( ρ ) ) , where ρ ∈ X. The mappings μ A and ν A map elements of X to the unit closed interval [0, 1], with the restriction that their sum must not exceed unity, are respectively termed as degrees of association and non-association of the element ρ to A . This set in set builder form is usually symbolized as A = { ( ρ , μ A ( ρ ) , ν A ( ρ ) ) : ρ ∈ X }

Definition 2.3. [29] Supposing X to be a crisp set and E a non-void assembly of attributes. Further assume that 2 ^X is the power set of X and A is a non-void sub-collection of E. A doublet (ψ, A), where ψ : A → 2 ^X is a mapping, is termed as soft set over X. Thus, in set-builder form, we write ( ψ , A ) = { ( e , ψ ( e ) ) : e ∈ A , ψ ( e ) ∈ 2 X } The pair (ψ, A) is also shortly written ψ _A .

Definition 2.4. [13] Take X to be a crisp set and A the sub-family of attribute’s set E. Suppose that F ( X ) stands for assembly of all fuzzy sets defined over X. A collection of ordered doublets Γ A = { ( e , γ A ( e ) ) : e ∈ E , γ A ∈ F ( X ) } is termed as fuzzy soft set (or an fs-set) over X. Here γ A : E → F ( X ) such that γ _A  (e) = φ whenever e ∉ A, is known as fuzzy approximate function of Γ _A . The value γ _A  (e) is a set commonly referred to as e-element of the fs-set, for all e ∈ E.

The aggregate of all fuzzy soft sets over X is designated as FS (X).

Definition 2.5. [26] Understand that X is a crisp set endowed with A as the sub-family of attribute’s set E. Assume that IF ^X stands for assembly of all IF-subsets defined over X. A multi-valued mapping ψ that drags elements of A to IF ^X is termed as an intuitionistic fuzzy soft set (IFS-set) over X is designated as (ψ, A) or sometimes as ψ _A . For any attribute e ∈ A, ψ (e) yields the value in the form of an an IFS.

The set of all IFS-sets over the universe X having set of attributes from E is acknowledged as IFS-class, designated as IFS (X, E).

Definition 2.6. [48–50] A Pythagorean fuzzy set, abbreviated as PFS, is a family of the form P = { < ρ , μ P ( ρ ) , ν P ( ρ ) > : ρ ∈ X } where μ _P and ν _P are mappings from some crisp set X to the unit interval with the restriction that sum of their squares should not exceed unity i.e. 0 ≤ μ P 2 ( ρ ) + ν P 2 ( ρ ) ≤ 1 , called correspondingly the grade of association and non-association of ρ ∈ X to the set P. The doublet (μ _p , ν _p ) is called Pythagorean fuzzy number, abbreviated as PFN.

The space for a PFS is a unit circular arc in the first quadrant whereas it was a right isosceles triangle having length of the base and altitude each equal to unity in case of an IFS. Hence we have an enlarged space for PFSs as compared to IFSs as can be seen in Fig. 1. In fact we have included 2 π - 1 8 extra area for PFSs.

OPEN IN VIEWER

Fig. 1

Spaces of PFSs & IFSs

Definition 2.7. [57] The score function for any PFN p = (μ _p , ν _p ) is defined as s ( p ) = μ p 2 - ν p 2 where -1 ≤ s (p) ≤1. If p _i and p _j are two Pythagorean fuzzy numbers, then

If s (p _i ) < s (p _j ) then p _i precedes p _j i.e. p _i  ≺ p _j ,

Remark. If we choose p₁ = (0.1, 0.1) and p₂ = (0.7, 0.7), then according to Definition 2.7, we have p₁ ∼ p₂ which does not seem to be reasonable. So the idea of accuracy function is floated to amend the comparison rules.

Definition 2.8. [57] The accuracy function for any PFN p = (μ _p , ν _p ) is defined as a ( p ) = μ p 2 + ν p 2 where 0 ≤ a (p) ≤1. Assume that p _i and p _j are two PFNs.

If s (p _i ) and s (p _j ) coincide and a (p _i ) exceeds a (p _j ) then p _i  ≻ p _j ,

Definition 2.9. Let X be the universe and E the set parameters. Assume that A ⊆ E and PF ^X represent the class of all Pythagorean fuzzy subsets over X. A Pythagorean fuzzy soft set (PFSS) on X is denoted as (ψ, A) or ψ _A , where ψ : A → PF ^X is a mapping, and defined by

( ψ , A ) = { ( e , { ρ , μ ψ A ( ρ ) , ν ψ A ( ρ ) } ) : e ∈ A , ρ ∈ X } = { ( e , { ρ ( μ ψ A ( ρ ) , ν ψ A ( ρ ) ) } ) : e ∈ A , ρ ∈ X } = { ( e , { ( μ ψ A ( ρ ) , ν ψ A ( ρ ) ) ρ } ) : e ∈ A , ρ ∈ X }

where μ _{ψ A} and ν _{ψ A} are well-defined maps that drag elements of X to the unit closed interval [0, 1] along with the property that sum of their squared values should not exceed unity are termed as Pythagorean fuzzy numbers (PFNs). In particular, μ _{ψ A}  (ρ) represents the grade of association and ν _{ψ A}  (ρ) denotes degree of non-association of the constituent ρ ∈ X to the set (ψ, A).

For any attribute e, ψ (e) yields the output of multi-valued map at e which is all the time a Pythagorean fuzzy set. Hence, a PFSS (ψ, A) is described by the multi-valued mapping ψ : A → 2 ^X .

The throng of all PFSSs over X endued with assortment of traits from E is called PFS-class and is designated as PFS (X, E).

If we write μ _ij  = μ _{ψ A}  (e _j ) (ρ _i ) and ν _ij  = ν _{ψ A}  (e _j ) (ρ _i ) where i and j run, respectively, from 1 to m and from 1 to n then the PFSS ψ _A may be represented in tabular form as in Table 1.

The corresponding matrix form is ( ψ , A ) = [ ( μ ij , ν ij ) ] m × n = ( ( μ 11 , ν 11 ) ( μ 12 , ν 12 ) ⋯ ( μ 1 n , ν 1 n ) ( μ 21 , ν 21 ) ( μ 22 , ν 22 ) ⋯ ( μ 2 n , ν 2 n ) ⋮ ⋮ ⋱ ⋮ ( μ m 1 , ν m 1 ) ( μ m 2 , ν m 2 ) ⋯ ( μ mn , ν mn ) ) This matrix is called Pythagorean fuzzy soft matrix or shortly PFS-matrix.

Definition 2.10. [30] Let ψ A 1 ( 1 ) and ψ A 2 ( 2 ) be PFSSs over X. Then ψ A 1 ( 1 ) is PFS-subset of ψ A 2 ( 2 ) i.e. ψ A 1 ( 1 ) ⊆ ˜ ψ A 2 ( 2 ) , if

A₁ ⊆ A₂, and

Definition 2.11. [30] Let (ψ₁, A₁) and (ψ₂, A₂) be PFSSs defined over X. Then their union is defined as ( ψ , A ) = ( ψ 1 , A 1 ) ∪ ˜ ( ψ 2 , A 2 ) , where A = A₁ ∪ A₂ and for all e ∈ A, ψ ( e ) = { ψ 1 ( e ) , if e ∈ A 1 ∖ A 2 ψ 2 ( e ) , if e ∈ A 2 ∖ A 1 ψ 1 ( e ) ∪ ψ 2 ( e ) , if e ∈ A 1 ∩ A 2 where ψ₁ (e) ∪ ψ₂ (e) is the union of two PFSSs.

Definition 2.12. [30] The intersection of two PFSSs (ψ₁, A₁) and (ψ₂, A₂) is a PFSS ( ψ , A ) = ( ψ 1 , A 1 ) ∩ ˜ ( ψ 2 , A 2 ) , where A = A₁ ∩ A₂ ≠ φ and ψ (e) = ψ₁ (e) ∩ ψ₂ (e) for all e ∈ A.

Definition 2.13. [30] The difference (ψ, A) of two PFSSs (ψ₁, A₁) and (ψ₂, A₂) over X is described by (ψ, e) = (ψ₁, e) \ (ψ₂, e) ; ∀ e ∈ E and is designated as ( ψ 1 , A ) \ ˜ ( ψ 2 , A ) . Hence,

( ψ 1 , A 1 ) \ ˜ ( ψ 2 , A 2 ) = { ( e , { ρ , μ , ν } ) : ρ ∈ X , e ∈ E }

where μ = min { μ ψ 1 ( e ) ( ρ ) , ν ψ 2 ( e ) ( ρ ) } , and ν = max { ν ψ 1 ( e ) ( ρ ) , μ ψ 2 ( e ) ( ρ ) } .

Definition 2.14. [30] The complement of a PFSS (ψ, A) is a mapping ψ ^c  : A → PF ^X given by ψ ^c  (e) = [ψ (e)]  ^c , for all e ∈ A. It is represented as (ψ, A)  ^c or sometimes by (ψ ^c , A). Thus, if ψ ( e ) = { ( ρ , μ ψ ( e ) ( ρ ) , ν ψ ( e ) ( ρ ) ) : ρ ∈ X } then ψ c ( e ) = { ( ρ , ν ψ ( e ) ( ρ ) , μ ψ ( e ) ( ρ ) ) : ρ ∈ X } for all e ∈ A.

Definition 2.15. [30] A PFSS (ψ, E) over X is known as null PFSS, symbolized as ψ _φ or Φ, if ∀ e ∈ E we have ψ ( e ) = 0 ˜ , where 0 ˜ denotes null Pythagorean fuzzy set. Hence, ψ φ = { ( e , { ρ , 0 , 1 } ) : e ∈ E , ρ ∈ X }

Definition 2.16. [30] A PFSS (ψ, E) over X is called absolute PFSS, denoted by X ˘ , if ψ ( e ) = 1 ˜ , ∀ e ∈ E . Here 1 ˜ denotes absolute Pythagorean fuzzy set. Hence, X ˘ = { ( e , { ρ , 1 , 0 } ) : e ∈ E , ρ ∈ X }

Definition 2.17. [30] A Pythagorean fuzzy soft set PFSS (ψ, A) is called Pythagorean fuzzy soft point (PFS-point), denoted as ϑ _ψ , if for the element ϑ ∈ A we have

ψ (ϑ) ≠ ψ _φ , and

Definition 2.18. [30] A PFS-point ϑ _ψ belongs to a PFSS (ψ₁, A) i.e. ϑ ψ ∈ ˜ ( ψ 1 , A ) if ϑ ∈ A ⇒ ψ (ϑ) ⊆ ψ₁ (ϑ).

Example 2.19. Let X = {c, m, t} and A = {ϑ₁, ϑ₂}, then ϑ ψ ( 1 ) = { ( ϑ 1 , { ( c , 0.52 , 0.31 ) , ( m , 0.47 , 0.19 ) } ) } , and ϑ ψ ( 2 ) = { ( ϑ 2 , { ( c , 0.76 , 0.35 ) , ( t , 0.81 , 0.13 ) } ) } are two distinct PFS-points contained in the PFSS ψ _A  = {(ϑ₁, {(c, 0.52, 0.31) , (m, 0.47, 0.19)}) ,                (ϑ₂, {(c, 0.76, 0.35) , (t, 0.81, 0.13)})} Observe that ψ A = ϑ ψ ( 1 ) ∪ ˜ ϑ ψ ( 2 ) i.e. a PFSS is union of its Pythagorean fuzzy soft points.

It is important to note that any PFS-point ϑ ψ ( 1 ) (say) is also known as a singleton subset of the PFSS ψ _A . It is so because it contains one parameter and its ψ-approximate element.

In this section, the notion of PFS-topology is defined on a PFSS. The concepts of PFS-extended union and PFS-restricted intersection are used in the construction of PFS-topology. Certain properties of PFS-topology are defined and related illustrations are well presented.

Definition 3.1. Let PFS (X, E) be the collection of all PFS-subsets of the absolute PFSS X ˘ E . For A, B ⊆ E, a subcollection τ ˜ of PFS (X, E) is called Pythagorean fuzzy soft topology (PFS-topology) on X ˘ E if following properties hold:

Φ E , X ˘ E ∈ ˜ τ ˜ ,

Definition 3.2. Assume that ( X ˜ E , τ ˜ X ) is a PFS-topology. For some Y ⊆ X bestowed with E as set of parameters, τ ˜ Y is a PFS-topology on Y whose PFS-open sets are ψ B ( 2 ) = ψ A ( 1 ) ∩ ˜ Y ˘ E , where ψ A ( 1 ) are PFS-open sets of τ ˜ X , ψ B ( 2 ) are PFS-open sets of τ ˜ Y and Y ˘ E is absolute PFSS on Y. Then τ ˜ Y is taken as the PFS-subspace (or PFS sub-topology) of τ ˜ X . It can be written as τ ˜ Y = { ψ B ( 2 ) : ψ B ( 2 ) = ψ A ( 1 ) ∩ ˜ Y ˘ E , ψ A ( 1 ) ∈ τ ˜ X } .

Example 3.3. Let X = {ρ₁, ρ₂, ρ₃} be the universe and E = {e _i  : i = 1, 2, ⋯ , 4} be the aggregate of decision variates. Take two subcollections A = {e₁, e₂} and B = {e₁, e₂, e₃} of E. Assume that ψ A ( 1 ) = { ( e 1 , { ρ 1 ( 0.34 , 0.79 ) } , { ρ 2 ( 0.37 , 0.89 ) } ) , ( e 2 , { ρ 1 ( 0.35 , 0.23 ) } , { ρ 2 ( 0.37 , 0.23 ) } ) } , and ψ B ( 2 ) = { ( e 1 , { ρ 1 ( 0.35 , 0.23 ) } , { ρ 2 ( 0.71 , 0.52 ) } ) , ( e 2 , { ρ 2 ( 0.35 , 0.21 ) } , { ρ 3 ( 0.38 , 0.21 ) } ) , ( e 3 , { ρ 1 ( 0.33 , 0.25 ) } , { ρ 2 ( 0.32 , 0.26 ) } ) } . Then, τ ˜ X = { Φ E , X ˘ E , ψ A ( 1 ) , ψ B ( 2 ) } is a PFS-topology on X.

Now, absolute PFSS on Y = {ρ₂, ρ₃} ⊆ X is Y ˘ E = { ( e i , { ρ 2 ( 1 , 0 ) } , { ρ 3 ( 1 , 0 ) } ) : 1 ≤ i ≤ 4 } Since Y ˘ E ∩ ˜ Φ E = Φ E , Y ˘ E ∩ ˜ ψ A ( 1 ) = { ( e 1 , { ρ 2 ( 0.37 , 0.89 ) } ) , ( e 2 , { ρ 2 ( 0.35 , 0.23 ) } , { ρ 3 ( 0.37 , 0.23 ) } ) } = Ψ A ( 1 ) , Y ˘ E ∩ ˜ ψ B ( 2 ) = { ( e 1 , { ρ 2 ( 0.71 , 0.52 ) } , ( e 2 , { ρ 2 ( 0.35 , 0.21 ) } , { ρ 3 ( 0.38 , 0.21 ) } ) , ( e 3 , { ρ 2 ( 0.32 , 0.26 ) } ) } = Ψ B ( 2 ) , Y ˘ E ∩ ˜ X ˘ E = Y ˘ E , so τ ˜ Y = { Φ E , Ψ A ( 1 ) , Ψ B ( 2 ) , Y ˘ E } is a PFS sub-topology of τ ˜ X .

Definition 3.4. Let ( X ˜ E , τ ˜ ) be a PFS-topological space and ψ A ⊆ ˜ X ˜ E .

The interior ψ A ∘ of ψ _A is PFS-union of all PFS-open subsets of ψ _A . It is worth noticing that ψ A ∘ is the largest PFS-open subset of ψ _A .

We elaborate these notions with the assistance of following example.

Example 3.5. Take X = {ρ₁, ρ₂} as the universal set with E = {e₁, e₂} as the assembly of attributes. If ψ E ( 1 ) = { ( e 1 , { ρ 1 ( 0.39 , 0.41 ) } , { ρ 2 ( 0.71 , 0.32 ) } ) , ( e 2 , { ρ 1 ( 0.37 , 0.25 ) } , { ρ 2 ( 0.77 , 0.11 ) } ) } ; ψ E ( 2 ) = { ( e 1 , { ρ 1 ( 0.87 , 0.28 ) } , { ρ 2 ( 0.75 , 0.17 ) } ) , ( e 2 , { ρ 1 ( 0.86 , 0.25 ) } , { ρ 2 ( 0.78 , 0.08 ) } ) } ; ψ E ( 3 ) = { ( e 1 , { ρ 1 ( 0.12 , 0.49 ) } , { ρ 2 ( 0.67 , 0.39 ) } ) , ( e 2 , { ρ 1 ( 0.13 , 0.47 ) } , { ρ 2 ( 0.68 , 0.29 ) } ) } then the collection τ ˜ = { Φ E , X ˘ E , ψ E ( 1 ) , ψ E ( 2 ) , ψ E ( 3 ) } of PFS subsets of X ˜ E is a PFS-topology. The members of τ ˜ are obviously PFS open sets. The corresponding closed PFSSs are ( Φ E ) c = X ˘ E ; ( X ˘ E ) c = Φ E ; ( ψ E ( 1 ) ) c = { ( e 1 , { ρ 1 ( 0.41 , 0.39 ) } , { ρ 2 ( 0.32 , 0.71 ) } ) , ( e 2 , { ρ 1 ( 0.25 , 0.37 ) } , { ρ 2 ( 0.11 , 0.77 ) } ) } ; ( ψ E ( 2 ) ) c = { ( e 1 , { ρ 1 ( 0.28 , 0.87 ) } , { ρ 2 ( 0.17 , 0.75 ) } ) , ( e 2 , { ρ 1 ( 0.25 , 0.86 ) } , { ρ 2 ( 0.08 , 0.78 ) } ) } ; ( ψ E ( 3 ) ) c = { ( e 1 , { ρ 1 ( 0.49 , 0.12 ) } , { ρ 2 ( 0.39 , 0.67 ) } ) , ( e 2 , { ρ 1 ( 0.47 , 0.13 ) } , { ρ 2 ( 0.29 , 0.68 ) } ) } . Consider the PFSS ψ E = { ( e 1 , { ρ 1 ( 0.58 , 0.32 ) } , { ρ 2 ( 0.73 , 0.29 ) } ) , ( e 2 , { ρ 1 ( 0.71 , 0.25 ) } , { ρ 2 ( 0.78 , 0.09 ) } ) } so that ( ψ E ) c = { ( e 1 , { ρ 1 ( 0.32 , 0.58 ) } , { ρ 2 ( 0.29 , 0.73 ) } ) , ( e 2 , { ρ 1 ( 0.25 , 0.71 ) } , { ρ 2 ( 0.09 , 0.78 ) } ) } . The PFS-interior of ψ _E is ψ E ∘ = Φ E ∪ ˜ ψ E ( 1 ) ∪ ˜ ψ E ( 3 ) = { ( e 1 , { ρ 1 ( 0.39 , 0.41 ) } , { ρ 2 ( 0.71 , 0.32 ) } ) , ( e 2 , { ρ 1 ( 0.37 , 0.25 ) } , { ρ 2 ( 0.77 , 0.11 ) } ) } .

The PFS-closure of ψ _E is ψ E ¯ = X ˘ E Now, ψ E c ¯ = X ˘ E ∩ ˜ ( ψ E ( 1 ) ) c ∩ ˜ ( ψ E ( 3 ) ) c = { ( e 1 , { ρ 1 ( 0.41 , 0.39 ) } , { ρ 2 ( 0.32 , 0.71 ) } ) , ( e 2 , { ρ 1 ( 0.25 , 0.37 ) } , { ρ 2 ( 0.11 , 0.77 ) } ) } . Hence, PFS-frontier of ψ _E is Fr ( ψ E ) = ψ E ¯ ∩ ˜ ψ E c ¯ = { ( e 1 , { ρ 1 ( 0.41 , 0.39 ) } , { ρ 2 ( 0.32 , 0.71 ) } ) , ( e 2 , { ρ 1 ( 0.25 , 0.37 ) } , { ρ 2 ( 0.11 , 0.77 ) } ) } . Finally, the PFS-exterior of ψ _E is Ext ( ψ E ) = ( ψ E c ) ∘ = Φ E .

Theorem 3.6. Let ( X ˜ E , τ ˜ ) be a PFS-topological space and ψ A ⊆ ˜ X ˜ E , then

( ψ A ∘ ) c = ( ψ A c ) ¯ , and

Proof. Straight forward.□

Theorem 3.7. Let ( X ˜ E , τ ˜ ) be a PFS-topological space and ψ A ⊆ ˜ X ˜ E , then Fr ( ψ A ) = Fr ( ψ A c ) .

Proof. In accordance with definition, Fr ( ψ A ) = ( ψ A ) ¯ ∩ ˜ ( ψ A c ) ¯ = ( ψ A c ) ¯ ∩ ˜ ( ψ A ) ¯ = ( ψ A c ) ¯ ∩ ˜ [ ( ψ A c ) ] c ¯ = Fr ( ψ A c )

□ Definition 3.8. Believe that X is a crisp set having E as the aggregate of attributes, then τ ˜ discrete = PFS (X, E) and τ ˜ indiscrete = { Φ E , X ˘ E } are termed as, in order, discrete and indiscrete PFS-topologies. Notice that τ ˜ discrete is the largest whereas τ ˜ indiscrete is the smallest PFS-topology on X ˜ E .

Remark. The intersection of two PFS-topological spaces is always a PFS-topological space but their union need not to be so. Next example advocates this observation.

Example 3.9. Let X = {ρ₁, ρ₂} be the universe and E = {e _i  : i = 1, 2, 3, 4} be the collection of attributes. Let A = {e₁, e₂} , B = {e₃, e₄} ⊆ E with corresponding PFSSs ψ A ( 1 ) = { ( e 1 , { ρ 1 ( 0.39 , 0.47 ) } , { ρ 2 ( 0.79 , 0.37 ) } ) , ( e 2 , { ρ 1 ( 0.99 , 0.17 ) } , { ρ 2 ( 0.79 , 0.17 ) } ) } , and ψ B ( 2 ) = { ( e 3 , { ρ 1 ( 0.89 , 0.27 ) } , { ρ 2 ( 0.79 , 0.17 ) } ) , ( e 4 , { ρ 1 ( 0.89 , 0.27 ) } , { ρ 2 ( 0.79 , 0.17 ) } ) } . Then, τ ˜ 1 = { Φ E , ψ A ( 1 ) , X ˘ E } and τ ˜ 2 = { Φ E , ψ B ( 2 ) , X ˘ E } are PFS-topologies on X but τ ˜ 1 ∪ ˜ τ ˜ 2 = { Φ E , X ˘ E , ψ A ( 1 ) , ψ B ( 2 ) } is not so.

Definition 3.10. Let τ ˜ 1 and τ ˜ 2 be PFS-topological spaces. If τ ˜ 1 ⊆ ˜ τ ˜ 2 then τ ˜ 1 is acknowledged as coarser or weaker than τ ˜ 2 and τ ˜ 2 is referred to as finer or stronger than τ ˜ 1 . In either case τ ˜ 1 and τ ˜ 2 are called comparable.

Example 3.11. Let X = {ρ₁, ρ₂} be the universe and E = {e₁, e₂, e₃} be the set of decision variables with A = {e₁, e₂} ⊂ E. Consider PFSSs ψ A ( 1 ) = { ( e 1 , { ρ 1 ( 0.31 , 0.47 ) } , { ρ 2 ( 0.71 , 0.37 ) } ) , ( e 2 , { ρ 1 ( 0.31 , 0.27 ) } , { ρ 2 ( 0.71 , 0.17 ) } ) } ; ψ A ( 2 ) = { ( e 1 , { ρ 1 ( 0.81 , 0.27 ) } , { ρ 2 ( 0.71 , 0.17 ) } ) , ( e 2 , { ρ 1 ( 0.81 , 0.27 ) } , { ρ 2 ( 0.71 , 0.17 ) } ) } ; and ψ A ( 3 ) = { ( e 1 , { ρ 1 ( 0 , 0.47 ) } , { ρ 2 ( 0.61 , 0.37 ) } ) , ( e 2 , { ρ 1 ( 0 , 0.47 ) } , { ρ 2 ( 0.61 , 0.27 ) } ) } . It can be seen that τ ˜ 1 = { Φ E , ψ A ( 1 ) , X ˘ E } and τ ˜ 2 = { φ E , ψ A ( 1 ) , ψ A ( 2 ) , ψ A ( 3 ) , X ˘ E } are two PFS-topologies. Since τ ˜ 1 ⊂ ˜ τ ˜ 2 so τ ˜ 1 is coarser than τ ˜ 2 .

Remark. The well-known laws "law of excluded middle" and "law of contradiction" of crisp set theory do not make sense for IFSs. This characteristic is inherited in PFSS theory due to which some results in PFS-topology deviate from classical topology. The forthcoming illustration advocates our view-point.

Example 3.12. Let X = {ρ₁, ρ₂, ρ₃} be the universe and E = {e _i  : i = 1, 2, 3, 4} be the family of attributes with A = {e₁, e₂} ⊆ E. Assume that ψ A = { ( e 1 , { ρ 1 ( 0.31 , 0.47 ) } , { ρ 2 ( 0.71 , 0.37 ) } ) , ( e 2 , { ρ 1 ( 0.31 , 0.27 ) } , { ρ 2 ( 0.71 , 0.17 ) } ) } . Clearly τ ˜ = { Φ E , X ˘ E , ψ A , ψ A c } fails to be a PFS-topology on X for neither ψ A ∩ ˜ ψ A c ∈ ˜ τ ˜ nor ψ A ∪ ˜ ψ A c ∈ ˜ τ ˜ .

Definition 3.13. Let ( X ˜ E , τ ˜ ) be a PFS-topological space. Then B ⊆ ˜ τ ˜ is acknowledged as a PFS-basis for τ ˜ if for each ψ A ∈ ˜ τ ˜ , there exists B ∈ B such that ψ A = ∪ ˜ B .

Example 3.14. Let X = {ρ₁, ρ₂} be the universe and E = {e₁, e₂} be attributes’ set with A ⊆ E. Think through PFSSs ψ A ( 1 ) = { ( e 1 , { ρ 1 ( 1 , 0 ) } , { ρ 2 ( 0.27 , 0.78 ) } ) , ( e 2 , { ρ 1 ( 0.37 , 0.26 ) } , { ρ 2 ( 1 , 0 ) } ) } ; ψ A ( 2 ) = { ( e 1 , { ρ 1 ( 0.39 , 0.84 ) } , { ρ 2 ( 1 , 0 ) } ) , ( e 2 , { ρ 1 ( 0.74 , 0.38 ) } , { ρ 2 ( 0.89 , 0.12 ) } ) } ; and ψ A ( 3 ) = { ( e 1 , { ρ 1 ( 0.33 , 0.28 ) } , { ρ 2 ( 0.75 , 0.29 ) } ) , ( e 2 , { ρ 1 ( 1 , 0 ) } , { ρ 2 ( 0.24 , 0.91 ) } ) } . The collection B = { ψ A ( 1 ) , ψ A ( 2 ) , ψ A ( 3 ) } serves as a PFS-basis for the PFS-topology τ ˜ = { Φ E , X ˘ E , ψ A ( 1 ) , ψ A ( 2 ) , ψ A ( 3 ) } .

Theorem 3.15. If B is a PFS-basis for τ ˜ then, for each e ∈ E, B e = { ψ ( e ) : ψ E ∈ B } serves as a PFS-basis for the PFS-topology τ ˜ ( e ) = { ψ ( e ) : ψ E ∈ τ ˜ } .

Proof. Let ψ⁽¹⁾ (e) ∈ τ (e) for some ψ E ( 1 ) ∈ τ . Since B is a PFS-basis for τ ˜ so, in accordance with definition, there exists B ′ ∈ B such that ψ E ( 1 ) = ∪ B ′ and hence ψ ( 1 ) ( e ) = ∪ B e ′ , where B e ′ = { ψ ( e ) : ψ E ∈ B ′ } ⊆ B e .□

Theorem 3.16. Let ( X ˜ E , τ ˜ ) be a PFS-topological space. A collection B = { ψ B α | α ∈ Ω } ⊆ ˜ τ ˜ is a PFS-basis for τ ˜ if and only if for any PFSS ψ _U and any PFS-element ϑ _ψ  ∈ ψ _U , there is a ψ B α ∈ B such that ϑ ψ ∈ ψ B α ⊆ ˜ ψ U .

Proof. Suppose that B is a PFS-basis for τ ˜ . Then for any PFS open set ψ _U of τ ˜ , there are PFSSs ψ _{B γ} , γ ∈ Ω ′ ⊆ ˜ Ω such that ψ U = ∪ ˜ ψ B γ . Thus, for any P e x ∈ ψ U , there is a B _γ such that ϑ ψ ∈ ψ B γ ⊆ ˜ ψ U Conversely, suppose that for every PFS-open set ψ _U and any ϑ _ψ  ∈ ψ _U , there is a ψ _B in B such that ϑ ψ ∈ ψ B ⊆ ˜ ψ U . Then, ψ U = ∪ ˜ ϑ ψ ∈ ψ U { x } ⊆ ˜ ∪ ˜ ϑ ψ ∈ ψ U { ψ B } ⊆ ˜ ψ U so that ψ U = ∪ ˜ ϑ ψ ∈ ψ U { ψ B } which is union of PFSSs in B .□

4 PFS-separation axioms

In this section, the concept of PFS-soft points, PFS-open sets and PFS-closed sets are used to define PFS-separation axioms.

Definition 4.1. A PFS-topological space ( X ˜ E , τ ˜ ) is called as a PFS T₀-space if for every pair of distinct PFS-points ϑ ψ ( 1 ) , ϑ ψ ( 2 ) there exists at least one PFS-open set ψ _A containing exactly one of the PFS-points.

Example 4.2. Every discrete PFS-topological space PFS (X, E) is a PFS T₀-space for there exists an open set { ϑ ψ ( 1 ) } that clearly contains ϑ ψ ( 1 ) but not ϑ ψ ( 2 ) .

Remark. The property of being a PFS T₀-space of any PFS-topological space ( X ˜ E , τ ˜ ) is hereditary. i.e. every subspace of a PFS T₀-space is PFS T₀-space.

Definition 4.3. A PFS-space is PFS T₁-space if for any two distinct PFS-points ϑ ψ ( 1 ) , ϑ ψ ( 2 ) of X ˜ E , there exist two PFS-open sets ψ _C and ψ _D such that ϑ ψ ( 1 ) ∈ ˜ ψ C , ϑ ψ ( 2 ) ∉ ˜ ψ C and ϑ ψ ( 1 ) ∉ ˜ ψ D , ϑ ψ ( 2 ) ∈ ˜ ψ D .

Example 4.4. Every discrete PFS-topological space PFS (X, E) is a PFS T₁-space. If ϑ ψ ( 1 ) and ϑ ψ ( 2 ) are two distinct points in X ˜ E , then there are PFS open sets { ϑ ψ ( 1 ) } and { ϑ ψ ( 2 ) } such that ϑ ψ ( 1 ) ∈ { ϑ ψ ( 1 ) } whereas ϑ ψ ( 2 ) ∉ { ϑ ψ ( 1 ) } .

Theorem 4.5. The following statements about a PFS-topological space ( X ˜ E , τ ˜ ) are equivalent:

( X ˜ E , τ ˜ ) is a PFS T₁-space.

Proof. Straight forward.□

Remark. The property of being a PFS T₁-space of any PFS-topological space ( X ˜ E , τ ˜ ) is hereditary. i.e. every subspace of a PFS T₁-space is PFS T₁-space.