On pythagorean fuzzy soft topological spaces

Abstract

Imprecision in the decision-making process is an essential consideration. In order to navigate the imprecise decision-making framework, measuring tools and methods have been developed. Pythagorean fuzzy soft sets are one of the new methods for dealing with imprecision. Pythagorean fuzzy soft topological spaces is an extension of intuitionistic fuzzy soft topological spaces. These sets generalizes intuitionistic fuzzy sets for a broader variety of implementations. This work is a gateway to study such a problem. The concept of Pythagorean fuzzy soft topological spaces(PyFSTS), interior, closure, boundary, neighborhood of Pythagorean fuzzy soft spaces PyFSS, base and subspace of PyFSTSs are presented and its properties are figured out. We established an algorithm under uncertainty based on PyFSTS for multi-attribute decision-making (MADM) and to validate this algorithm, a numerical example is solved for suitable brand selection. Finally, the benefits, validity, versatility and comparison of our proposed algorithms with current techniques are discussed.The advantage of the proposed work is to detect vagueness with more sizably voluminous valuation space than intuitionistic fuzzy sets.

Keywords

Pythagorean fuzzy soft sets Pythagorean fuzzy soft topology Pythagorean fuzzy soft interior and soft closure multi-attribute decision making

1 Introduction

Economics, medical science researchers and many other areas struggle with the unclear, imprecise and sometimes inadequate knowledge of ambiguous data modeling on a daily basis. To overcome this difficulties, Fuzzy Set (FS) theory was developed by Zadeh [37] to deal the ambiguity. There are proposals for non-classical and higher order fuzzy sets after the proposal of fuzzy set theory for various specialized purposes. Intuitionistic fuzzy set (IFS) is one of the generalization of FS theory, which deals with both membership and non-membership values and it is introduced by Atanassov [4]. Another extension of FS and IFS is Pythagorean fuzzy set (PyFS) and it is developed by Yager [36] in 2013. This set theory established with a new class of non-standard fuzzy subsets and which has got several social and natural science applications [1 , 37] Pythagorean fuzzy topological (PyFTS) spaces was proposed and investigated by Olgun et al. [17] in 2019.

Molodtsov [15] introduced the soft set theoretical concept as a fully generic mathematical method free of the parametrization insufficiency ailment of various theories concerning uncertainty concepts. In fact, this principle is very accurate, convenient, comfortable and easy, applicable in the real time problem solving technique. For the case of soft set theory applications, Molodtsov [15] successfully applied it in different fields, like game theory, operation analysis, probability theory, Riemann integration, smoothness of functions, etc. Soft topological spaces were proposed by Shabir and Naz [29] On these general topological spaces and separation axioms, some notions of soft sets were defined and studied. Maji [13, 14] introduced fuzzy & intuitionistic fuzzy soft set by joining two concepts namely, soft set with IFS. Neutrosophic set [30 –32], neutrosophic topology [19 , 28], neutrosophic soft set [24], complex intuitionistic fuzzy sets (CIFS)[2] and complex neutrosophic set(CNS) [10] are the further extensions of IFS and FS theory.

In the entire manuscript, step by step, the motivation and objectives of this expanded and hybrid work are given. We Create that other fuzzy-set hybrid systems become special Pythagorean fuzzy soft topological space (PyFSTS) with some specific conditions. The flexibility, robustness, simplicity and supremacy of our proposed algorithm and model are discussed. This model is the most common method of use that is applicable in artificial intelligence, agriculture, engineering, medicine, and other everyday living situations to gather data on a broad scale. This study will easily be done with other methods and multiple forms of integrated systems in the future. Pythagorean fuzzy topological space [16], complex Pythagorean fuzzy topological space [3] and soft set [11] motivated us to propose this new concept. MADM played a prominent role in decision making. This methodology helps to choose potential alternative. So we proposed an algorithm which generates Pythagorean fuzzy soft topology and with the aid of score function best alternative has been selected.

The manuscript is arranged in this fashion: In section 2, important preliminary definitions are presented. Definition of PyFSTS, neighborhood, closure and interior of Pythagorean fuzzy soft set(in short PyFSSs) and their properties are discussed in section 3. In section 4, introduces an algorithm with a numerical example and in addition, the advantages, limitations, simplicity, versatility and validation of the proposed algorithm with the existing theories are explained clearly. Section 5, concludes the paper.

2 Some Basic Notions

Definition 2.1. [35] The set $A = {(x, μ_{A} (x)) : x \in X}$ is said to be a fuzzy set on $X$ , where $X$ be a non void set and $μ_{A} (x) \in [0, 1]$ is the membership degree function.

Definition 2.2. [34] The set $A = {(x, μ_{A} (x), ν_{A} (x)) : x \in X}$ is called an intuitionistic fuzzy set on $X$ , where $X$ be a non void set and $0 \leq μ_{A} (x) + ν_{A} (x)) \leq 1$ for all $x \in X$ and $μ_{A} (x), ν_{A} (x) \in [0, 1]$ are the membership and non membership degree functions, respectively.

Definition 2.3. [34] A PyFS $S$ on $U$ is defined as: $S = {(ζ, 〈 α (ζ), β (ζ) 〉) | ζ \in U}$

where $α : U \to [0, 1]$ is called the positive membership function (PMF) and $β : U \to [0, 1]$ is called the negative membership function (NMF). The condition for a PyFS is that 0 ≤ α² + β² ≤ 1 and the refusal degree function is given as γ² (ζ) =1 - (α² (ζ) + β² (ζ)). A Pythagorean fuzzy number (PyFN) is defied as the doublet (α, β).

Definition 2.4. [34] Two objects $S_{1} = {(ζ, 〈 α_{S_{1}} (ζ), β_{S_{1}} (ζ) 〉) : ζ \in U}$ and $S_{2} = {(ζ, 〈 α_{S_{2}} (ζ), β_{S_{2}} (ζ) 〉) : ζ \in U}$ are two PyFS described on $U$ , the universe of discourse, then

The intersection of $S_{1}$ and $S_{2}$ is $S_{1} \cap S_{2} = {(ζ, 〈 α_{S_{1}} (ζ) \land α_{S_{2}} (ζ), β_{S_{1}} (ζ) \lor β_{S_{2}} (ζ) 〉) : ζ \in U},$

The union of $S_{1}$ and $S_{2}$ is $S_{1} \cup S_{2} = {(ζ, 〈 α_{S_{1}} (ζ) \lor α_{S_{2}} (ζ), β_{S_{1}} (ζ) \land β_{S_{2}} (ζ) 〉) : ζ \in U},$

$S_{1} \subseteq S_{2}$ iff $α_{S_{1}} (ζ) \leq α_{S_{2}} (ζ), β_{S_{1}} (ζ) \geq β_{S_{2}} (ζ), \forall ζ \in U .$

Definition 2.5. [34] The PyFSS is not a set, but it is a specified unit of some elements of the set $PyF (\hat{U})$ and hence it can be put down as a set of ordered pairs: $(\hat{F}, \hat{E}) = {(ξ, 〈 ζ, α_{\hat{F} (ξ)} (ζ), β_{\hat{F} (ξ)} (ζ) 〉 | ζ \in U) : ξ \in \hat{E}}$ , where $α_{\hat{F} (ξ)} (ζ)$ and $β_{\hat{F} (ξ)} (ζ)$ are the PMFs and NMFs, respectively. If $ζ \in \hat{U}$ , $0 \leq α_{\hat{F} (ξ)}^{2} (ζ) + β_{\hat{F} (ξ)}^{2} (ζ) \leq 1$ .

Definition 2.6. [16] If τ is a family of Pythagorean fuzzy subsets of a non empty set $X$ and if

$1_{X}, 0_{X} \in τ$ ,

for any $A_{1}, A_{2} \in τ$ , we have $A_{1} \cap A_{2} \in τ$

for any $A_{i} \in τ$ , we have $\cup_{i} A_{i} \in τ$ where $i \in Δ$

then τ is said to be a Pythagorean fuzzy topological space(PyFTS) on

X

and the pair

(X, τ)

is said to be a Pythagorean fuzzy topological space.

3 Pythagorean Fuzzy Soft Topological Spaces

The notion of PyFSTS are introduced and further we investigated its properties.

Let $\hat{U}$ and $PyF (\hat{U}, \hat{E})$ denotes, origination of the universal set and the family of PyFSs on $\hat{U}$ , respectively.

Definition 3.1. A PyFSS $(\hat{F}, \hat{E})$ over $\hat{U}$ is called a void PyFSS (in short, $\hat{0}$ ), iff ∀ $ξ \in \hat{E}$ , $(\hat{F}, \hat{E}) (ξ) = (\tilde{0}, \tilde{1})$ , where $\tilde{0}, \tilde{1}$ are the value of the PMF and the value of the NMFs, respectively of the null and absolute Pythagorean fuzzy sets over $\hat{U}$ .

Definition 3.2. A PyFSS $(\hat{F}, \hat{E})$ over $\hat{U}$ is called an absolute PyFSS (in short, $\hat{1}$ ), iff ∀ $ξ \in \hat{E}$ , $(\hat{F}, \hat{E}) (ξ) = (\tilde{1}, \tilde{0})$ , where $\tilde{1}, \tilde{0}$ are MF, and negative MFs respectively of the absolute and null fuzzy sets over $\hat{U}$ .

Definition 3.3. Let $\hat{T} \subset PyF (\hat{U}, \hat{E})$ , then $\hat{T}$ is said to be a PyFSTS on $\hat{U}$ if

$\hat{0}$ and $\hat{1}$ are member of $\hat{T}$ .

the intersection of any two PyFSS in $\hat{T}$ pertains to $\hat{T}$ .

the union of any number of PyFSS in $\hat{T}$ pertains to $\hat{T}$ ,

The triple

(\hat{U}, \hat{T}, \hat{E})

is said to be a PyFSTS over

\hat{U}

Every member of $\hat{T}$ is said to be a $\hat{T}$ -open PyFSS.

The complement of a $\hat{T}$ -open is said to be a $\hat{T}$ -closed PyFSS.

{\hat{T}}_{1} \subset {\hat{T}}_{2}

, then

{\hat{T}}_{1}

is called Pythagorean fuzzy soft strictly coarser (weaker) than

{\hat{T}}_{2}

(or)

{\hat{T}}_{2}

is Pythagorean fuzzy soft strict finer than

{\hat{T}}_{1}

, where

{\hat{T}}_{1}

and

{\hat{T}}_{2}

are two topologies on

(\hat{U}, \hat{E})

. Besides

PyF (\hat{U}, \hat{E})

is a PyFSTS on

(\hat{U}, \hat{E})

. A number of topologies on

(\hat{U}, \hat{E})

may be there.

Example 3.1. Let $\hat{U} = {ζ_{1}, ζ_{2}, ζ_{3}}$ be the reference set (candidates who have been nominated for the promotion) and $\hat{E} = {ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}}$ be the parameters or attributes set, where ξ₁=intelligent, ξ₂=experience, ξ₃=attitude, ξ₄=competence. Let $\hat{A} = {ξ_{1}, ξ_{2}} \subset \hat{E}$ and $\hat{B} = {ξ_{2}} \subset \hat{E}$ . Then we consider two PyFSS $(\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ are given by: $(\hat{F}, \hat{A}) = {(ξ_{1}, \hat{F} (ξ_{1})), (ξ_{2}, \hat{F} (ξ_{2}))}$ , and $(\hat{G}, \hat{B}) = {(ξ_{2}, \hat{G} (ξ_{2}))},$ where $\hat{F} (ξ_{1}) = {ζ_{1} = 〈 0.6, 0.5 〉, ζ_{2} = 〈 0.7, 0.4 〉, ζ_{3} = 〈 0.7, 0.1 〉}$ $\hat{F} (ξ_{2}) = {ζ_{1} = 〈 0.2, 0.3 〉, ζ_{2} = 〈 0.5, 0.6 〉, ζ_{3} = 〈 0.7, 0.2 〉}$ $\hat{G} (ξ_{2}) = {ζ_{1} = 〈 0.4, 0.8 〉, ζ_{2} = 〈 0.3, 0.8 〉, ζ_{3} = 〈 0.3, 0.4 〉}$

Here,

$\hat{T} = {\hat{1}, \hat{0}, (\hat{F}, \hat{A}), (\hat{G}, \hat{B})}$ is a PyFST over $\hat{U}$ .

${\hat{T}}_{1} = {\hat{1}, \hat{0}, (\hat{F}, \hat{A})}$ and ${\hat{T}}_{2} = {\hat{0}, \hat{1}, (\hat{F}, \hat{A}), (\hat{G}, \hat{B})}$ are two PyFSTs over $\hat{U}$ . It is straightforward that ${\hat{T}}_{1} \subseteq {\hat{T}}_{2}$ . Thus, ${\hat{T}}_{1}$ is called PyFS-coarser or weaker and ${\hat{T}}_{2}$ is called PyFS-finer or stronger than ${\hat{T}}_{1}$ .

Theorem 3.1. If $\hat{T_{2}} \cap \hat{T_{2}} = {\hat{D} \in PyFSSs (\hat{U}, \hat{E}) : \hat{D} \in \hat{T_{2}} \cap \hat{T_{2}}}$ , where $(\hat{U}, \hat{T_{1}}, \hat{E})$ and $(\hat{U}, \hat{T_{2}}, \hat{E})$ are two PyFSTSs over $(\hat{U}, \hat{E})$ , then $\hat{T_{2}} \cap \hat{T_{2}}$ is also a PyFSTS on $(\hat{U}, \hat{E})$ .

Proof. (i) Obviously, $\hat{1}, \hat{0} \in \hat{T_{1}}, \hat{T_{2}}$ (ii) Let $\hat{D_{1}}, \hat{D_{2}} \in \hat{T_{1}} \cap \hat{T_{2}}$ . This implies that $\hat{D_{1}}, \hat{D_{2}} \in \hat{T_{1}}$ and $\hat{D_{1}}, \hat{D_{2}} \in \hat{T_{2}}$ , ⇒ $\hat{D_{1}} \cap \hat{D_{2}} \in \hat{T_{1}}$ and $\hat{D_{1}} \cap \hat{D_{2}} \in \hat{T_{2}}$ , ⇒ $\hat{D_{1}} \cap \hat{D_{2}} \in \hat{T_{1}} \cap \hat{T_{2}}$ .

(iii) Let ${\hat{D_{i}} : i \in Γ} \in \hat{T_{1}} \cap \hat{T_{2}}$ . This implies that ${\hat{D_{i}}} \in \hat{T_{1}}$ and ${\hat{D_{i}}} \in \hat{T_{2}}$ , ⇒ $\cup_{i} \hat{D_{i}} \in \hat{T_{1}}$ and $\cup_{i} \hat{D_{i}} \in \hat{T_{2}}$ , ⇒ $\cup_{i} \hat{D_{i}} \in \hat{T_{1}} \cap \hat{T_{2}}$ .

Hence it is proved that $\hat{T_{1}} \cap \hat{T_{2}}$ is a PyFSTS on $(\hat{U}, \hat{E})$ .

Remark 3.1. The union of two PyFSTSs may not be so.

Let $\hat{U} = {ζ_{1}, ζ_{2}, ζ_{3}}$ be the reference set and $\hat{E} = {ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}}$ be the set of parameters or attributes. Let $\hat{A} = {ξ_{1}, ξ_{2}, ξ_{3}} \subset \hat{E}$ and $\hat{B} = {ξ_{3}, ξ_{4}, ξ_{5}} \subset \hat{E}$ . Then we consider two PyFSSs $(\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ are given by: $(\hat{F}, \hat{A}) = {(ξ_{1}, \hat{F} (ξ_{1})), (ξ_{2}, \hat{F} (ξ_{2})), (ξ_{3}, \hat{F} (ξ_{3}))}$ , and $(\hat{G}, \hat{B}) = {(ξ_{3}, \hat{G} (ξ_{3})), (ξ_{4}, \hat{F} (ξ_{4})), (ξ_{5}, \hat{F} (ξ_{5}))},$ where $\hat{F} (ξ_{1}) = {ζ_{1} = 〈 0.2, 0.5 〉, ζ_{2} = 〈 0.5, 0.1 〉, ζ_{3} = 〈 0.3, 0.2 〉}$ $\hat{F} (ξ_{2}) = {ζ_{1} = 〈 0.5, 0.1 〉, ζ_{2} = 〈 0.2, 0.3 〉, ζ_{3} = 〈 0.6, 0.5 〉}$ $\hat{F} (ξ_{3}) = {ζ_{1} = 〈 0.4, 0.2 〉, ζ_{2} = 〈 0.3, 0.3 〉, ζ_{3} = 〈 0.6, 0.1 〉}$ , $\hat{G} (ξ_{3}) = {ζ_{1} = 〈 0.3, 0.5 〉, ζ_{2} = 〈 0.6, 0.5 〉, ζ_{3} = 〈 0.3, 0.2 〉}$ $\hat{G} (ξ_{4}) = {ζ_{1} = 〈 0.4, 0.4 〉, ζ_{2} = 〈 0.6, 0.2 〉, ζ_{3} = 〈 0.3, 0.2 〉}$ $\hat{G} (ξ_{5}) = {ζ_{1} = 〈 0.7, 0.2 〉, ζ_{2} = 〈 0.5, 0.3 〉, ζ_{3} = 〈 0.4, 0.5 〉}$ .

Then ${\hat{T}}_{1} = {\hat{1}, \hat{0}, (\hat{F}, \hat{A})}$ and ${\hat{T}}_{2} = {\hat{1}, \hat{0}, (\hat{G}, \hat{B})}$ are two PyFSTSs over $\hat{U}$ . On the other hand, since $(\hat{F}, \hat{A}), (\hat{G}, \hat{B}) \in {\hat{T}}_{1} \cup {\hat{T}}_{2}$ . But $(\hat{F}, \hat{A}) \cup (\hat{G}, \hat{B}), (\hat{F}, \hat{A}) \cap (\hat{G}, \hat{B}) \notin {\hat{T}}_{1} \cup {\hat{T}}_{2}$ . Thus, ${\hat{T}}_{1} \cup {\hat{T}}_{2}$ is not a PyFSTS on $\hat{U}$ . But ${\hat{T}}_{1} \cap {\hat{T}}_{2}$ is a PyFSTS on $\hat{U}$ .

Definition 3.4. Let $\hat{T}$ be a PyFSTS on $(\hat{U}, \hat{E})$ and ${\hat{D}}_{1}, {\hat{D}}_{2} \in PyFSS (\hat{U}, \hat{E})$ . Then ${\hat{D}}_{2}$ is said to be a neighbourhood (nbd) of ${\hat{D}}_{1}$ , if there exists a PyFSOS $\hat{β}$ (i.e., $\hat{β} \in \hat{T}$ ) ∋ ${\hat{D}}_{1} \subset \hat{β} \subset {\hat{D}}_{1}$ .

Theorem 3.2. A PyFSS $\hat{β} \in PyFSSs (\hat{U}, \hat{E})$ is a PyFSOS iff $\hat{β}$ is a nbd of each PyFSS ${\hat{D}}_{1} \subset \hat{β}$ .

Proof. Let $\hat{β}$ be a PyFSOS and ${\hat{D}}_{1}$ any PyFSSs in $\hat{β}$ . Since we have ${\hat{D}}_{1} \subset \hat{β} \subset \hat{β}$ , this implies that $\hat{β}$ is a nbd of ${\hat{D}}_{1}$ . Consequently, if suppose $\hat{β}$ is a nbd of every PyFSS $\subseteq \hat{β}$ . Since $\hat{β} \subset \hat{β}$ , ∃ a PyFSOS ${\hat{D}}_{2}$ ∋ $\hat{β} \subset {\hat{D}}_{2} \subset \hat{β}$ . Therefore, $\hat{β} = {\hat{D}}_{2}$ and $\hat{β}$ is open.

Theorem 3.3. Let $(\hat{U}, \hat{T_{1}}, \hat{E})$ be a PyFSTS on $(\hat{U}, \hat{E})$ and $\hat{β} \in (\hat{U}, \hat{E})$ . $\hat{β}$ is called the nbd filter or nbd system of $\hat{β}$ , the family of all nbds, up to topology $\hat{T_{1}}$ (in short, $PyFnbd (\hat{β})$ ).

Theorem 3.4. Let the nbd system of the PyFSS $\hat{β}$ be $PyFnbd (\hat{β})$ . Then,

finite intersections of the members of $PyFnbd (\hat{β}) \in PyFnbd (\hat{β})$ .

each PyFSS containing a member of $PyFnbd (\hat{β}) \in PyFnbd (\hat{β})$ .

Proof.

Let ${\hat{D}}_{1}, {\hat{D}}_{2} \in PyFnbd (\hat{β})$ . Then $\exists {\hat{D}}_{1}^{'}, {\hat{D}}_{2}^{'} \in \hat{T}$ ∋ $\hat{β} \subset {\hat{D}}_{1}^{'} \subset {\hat{D}}_{1}$ and $\hat{β} \subset {\hat{D}}_{2}^{'} \subset {\hat{D}}_{2}$ . Since, ${\hat{D}}_{1}^{'} \cap {\hat{D}}_{2}^{'} \in \hat{T}$ , we have, $\hat{β} \subset {\hat{D}}_{1}^{'} \cap {\hat{D}}_{2}^{'} \subset {\hat{D}}_{1} \cap {\hat{D}}_{2}$ . Thus, ${\hat{D}}_{1} \cap {\hat{D}}_{2} \in PyFnbd (\hat{β})$ .

If ${\hat{D}}_{1} \in PyFnbd (\hat{β})$ and ${\hat{D}}_{2}$ be a PyFSS containing ${\hat{D}}_{1}$ , then $\exists {\hat{D}}_{1}^{'} \in \hat{T}$ ∋ $\hat{β} \subset {\hat{D}}_{1}^{'} \subset {\hat{D}}_{1} \subset {\hat{D}}_{2}$ . This proves that ${\hat{D}}_{2} \in PyFnbd (\hat{β})$

Definition 3.5. Let $\hat{D} \in PyFSS (\hat{U}, \hat{E})$ be an arbitrary PyFSS and let $(\hat{U}, \hat{T}, \hat{E})$ be a PyFSTS over $(\hat{U}, \hat{E})$ . Then

${\hat{D}}^{PyFSo}$ = interior of $\hat{D} = \cup {\hat{G} : \hat{G}$ is PyFSO and $\hat{G} \subseteq \hat{D}}$ ,

${\hat{D}}^{PyFS -}$ = closure of $\hat{D} = \cap {\hat{G} : \hat{G}$ is PyFSC and $\hat{G} \supseteq \hat{D}}$ .

Remark 3.2. For any PyFSS $\hat{D}$ in $(\hat{U}, \hat{T}, \hat{E})$ , we have

$[{\hat{D}}^{c}]^{PyFS -}$ = $[{\hat{D}}^{PyFSo}]^{c}$ .

$[{\hat{D}}^{c}]^{PyFSo}$ = $[{\hat{D}}^{PyFS -}]^{c}$ .

$\hat{D}$ is a PyFSCS iff ${\hat{D}}^{PyFS -} = \hat{D}$ .

$\hat{D}$ is a PyFSOS iff ${\hat{D}}^{PyFSo} = \hat{D}$ .

${\hat{D}}^{PyFS -}$ is a PyFSCS in $(\hat{U}, \hat{E})$ .

${\hat{D}}^{PyFSo}$ is a PyFSOS in $(\hat{U}, \hat{E})$ .

Theorem 3.5. Let $(\hat{U}, \hat{T}, \hat{E})$ be a PyFSTS with respect to $(\hat{U}, \hat{E})$ . Let ${\hat{D}}_{1}$ and ${\hat{D}}_{2}$ be Pythagorean fuzzy soft subsets of $(\hat{U}, \hat{E})$ . Then the succeeding assertions holds:

$\hat{D} \subseteq {\hat{D}}^{PyFS -}$ .

$\hat{D}$ is a PyFSCS iff ${\hat{D}}^{PyFS -} = \hat{D}$ .

${\hat{0}}^{PyFS -}$ = $\hat{0}$ and ${\hat{1}}^{PyFS -}$ = $\hat{1}$ .

${\hat{D}}_{1} \subseteq {\hat{D}}_{2} \Rightarrow$ ${\hat{D}}_{1}^{PyFS -} \subseteq {\hat{D}}_{2}^{PyFS -}$ .

$({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFS -}$ = ${\hat{D}}_{1}^{PyFS -} \cup {\hat{D}}_{2}^{PyFS -}$ .

$({\hat{D}}_{1} \cap {\hat{D}}_{2})^{PyFS -}$ = ${\hat{D}}_{1}^{PyFS -} \cap {\hat{D}}_{2}^{PyFS -}$ .

$({\hat{D}}^{PyFS -})^{PyFS -}$ = ${\hat{D}}^{PyFS -}$ .

Proof.

From definition 3.5(ii), $\hat{D} \subseteq$ ${\hat{D}}^{PyFS -}$

If $\hat{D}$ is a Pythagorean fuzzy soft closed set(PyFSCS), then $\hat{D}$ is the littlest Pythagorean fuzzy soft closed set containing itself and hence ${\hat{D}}^{PyFS -} = \hat{D}$ . The other way around, if ${\hat{D}}^{PyFS -}$ = $\hat{D}$ , then $\hat{D}$ is the littlest Pythagorean fuzzy soft closed set containing itself and therefore $\hat{D}$ is a PyTFSCS.

Since $\hat{0}$ and $\hat{1}$ are PyFSCSs in $(\hat{U}, \hat{T}, \hat{E})$ , ${\hat{0}}^{PyFS -} = \hat{0}$ and ${\hat{1}}^{PyFS -} = \hat{1}$ .

If PyFSS ${\hat{D}}_{1}$ is a subset of PyFSS ${\hat{D}}_{2}$ , since PyFSS ${\hat{D}}_{2}$ is a subset of ${\hat{D}}_{2}^{PyFS -}$ , then PyFSS ${\hat{D}}_{1}$ is a subset of ${\hat{D}}_{2}^{PyFS -}$ . That is, ${\hat{D}}_{2}^{PyFS -}$ is a PyFSCS containing ${\hat{D}}_{1}$ . But ${\hat{D}}_{1}^{PyFS -}$ is the littlest PyFSCS containing ${\hat{D}}_{1}$ . Therefore, ${\hat{D}}_{1}^{PyFS -} \subseteq {\hat{D}}_{2}^{PyFS -}$

Since PyFSS ${\hat{D}}_{1}$ is a ⊂ of union of two PyFSSs ${\hat{D}}_{1}$ and ${\hat{D}}_{2}$ and PyFSS ${\hat{D}}_{2}$ is a ⊂ of union of two PyFSSs ${\hat{D}}_{1}$ and ${\hat{D}}_{2}$ , ${\hat{D}}_{1}^{PyFS -} \subseteq ({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFS -}$ . Then closure of PyFSS ${\hat{D}}_{1}$ is a subset of closure of union of two PyFSSs ${\hat{D}}_{1}$ and ${\hat{D}}_{2}$ and closure of PyFSS ${\hat{D}}_{2}$ is a subset of closure of union of two PyFSSs ${\hat{D}}_{1}$ and ${\hat{D}}_{2}$ . Hence, union of closure of PyFSSs ${\hat{D}}_{1}^{PyFS -}$ , ${\hat{D}}_{2}^{PyFS -}$ is a subset of closure of union of $({\hat{D}}_{1}^{PyFS -}$ , ${\hat{D}}_{2})^{PyFS -}$ . By the fact that ${\hat{D}}_{1} \cup {\hat{D}}_{2} \subseteq {\hat{D}}_{1}^{PyFS -} \cup {\hat{D}}_{2}^{PyFS -}$ , and since $({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFS -}$ is the littlest PyFSCS containing ${\hat{D}}_{1} \cup {\hat{D}}_{2}$ , so $({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFS -} \subseteq {\hat{D}}_{1}^{PyFS -} \cup {\hat{D}}_{2}^{PyFS -}$ . Thus, $({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFS -} = {\hat{D}}_{1}^{PyFS -} \cup {\hat{D}}_{2}^{PyFS -}$ .

Since ${\hat{D}}_{1} \cap {\hat{D}}_{2} \subseteq {\hat{D}}_{1}$ and ${\hat{D}}_{1} \cap {\hat{D}}_{2} \subseteq {\hat{D}}_{2}$ , $({\hat{D}}_{1} \cap {\hat{D}}_{2})^{PyFS -} \subseteq {\hat{D}}_{1}^{PyFS -} \cap {\hat{D}}_{2}^{PyFS -}$ .

Since ${\hat{D}}^{PyFS -}$ is a PyFSCS, then $({\hat{D}}^{PyFS -})^{PyFS -} = {\hat{D}}^{PyFS -}$ .

Theorem 3.6. $(\hat{U}, \hat{T}, \hat{E})$ be a PyFSTS over $(\hat{U}, \hat{E})$ . Let $\hat{D}$ be a Pythagorean fuzzy soft subset of $(\hat{U}, \hat{E})$ . Then

$\hat{1} - {\hat{D}}^{PyFSo}$ = $(\hat{1} - \hat{D})^{PyFS -}$ .

$\hat{1} - {\hat{D}}^{PyFS -}$ = $(\hat{1} - \hat{D})^{PyFSo}$ .

Theorem 3.7. Let $(\hat{U}, \hat{T}, \hat{E})$ be a PyFSTS with respect to $(\hat{U}, \hat{E})$ . Let ${\hat{D}}_{1}$ and ${\hat{D}}_{2}$ be Pythagorean fuzzy soft subsets of $(\hat{U}, \hat{E})$ . Then the succeeding assertions holds:

$\hat{D}$ is a PyFSOS open iff ${\hat{D}}^{PyFSo} = \hat{D}$ .

${\hat{0}}^{PyFSo} = \hat{0}$ and ${\hat{1}}^{PyFSo} = \hat{1}$ .

${\hat{D}}_{1} \subseteq {\hat{D}}_{2} \Rightarrow {\hat{D}}_{1}^{PyFSo} \subseteq {\hat{D}}_{2}^{PyFSo}$ .

$({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFSo} = {\hat{D}}_{1}^{PyFSo} \cup {\hat{D}}_{2}^{PyFSo}$ .

$({\hat{D}}_{1} \cap {\hat{D}}_{2})^{PyFSo} = {\hat{D}}_{1}^{PyFSo} \cap {\hat{D}}_{2}^{PyFSo}$ .

$({\hat{D}}^{PyFSo})^{PyFSo} = {\hat{D}}^{PyFSo}$ .

Proof.

$\hat{D}$ is a PyFSOS iff $\hat{1} - \hat{D}$ is a PyFSCS, iff $(\hat{1} - \hat{D})^{PyFS -} = \hat{1} - \hat{D}$ , iff $\hat{1} - (\hat{1} - \hat{D})^{PyFS -} = \hat{D}$ iff ${\hat{D}}^{PyFSo} = \hat{D}$ .

As $\hat{0}$ and $\hat{1}$ are PyFSOSs in $(\hat{U}, \hat{T}, \hat{E})$ , ${\hat{0}}^{PyFSo} = \hat{0}$ and ${\hat{1}}^{PyFSo} = \hat{1}$ .

If ${\hat{D}}_{1} \subseteq {\hat{D}}_{2}$ , since ${\hat{D}}_{2} \supseteq {\hat{D}}_{2}^{PyFSo}$ , then ${\hat{D}}_{1} \supseteq {\hat{D}}_{2}^{PyFSo}$ . That is, ${\hat{D}}_{2}^{PyFSo}$ is a PyFSOS containing ${\hat{D}}_{1}$ . But ${\hat{D}}_{1}^{PyFSo}$ is the largest PyFSOS contained in ${\hat{D}}_{1}$ . Therefore, ${\hat{D}}_{1}^{PyFSo} \subseteq {\hat{D}}_{2}^{PyFSo}$

Since ${\hat{D}}_{1} \subseteq {\hat{D}}_{1} \cup {\hat{D}}_{2}$ and ${\hat{D}}_{2} \subseteq {\hat{D}}_{1} \cup {\hat{D}}_{2}$ , ${\hat{D}}_{1}^{PyFSo} \subseteq ({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFSo}$ and ${\hat{D}}_{2}^{PyFSo} \subseteq ({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFSo}$ . Therefore, ${\hat{D}}_{1}^{PyFSo} \cup {\hat{D}}_{2}^{PyFSo} \subseteq ({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFSo}$ . By the fact that ${\hat{D}}_{1} \cup {\hat{D}}_{2} \subseteq {\hat{D}}_{1}^{PyFSo} \cup {\hat{D}}_{2}^{PyFSo}$ , and since $({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFSo}$ is the largest PyFSOS containing ${\hat{D}}_{1} \cup {\hat{D}}_{2}$ , so $({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFSo} \subseteq {\hat{D}}_{1}^{PyFSo} \cup {\hat{D}}_{2}^{PyFSo}$ . Thus, $({\hat{D}}_{1} \cup {\hat{D}}_{2})^{PyFSo} = {\hat{D}}_{1}^{PyFSo} \cup {\hat{D}}_{2}^{PyFSo}$ .

Since ${\hat{D}}_{1} \cap {\hat{D}}_{2} \subseteq {\hat{D}}_{1}$ and ${\hat{D}}_{1} \cap {\hat{D}}_{2} \subseteq {\hat{D}}_{2}$ , $({\hat{D}}_{1} \cap {\hat{D}}_{2})^{PyFSo} \subseteq {\hat{D}}_{1}^{PyFSo} \cap {\hat{D}}_{2}^{PyFSo}$ .

Since ${\hat{D}}^{PyFSo}$ is a PyFSOS, then $({\hat{D}}^{PyFSo})^{PyFSo}$ = ${\hat{D}}^{PyFSo}$ .

Definition 3.6. Let $(\hat{U}, \hat{T}, \hat{E})$ be a PyFSTS over $(\hat{U}, \hat{E})$ and $\hat{D} \in PyFSSs (\hat{U}, \hat{E})$ . Then PyFS boundary (frontier) of $\hat{D}$ is denoted by $PyFSB (\hat{D})$ and is defined by $PyFSB (\hat{D}) = {\hat{D}}^{PyFS -} \cap ({\hat{D}}^{c})^{PyFS -}$ .

Theorem 3.8. Let $(\hat{U}, \hat{T}, \hat{E})$ be a PyFSTS over $(\hat{U}, \hat{E})$ and $\hat{D} \in PyFSSs (\hat{U}, \hat{E})$ . Then,

${\hat{D}}^{PyFSo} \cap PyFSB (\hat{D}) = \hat{0}$

${\hat{D}}^{PyFS -} = {\hat{D}}^{PyFSo} \cup PyFSB (\hat{D})$

$PyFSB (\hat{D}) = \hat{0}$ iff $\hat{D}$ is both open and closed.

$PyFSB (\hat{D}) = {\hat{D}}^{PyFS -} \cap ({\hat{D}}^{PyFSo})^{c} = \hat{0}$

Proof.

${\hat{D}}^{PyFSo} \cap PyFSB (\hat{D}) = {\hat{D}}^{PyFSo} \cap ({\hat{D}}^{PyFS -} \cap ({\hat{D}}^{c})^{PyFS -}) = {\hat{D}}^{PyFSo} \cap ({\hat{D}}^{PyFS -} \cap ({\hat{D}}^{PyFSo})^{c}) = {\hat{D}}^{PyFSo} \cap ({\hat{D}}^{PyFSo})^{c} \cap {\hat{D}}^{PyFS -} = \hat{0} \cap {\hat{D}}^{PyFS -} = \hat{0}$ .

${\hat{D}}^{PyFSo} \cup PyFSB (\hat{D}) = {\hat{D}}^{PyFSo} \cup ({\hat{D}}^{PyFS -} \cap ({\hat{D}}^{c})^{PyFS -}) = {\hat{D}}^{PyFSo} \cup ({\hat{D}}^{PyFS -} \cap ({\hat{D}}^{PyFSo})^{c}) = ({\hat{D}}^{PyFSo} \cup {\hat{D}}^{PyFS -}) \cap ({\hat{D}}^{PyFSo} \cup ({\hat{D}}^{PyFSo})^{c}) = ({\hat{D}}^{PyFSo} \cup {\hat{D}}^{PyFS -}) \cap \hat{1} = ({\hat{D}}^{PyFSo} \cup {\hat{D}}^{PyFS -}) = {\hat{D}}^{PyFS -}$ . Since ${\hat{D}}^{PyFSo} \subset \hat{D} \subset {\hat{D}}^{PyFS -}$ .

$PyFSB (\hat{D}) = {\hat{D}}^{PyFS -} \cap ({\hat{D}}^{c})^{PyFS -} = \hat{0}$ $\Rightarrow {\hat{D}}^{PyFS -} \cap ({\hat{D}}^{PyFSo})^{c} = \hat{0}$ $\Rightarrow {\hat{D}}^{PyFS -} \cap (({\hat{D}}^{PyFSo})^{c})^{c} = \hat{0}$ $\Rightarrow {\hat{D}}^{PyFS -} \cap {\hat{D}}^{PyFSo} = \hat{0}$ $\Rightarrow {\hat{D}}^{PyFS -} \subset {\hat{D}}^{PyFSo}$ i.e., $\hat{D} \subset {\hat{D}}^{PyFS -} \subset {\hat{D}}^{PyFSo} \Rightarrow \hat{D} \subset {\hat{D}}^{PyFSo}$ .

Also we know that ${\hat{D}}^{PyFSo} \subset \hat{D}$ . Thus ${\hat{D}}^{PyFSo} = \hat{D}$ . This shows that $\hat{D}$ is open.

Further more ${\hat{D}}^{PyFS -} \subset {\hat{D}}^{PyFSo} \subset \hat{D} \Rightarrow {\hat{D}}^{PyFS -} \subset \hat{D}$ . Also we know that $\hat{D} \subset {\hat{D}}^{PyFS -}$ . Thus ${\hat{D}}^{PyFS -} = \hat{D}$ . This shows that $\hat{D}$ is closed.

Conversely, if $\hat{D}$ is open and closed, then ${\hat{D}}^{PyFSo} = \hat{D}$ and ${\hat{D}}^{PyFS -} = \hat{D}$ . Now, $PyFSB (\hat{D}) = {\hat{D}}^{PyFS -} \cap ({\hat{D}}^{c})^{PyFS -} = {\hat{D}}^{PyFS -} \cap ({\hat{D}}^{PyFSo})^{c} = {\hat{D}}^{PyFS -} \cap {\hat{D}}^{c} = \hat{0}$ .

$PyFSB (\hat{D}) = {\hat{D}}^{PyFS -} \cap ({\hat{D}}^{c})^{PyFS -} = {\hat{D}}^{PyFS -} \cap ({\hat{D}}^{PyFSo})^{c}$

Definition 3.7. A collection $\hat{B} \subset \hat{T}$ is said to be a base for $\hat{T}$ , where $(\hat{U}, \hat{T}, \hat{E})$ is a PyFSTS over $(\hat{U}, \hat{E})$ . If $\forall \hat{A} \in \hat{T}$ can be put down as the arbitrary Pythagorean fuzzy soft union of some elements of $\hat{B}$ , then $\hat{B}$ is said to be a Pythagorean fuzzy soft basis (PyFSB) for the PyFST $\hat{T}$ . The elements of $\hat{B}$ are said to be Pythagorean fuzzy basic open sets.

Theorem 3.9. Let $(\hat{U}, \hat{T}, \hat{E})$ be a PyFSTS over $(\hat{U}, \hat{E})$ and $\hat{B}$ a PyFSB for $\hat{T}$ . Then $\hat{T}$ equals the collection of Pythagorean fuzzy soft unions of elements of $\hat{B}$ .

Proof. The proof is straight forward.

Theorem 3.10. Let $(\hat{U}, {\hat{T}}_{1}, \hat{E})$ and $(\hat{U}, {\hat{T}}_{2}, \hat{E})$ be two PyFSTS over $(\hat{U}, \hat{E})$ and ${\hat{B}}_{1}$ be a PyFSB for ${\hat{T}}_{1}$ and ${\hat{B}}_{2}$ be a PyFSB for ${\hat{T}}_{2}$ . If ${\hat{B}}_{1} \subset {\hat{B}}_{2}$ , then ${\hat{T}}_{1} \subset {\hat{T}}_{2}$ .

Proof. The proof is straight forward.

4 Algorithm and Numerical Example on MADM using PyFS-topology

This part of the paper, exhibits an application of PyFSSs and PyFSTS in MADM problems by using an algorithm and we compare the results of algorithm with the existing ones.

4.1 Proposed Algorithm

Algorithm: (Decision making with PyFSSs and PyFSTSs)

Step-1: Input the PyFSSs $(\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ .

Step-2: Frame the PyFSTS $\hat{T}$ , $∋ (\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ are two open sets in $\hat{T}$ .

Step-3: Calculate the score values corresponding to each PyFOS in $\hat{T}$ using the formula $S (ζ) = | α_{\hat{F} (ξ)}^{2} (ζ) - β_{\hat{F} (ξ)}^{2} (ζ) |$ , where $S \in [0, 1]$ .

Step-4: The decision values for each PyFS-open set can be calculated by taking the average of each PyFS-open set for each ζ_j and it is denoted as δ_j

Step-5: Find the sum of the corresponding values of PyFS-open sets $(\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ . This will be considered as the final decision values.

Step-6: The optimum alternative ζ_j will be selected by taking the maximum value in final decision from Step-5.

The proposed Algorithm for MADM is represented as a flow chart in the Figure 1.

Fig. 1

Flow chart of the proposed Algorithm.

4.2 Numerical Example

The owner of a popular store of chennai wants to make a contract with the multi national company to market the wrist watch products. The different popular brands available are $\hat{U} = {ζ_{1}, ζ_{2}, ζ_{3}, ζ_{4}, ζ_{5}, ζ_{6}}$ , where ζ₁=Rolex, ζ₂=Casio, ζ₃=Omega, ζ₄=Citizen, ζ₅=Seiko, ζ₆=Avant-Garde. Define a set of criteria for the selection of a suitable brand for their store as follows $\hat{E} = {ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}, ξ_{6}}$ , where ξ₁=Delivery Performance, ξ₂=Recovery services in damage cases, ξ₃=Stability of price, ξ₄=Product quality, ξ₅=Distribution plans (in-store furniture), ξ₆= customer-brand relationship,.

Further the manager divides the criteria into two subsets, $\hat{A} = {ξ_{1}, ξ_{2}, ξ_{5}}$ (Category-1: Brand service) & $\hat{B} = {ξ_{3}, ξ_{4}, ξ_{6}}$ (Category-2: Product nature).

Step-1: Input the PyFSSs $(\hat{F}, \hat{A})$ and $(\hat{G}, \hat{B})$ , given by $(\hat{F}, \hat{A}) = {(ξ_{1}, \hat{F} (ξ_{1})), (ξ_{2}, \hat{F} (ξ_{2})), (ξ_{5}, \hat{F} (ξ_{5}))}$ and $(\hat{G}, \hat{B}) = {(ξ_{3}, \hat{G} (ξ_{3})), (ξ_{4}, \hat{G} (ξ_{4})), (ξ_{6}, \hat{G} (ξ_{6}))}$ and where $\hat{F} (ξ_{1}) = {ζ_{1} = 〈 0.5, 0.3 〉, ζ_{2} = 〈 0.2, 0.3 〉, ζ_{3} = 〈 0.6, 0.1 〉, ζ_{4} = 〈 0.2, 0.7 〉, ζ_{5} = 〈 0.4, 0.4 〉, ζ_{6} = 〈 0.4, 0.3 〉}$ $\hat{F} (ξ_{2}) = {ζ_{1} = 〈 0.4, 0.7 〉, ζ_{2} = 〈 0.6, 0.6 〉, ζ_{3} = 〈 0.3, 0.2 〉, ζ_{4} = 〈 0.4, 0.3 〉, ζ_{5} = 〈 0.6, 0.2 〉, ζ_{6} = 〈 0.5, 0.3 〉}$ $\hat{F} (ξ_{5}) = {ζ_{1} = 〈 0.7, 0.2 〉, ζ_{2} = 〈 0.5, 0.6 〉, ζ_{3} = 〈 0.7, 0.3 〉, ζ_{4} = 〈 0.6, 0.2 〉, ζ_{5} = 〈 0.7, 0.1 〉, ζ_{6} = 〈 0.5, 0.4 〉}$ .

$\hat{G} (ξ_{3}) = {ζ_{1} = 〈 0.4, 0.2 〉, ζ_{2} = 〈 0.3, 0.3 〉, ζ_{3} = 〈 0.6, 0.1 〉, ζ_{4} = 〈 0.3, 0.3 〉, ζ_{5} = 〈 0.5, 0.3 〉, ζ_{6} = 〈 0.7, 0.2 〉}$ $\hat{G} (ξ_{4}) = {ζ_{1} = 〈 0.2, 0.2 〉, ζ_{2} = 〈 0.5, 0.3 〉, ζ_{3} = 〈 0.3, 0.2 〉, ζ_{4} = 〈 0.5, 0.6 〉, ζ_{5} = 〈 0.2, 0.3 〉, ζ_{6} = 〈 0.6, 0.7 〉}$ $\hat{G} (ξ_{6}) = {ζ_{1} = 〈 0.3, 0.1 〉, ζ_{2} = 〈 0.6, 0.2 〉, ζ_{3} = 〈 0.3, 0.2 〉, ζ_{4} = 〈 0.6, 0.2 〉, ζ_{5} = 〈 0.2, 0.2 〉, ζ_{6} = 〈 0.1, 0.1 〉}$

Step-2: Framing the PyFSTS $\hat{T} = {\hat{1}, \hat{0}, (\hat{F}, \hat{A}), (\hat{G}, \hat{B})}$ , where $\hat{1}$ and $\hat{0}$ are PyFS-full and PyFS-null set, respectively.

Step-3: Compute the score values of PyFS-open sets, $(\hat{F}, \hat{A}), (\hat{G}, \hat{B})$ , $\hat{1}$ and $\hat{0}$ are as follows $\hat{F} (ξ_{1}) = {S (ζ_{1}) = 0.16, S (ζ_{2}) = 0.05, S (ζ_{3}) = 0.35, S (ζ_{4}) = 0.45, S (ζ_{5}) = 0.0, S (ζ_{6}) = 0.07}$ , $\hat{F} (ξ_{2}) = {S (ζ_{1}) = 0.33, S (ζ_{2}) = 0.0, S (ζ_{3}) = 0.05, S (ζ_{4}) = 0.07, S (ζ_{5}) = 0.32, S (ζ_{6}) = 0.16}$ , $\hat{F} (ξ_{5}) = {S (ζ_{1}) = 0.45, S (ζ_{2}) = 0.11, S (ζ_{3}) = 0.4, S (ζ_{4}) = 0.32, S (ζ_{5}) = 0.48, S (ζ_{6}) = 0.09}$ , $\hat{G} (ξ_{3}) = {S (ζ_{1}) = 0.12, S (ζ_{2}) = 0.0, S (ζ_{3}) = 0.35, S (ζ_{4}) = 0.0, S (ζ_{5}) = 0.16, S (ζ_{6}) = 0.45}$ , $\hat{G} (ξ_{4}) = {S (ζ_{1}) = 0.0, S (ζ_{2}) = 0.16, S (ζ_{3}) = 0.05, S (ζ_{4}) = 0.11, S (ζ_{5}) = 0.05, S (ζ_{6}) = 0.13}$ , $\hat{G} (ξ_{6}) = {S (ζ_{1}) = 0.08, S (ζ_{2}) = 0.32, S (ζ_{3}) = 0.05, S (ζ_{4}) = 0.32, S (ζ_{5}) = 0.0, S (ζ_{6}) = 0.0}$ . The score values of $\hat{1}$ and $\hat{0}$ are 1 and 1 for each element of the reference set.

Step-4: Find decision values of $(\hat{F}, \hat{A})$ , and $(\hat{G}, \hat{B})$ and no need to find the decision values of $\hat{1}$ and $\hat{0}$ as this will not affect the ranking process.

The decision values of $(\hat{F}, \hat{A}) = {δ_{1} = 0.313, δ_{2} = 0.053, δ_{3} = 0.267, δ_{4} = 0.28, δ_{5} = 0.267, δ_{6} = 0.107}$ ,

The decision values of $(\hat{G}, \hat{B}) = {δ_{1} = 0.067, δ_{2} = 0.16, δ_{3} = 0.15, δ_{4} = 0.143, δ_{5} = 0.07, δ_{6} = 0.193}$ ,

Step-5: Computation of the absolute decision values by summing the decision values δ_j of $(\hat{F}, \hat{A})$ , and $(\hat{G}, \hat{B})$ . Hence, The required final decision values are δ₁ = 0.38, δ₂ = 0.213, δ₃ = 0.417, δ₄ = 0.423, δ₅ = 0.337, δ₆ = 0.3.

Step-6: From step-5, we see that brand 4 yields maximum score value(membership degree), i.e. 0.423.

Conclusion: The maximum score value=0.423 which corresponds to the brand 4. Therefore, the owner of the store can choose brand 4 = Citizen as it is the suitable brand for the store.

The results of MADM using the proposed algorithm of PyFSTSs is better than that of FSs or IFSs because of the special structural constrain of PyFSs, its parameterization and its close relationship to human nature. Furthermore, the data clustered by using PyFSTS cannot be aggregated by existing topological operators. The comparison table 1 show the difference between novel Pythagorean fuzzy soft topological spaces with existing work.

Table 1
Comparative analysis of proposed approach with existing theories

Sets Uncertainty Truth Value of an element False value of an element Parameterization Advantages Limitations

FS [32] ✓ ✓ - - Deal with the hesitations Do not collect full information about the input data.

IFS [4] ✓ ✓ ✓ - This framework elaborates uncertainty and vagueness Do not deal with the parameterizations

PyFS [31] ✓ ✓ ✓ - PyFT deals with uncertain information more sufficiently and accurately than IFS Do not deal with the parameterizations

PyFSTS (proposed) ✓ ✓ ✓ ✓ PyFST deals with uncertain information with parameterization Effective but calculations are heavy as compared to above existing theories.

Sets	Uncertainty	Truth Value of an element	False value of an element	Parameterization	Advantages	Limitations
FS [32]	✓	✓	-	-	Deal with the hesitations	Do not collect full information about the input data.
IFS [4]	✓	✓	✓	-	This framework elaborates uncertainty and vagueness	Do not deal with the parameterizations
PyFS [31]	✓	✓	✓	-	PyFT deals with uncertain information more sufficiently and accurately than IFS	Do not deal with the parameterizations
PyFSTS (proposed)	✓	✓	✓	✓	PyFST deals with uncertain information with parameterization	Effective but calculations are heavy as compared to above existing theories.

5 Conclusion

Soft Topology is an interesting and significant field of mathematical science and it also provides several relationships among other mathematical models and scientific environment. Recently, abounding number of researchers have implemented and researched the soft set theoretical concepts, which can be easily implemented to umpteen real time situations of uncertainties. We, firstly provided the definition of the "Pythagorean fuzzy soft topology" and then conferred its fundamental properties along with some numerical examples. We also developed an algorithm based on PyFSTS for MADM problems. Therefore, this paper can be considered as the initiation point for the research on "Pythagorean fuzzy soft topology". The proposed algorithm is easily altered and flexible according to the different situations of inputs and outputs. In the future, this work can be extended to Pythagorean fuzzy soft continuity, compactness, connectedness between information systems and soft sets, and the results deduced from the studies on PyFSTS can be used to improve these types of relations and to solve the MADM problems by using AHP, TOPSIS, VIKOR, PROMETHEE family and ELECTRE family using different hybrid structures of fuzzy, soft and rough sets.

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