EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6598 ISSN 1307-5543 – ejpam.com Published by New York Business Global Pythagorean Fuzzy Digital Fine Space Approach for Selecting Menstrual Hygiene Products in Rural Areas N. Preethi1, G. K. Revathi1,∗ 1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology Chennai, Chennai-127, Tamilnadu, India Abstract. Pythagorean fuzzy digital fine b≈-door space is a novel notion of generalized Pythagorean fuzzy digital fine topological space that is presented in this paper. Additionally, a number of characterizations of Pythagorean fuzzy digital fine b≈-door space are examined. Examples were presented to demonstrate the applicability of these ideas and also some characteristics and con- nections between other Pythagorean fuzzy digital fine topological spaces and Pythagorean fuzzy digital fine b≈-door space were examined. Furthermore, an application which utilizes Pythagorean fuzzy digital fine topological space to select the best menstrual hygiene product for rural women is investigated. 2020 Mathematics Subject Classifications: 54A40, 03E72 Key Words and Phrases: Pythagorean fuzzy digital fine topological space, Pythagorean fuzzy digital b≈f -Baire space, Pythagorean fuzzy digital b≈f D-Baire space Pythagorean fuzzy digital b≈f - Hausdorff space, Pythagorean fuzzy digital b≈f -door spaces 1. Introduction An extension of classical topology, fuzzy topology makes use of the idea of fuzziness to provide more complex and adaptable interpretations of mathematical ideas. It was created to address circumstances in which the conventional understanding of membership function that is, whether an element is a member of a set or not is too strict to adequately represent actual occurrences. The fundamental concept of fuzzy topology is fuzzy sets, which were initially introduced by Lotfi Zadeh [1] in 1965. Unlike classical sets, where an element is either a member or not (with membership values strictly 1 or 0), fuzzy sets permit varying degrees of membership across a continuum [2]. Developed by Kras- simir Atanassov [3] in 1986, intuitionistic fuzzy topology is an extension of fuzzy topology that incorporates the idea of intuitionistic fuzzy sets. This enhancement offers a more comprehensive framework for managing uncertainty by taking into account the degree of membership, the degree of non-membership and a hesitation margin. It is particularly effective in situations where uncertainty exists not only regarding membership value but ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6598 https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 2 of 17 also non-membership value. Pythagorean fuzzy topology is an advanced form of fuzzy and intuitionistic fuzzy topology. It uses Pythagorean fuzzy sets, which provide more flexibil- ity in handling uncertainty. This is especially useful in situations where it is important to balance membership and non-membership values carefully and where there is a high level of indeterminacy. Motivated by this need, Yager R. R. [4] introduced Pythagorean fuzzy sets under a specific constraint µ2 + ν2 ≤ 1., enabling a more refined mathematical framework for addressing uncertainty. As a result, Pythagorean fuzzy topology paves the way for new research directions and practical applications in areas where classical or even intuitionistic methods may be inadequate. Fuzzy Digital topology [5] integrates fuzzy set theory with digital topology [6]. It employs fuzzy sets to represent uncertainty in digital images and utilizes topological principles to analyze and process them. The main objec- tive is to develop techniques capable of effectively managing the inherent fuzziness present in digital imagery. This includes defining fuzzy topological spaces, fuzzy connectedness, and fuzzy boundaries within digital grid structures. To improve the recognition and cat- egorization of objects with partially hidden or ambiguous borders, digital fuzzy topology incorporates fuzziness into the topological modelling of digital images. This method works especially well in real-world applications, including biometric verification, automated vi- sual inspection, and autonomous vehicle navigation systems. There are numerous uses for the Pythagorean fuzzy topological space [7], which was created with Pythagorean fuzzy sets, in decision-making. Fuzzy door Space was first pro- posed by Anjalmose S. and Thangaraj G. [8], who also looked into the connections between fuzzy door space and some fuzzy topological space. The notion of b-open sets was pre- sented and examined by Andrijivic [9]. Additionally, Parameswari R.U. and Thangavelu P. [10], presented on the idea of a b#-open set and its fundamental characteristics. Using the intuitionistic fuzzy b-set as a basis, AbdulGawad, A. AL-Qubati, and Mohamed El Sayed [11] developed the concept of Intuitionistic fuzzy b-door space and investigated its properties. Harish Garg [12] explored confidence Pythagorean fuzzy weighted and ordered weighted operators, which are novel averaging and geometric operators. To illustrate their validity and effectiveness, a real-life application has been proposed. The analysis of door spaces is a well-known topic in fuzzy topology and intuitionistic fuzzy topology. However, it is not yet known whether it can be used to Pythagorean fuzzy digital fine topological spaces. In an intuitionistic fuzzy framework, a great deal of re- search has been done to produce theoretical findings regarding door spaces. The necessity for a thorough investigation of the function of door spaces with Pythagorean fuzzy digital fine topological space is highlighted by this gap. The following is the motivation behind this study: • For theoretical developments, it is crucial to comprehend how fine collections behave in a topological space. • To improve the study of Pythagorean fuzzy topology, the idea of door spaces is extended to Pythagorean fuzzy digital fine door spaces. • New insights into structural features are obtained by connecting Pythagorean fuzzy N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 3 of 17 digital fine topological spaces with Pythagorean fuzzy digital fine b≈-door space. • Filling this research void will advance Pythagorean fuzzy topology and enable its application to more general mathematics and practical issues. This paper introduces the concept of door space within the framework of Pythagorean fuzzy digital fine topology. Establishing and examining a variety of structures in Pythagorean fuzzy digital fine door spaces is what makes this study new. These structures include: • Pythagorean fuzzy digital fine topology • Pythagorean fuzzy digital fine b≈- door space • Pythagorean fuzzy digital fine b≈- first category space • Pythagorean fuzzy digital fine b≈- Baire space • Pythagorean fuzzy digital fine b≈- Hausdorff space A novel real world application titled as selection of the most suitable menstrual hy- giene product for rural women, is presented along with supporting numerical example. This practical case study illustrates how the Pythagorean fuzzy digital fine topology can effectively guide and influence the decision-making process in product selection and the complete structure of this study is given in the Figure 1. Figure 1: Flow of Pythagorean fuzzy digital space concepts The structure of the paper is as follows: the fundamental definitions are given in Section 2. The definition of Pythagorean fuzzy digital fine b≈- door space and a discussion N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 4 of 17 of several intriguing characteristics are covered in Section 3. The Pythagorean fuzzy digital b≈f -quasi compact is suggested and its characteristics are examined in Section 4. The Pythagorean fuzzy digital fine topological space selection algorithm is presented in Section 5. The validation of the present study is provided in Section 6. Finally, Section 7 summarises the findings of the study. 2. Preliminaries The basic concepts necessary for understanding this study are outlined in this section. Table 1 lists the abbreviations used throughout the work. Table 1: List of abbreviations and acronyms Abbreviations Acronyms Interior I Closure C Door Space DS Membership function MS function Non-membership function NMS function Fuzzy topological space FTS Fuzzy nowhere dense set FNDS Intuitionistic fuzzy b-set IF b-set Intuitionistic fuzzy b-open IF b−OS intuitionistic fuzzy b-closed IF b− CS Intuitionistic fuzzy b-Door space IF b−DS Pythagorean fuzzy set PFS Pythagorean fuzzy topology PFT Pythagorean fuzzy topological space PFTS Pythagorean fuzzy weighted average PFWA Pythagorean fuzzy weighted geometric PFWG Pythagorean fuzzy digital set (ω∼) PFDS(ω∼) Pythagorean fuzzy digital fine PFDf Pythagorean fuzzy digital fine set PFDfS Pythagorean fuzzy digital fine point PFDfP Pythagorean fuzzy digital fine number PFDfN Pythagorean fuzzy digital fine topology PFDfT Pythagorean fuzzy digital fine topological space PFDfTS Pythagorean fuzzy digital fine b≈-topological space PFDb≈f −DS Pythagorean fuzzy digital fine b≈ - open set PFDb≈f −OS Pythagorean fuzzy digital fine b≈ - closed set PFDb≈f − CS Pythagorean fuzzy digital fine b≈ - closure PFDb≈f −C Pythagorean fuzzy digital fine b≈− interior PFDb≈f −I N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 5 of 17 Definition 1. [7] Let ξ be an universal set. Then a PFS ω which is a set of ordered pair over ξ is defined by ω = {< x, µω(x), νω(x) > /x ∈ ξ} where µω(x) : ξ → [0, 1] and νω(x) : ξ → [0, 1] define the degree of MS and the degree of NMS respectively such that 0 ≤ µ2ω(x) + νω2(x) ≤ 1. Definition 2. [7] Suppose that ω = (µω, vω) and ς = (µς , vς) can be any two PFS of ξ. Then • ωC = (vω, µω) yields the complement of ω. • ω ∩ ς = {min (µω, µς) , max ((vω, vς)} yields the intersection of ω and ς. • ω ∪ ς = {max (µω, µς) , min(vω, vς) }yields the union of ω and ς. • If µω ≤ µς and vω ≥ vς then ω is a subset of ς(ω ⊆ ς) or ς contains ω(ς ⊇ ω). Definition 3. [7] Consider ξ is a set and τ is a family of PFS of ξ. • 0∼, 1∼ ∈ τ • For any {ωj/j ∈ I, ωj ∈ τ} , ⋂n j=1 ωj ∈ τ • ∪j=1ωj ∈ τ for any {ωj/j ∈ I, ωj ∈ τ}, the notation τ is a PFT on ξ when I is any arbitrary index set. This pair (ξ, τ) is referred to in this context as a PFTS. An open PFS is any member of τ , and the complement of an open PFS is a closed PFS. The topology containing all PFS is called a discrete PFTS. Definition 4. [7] In a PFTS (ξ, τ), let ω and ς be any two PFS. If there is an open PFS η in which ω ⊆ η ⊆ ς, then ς is regarded as a neighborhood of ω. Definition 5. [6, 13] Let X and Y be points in ξ, the set of integer-coordinate points arranged in a rectangular grid. A path Γ from X to Y is defined as a sequence X = X0, X1, X2, . . . , Xn = Y , such that for each i from 1 to n, the point Xi is adjacent to Xi−1, with adjacency being either 4 -adjacent or 8 -adjacent. Definition 6. [6, 13] In the set ξ, points X and Y are said to be connected when a path from X to Y includes all points of a subset P∗ of ξ. Definition 7. [14] Consider ξ as a rectangular grid of points with integer coordinates in the Euclidean plane Σ, and let ω = (µω, vω) be a PFS of Σ. The PFDS of ξ, denoted by ω ∼, assigns to each point X ∈ ξ a MS degree µω∼(X) and a NMS degree vω∼(X) defined as follows: µω∼(X) = max {µω(S) | S ∈ P ∗} and vω∼(X) = max {vω(S) | S ∈ P ∗}. Here S is the subset of the plane and P∗ is the open unit square has the center X . Definition 8. [8] In a FTS (ξ, τ), a fuzzy set ω is termed as fuzzy dense if C(ω) = 1∼. This implies there is no fuzzy closed set η in (ξ, τ) such that ω < η < 1. N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 6 of 17 Definition 9. [8] If there is no non-zero fuzzy open set ς in (ξ, τ) such that ς < C(ω), i.e., I(C(ω)) = 0∼, then a fuzzy set ω in a FTS (ξ, τ) is said to be a FNDS. Definition 10. [8] Consider the FTS (ξ, τ). The fuzzy first category set is a fuzzy set ω in ( ξ, τ ) if ω = ∨∞ n=1 ωn, where n’s are FNDS’s in (ξ, τ) . In (ξ, τ), a fuzzy set that is not a fuzzy first category set is referred to as a fuzzy second category set. Definition 11. [8] The fuzzy first category space is a FTS (ξ, τ) where 1 = ∨∞ n=1 ωn where n’s are FNDS’s in (ξ, τ). A topological space is considered to be of fuzzy second category space if it is not of fuzzy first category space. Definition 12. [8] Assume that the FTS is (ξ, τ). If I (V∞ n=1ωn) = 0∼, then ( ξ, τ ) is a fuzzy Baire space. Here ωn ’s are FNDS’s in (ξ, τ). Definition 13. [8] If every fuzzy first category set in (ξ, τ) is a FNDS in (ξ, τ), then the FTS (ξ, τ) is referred to as a fuzzy D-Baire space. Definition 14. [8] If I(ω) is fuzzy dense in (ξ, τ) for any fuzzy dense set ω in (ξ, τ) with I(ω) ̸= 0∼ (the null set), then the FTS (ξ, τ) is a fuzzy Quasi-maximal space. Definition 15. [8] Fuzzy Hausdorff spaces are FTS s (ξ, τ) if, for every ω, ς in (ξ, τ) and ω ̸= ς, we identify the fuzzy open sets η and δ such that that ω ≤ η, ς ≤ δ and η ∧ δ = 0∼. Definition 16. [8] Any fuzzy subset of a FTS (ξ, τ) that is either fuzzy open or fuzzy closed is referred to as a fuzzy door space. Definition 17. [9, 10] If ω = C(I(ω))∪I(C(ω)), then a subset ω of a space ξ is b#-open. Definition 18. [11] (i) If ω = (C(I(ω) ∪ (I(C(ω))), then an IFS ω of an IFTS (ξ, τ) is IF b-OS and (ii) if ω = (C(I(ω))∩ (I(C(ω))), then an IFS ω of an IFTS (ξ, τ) is IFb-CS. Definition 19. [15] Let (ξ, τ) be a TS and τ (Aα) = τα = { Gα( ̸= ξ);Gα ∩ Aα ̸= ϕ, for Aα ∈ τ Aα ̸= ϕ for some α ∈ J, J is an index set. And also define τf = { ϕ, ξ, ⋃ α∈J {τα} } . The fine topological space formed by the topology τ on ξ is denoted by (ξ, τ, τf ) and this collection of τf subsets of ξ, which is known as the fine collections of subsets of ξ. Fine open sets in (ξ, τ, τf ) are the elements of τf , while fine closed sets, which are repre- sented by τCf , are the complement of fine sets. Definition 20. [12] PFWA and PFWG are two averaging and geometric aggregating operators that have been developed for a set of Pythagorean fuzzy numbers Ji(1 ≤ i ≤ n) as follows PFWA (J1, J2, J3, . . . . . . .., Jn) = 〈√ 1− ∏n i=1 ( 1− µ2i )wi , ∏n i=1 ϑi wi 〉 And PFWG (J1, J2, J3, . . . . . . . . . ., Jn) = 〈∏n i=1 µ wi i , √ 1− ∏n i=1 ( 1− ϑ2i )wi 〉 where w = (w1,w2,w3, . . . . . . ,wn) T is the associated normalized weight vector of these Pythagorean fuzzy numbers such that wi(i = 1, 2, 3, . . . .n) ∈ [0, 1] and ∑n i=1wi = 1. N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 7 of 17 3. Pythagorean Fuzzy Digital Fine b≈-Door Spaces This section provides an introduction to PFDfT and PFDb≈f − DS, along with a detailed discussion of their properties. Definition 21. Let (ξ, τ) be a PFTS. Define τ (Aα) = τ̂α = { Gα( ̸= 1∼);Gα ∩Aα ̸= 0∼, for Aα ∈ τ Aα ̸= 0∼ for some α ∈ J, J is an index set. And also define τ̂f = { 0∼, 1∼, ⋃ α∈J Gα } . Here the operator “ ⋃ ” denotes the union of the set of all collections of Gα. The PFDfS of ξ are therefore denoted by τ̂f , and the PFDfTS produced by the topology τ on ξ is denoted by (ξ, τ, τ̂f ). All of the elements of ( ξ, τ, τ̂f ) are considered to be PFD fine open sets, and their corresponding complements are considered to be PFD fine closed sets. Definition 22. A PFDfTS (ξ, τ) and its PFDfS ω are referred to as • PFDb≈f −OS if ω = ((C(I(ω)) ∪ (I(C(ω))) • PFDb≈f − CS if ω = ((C(I(ω)) ∩ (I(C(ω))) Definition 23. Let ω be any PFDb≈f - set in PFDfTS (ξ, τ, τ̂f ) The PFDb≈f −C and PFDb≈f −I of ω is given as PFD b≈f − C(ω) = ∩{A : ω ⊆ A,A is PFDb≈f − CS in (ξ, τ, τ̂f )} PFDb≈f − I(ω) = ∪{B : B ⊆ ω,B is PFDb≈f −OS in (ξ, τ, τ̂f )} Definition 24. A PFDfTS (ξ, τ, τ̂f ) is called as the PFDb≈f − DS if every PFDb≈f set in ( ξ, τ, τ̂f ) is either PFDb≈f open or PFDb≈f closed. Example 1. Consider ξ = {l, m} be a non empty set and τ = {0∼, 1∼, ξ} is a PFDT on ξ where 0∼ = {< 0, 1 >,< 0, 1 >}, 1∼ = {< 1, 0 >,< 1, 0 >}, and ω =< x, (l/0.2, m/0.3), (l/0.9, m/0.8) >. Consider G =< x, (l/0.2, m/0.1), (l/0.9, m/0.9) >. Here ω ∩ G ̸= 0∼. Thus, (ξ, τ, τ̂f ) is the PFDfTS produced by the topology τ on ξ, and τ̂f = {0∼, 1∼, G} is the PFDf col- lections of subsets of ξ. Now I(ω) = ω; C(I(ω)) = ωc and C(ω) = ωc; I(C(ω)) = ω. Then ((C(I(ω) ∪ (I(C(ω))) = ωc and ((C(Iω)) ∩ (I(C(ω))) = ω. So ω is PFDb≈f − CS and ω is not PFDb≈f −OS. Hence (ξ, τ, τ̂f ) is a PFDb≈f −DS. Definition 25. Let (ξ, τ, τ̂f ) be a PFDfTS and let ω be a PFDb≈f set in (ξ, τ, τ̂f ). If there is no non-zero PFDb≈f -OS ς in (ξ, τ, τ̂f ) such that ς ⊆ PFDb≈f -C(ω), then ω is a PFDb≈f − nowhere dense set which means that PFDb≈f − I (C(ω)) = 0∼. Definition 26. If ω is a PFDb≈f - nowhere dense set in a PFDfTS (ξ, τ, τ̂f ) then (ξ, τ, τ̂f ) is denoted as a PFDb≈f -Baire space if PFDb≈f − I (∪ωα) = 0∼. Example 2. Let ξ = {l, m, n} be a non empty set and τ = {0∼, 1∼,X} is a PFDT on ξ where 0∼ = {< 0, 1 >,< 0, 1 >,< 0, 1 >}, 1∼ = {< 1, 0 >,< 1, 0 >,< 1, 0 >}, and ω =< N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 8 of 17 x, (l/0.6,m/0.6,n/0.6), (l/0.4,m/0.4,n/0.5) >. Consider G1 =< x, (l/0.6,m/0.6,n/0.6), (l/0.5,m/0.5,n/0.5) > and G2 =< x, ( l/0.4, m/0.4,n/0.5 ), ( l/0.7, m/0.7, n/0.6 ) > Here ω ∩ G ̸= 0∼. The PFDf collections of subsets of ξ are thus denoted by τ̂f = {0∼, 1∼, G1 ∪G2}, and the PFDfTS created by the topology τ on ξ is denoted by (ξ, τ, τ̂f ). Now PFDb≈f − I(C(ωc)) = 0∼ PFDb≈f − I(C(G2)) = 0∼ PFDb≈f − I(C(G1)) = 1∼ and PFDb≈f − I(C(ω)) = 1∼. So here ωc and G2 are PFDb≈f - nowhere dense sets and ω and G1 are not PFDb≈f - nowhere dense sets. Now PFDb≈f − I (ωc ∪G2) = 0∼. Hence (ξ, τ, τ̂f ) is a PFD b≈f -Baire space. Definition 27. Let (ξ, τ, τ̂f ) is a PFDfTS, then (ξ, τ, τ̂f ) is called a PFDb≈f -submaximal space if every PFDb≈f - dense set is a PFDb≈f −OS in (ξ, τ, τ̂f ) Definition 28. A PFDb≈f set ω in a PFDfTS (ξ, τ, τ̂f ) is said to be a PFDb≈f -first category if (∪∞ n=1ωn) = ω. Here ωn implies that the PFDb≈f - nowhere dense set in (ξ, τ, τ̂f ). A PFDb≈f set ω in a PFDfTS (ξ, τ, τ̂f ) which is not of the PFDb≈f -first category, is said to be of the PFDb≈f -second category. Definition 29. APFDfTS (ξ, τ, τ̂f ) is said to be a PFD b≈f -first category space if (∪∞ n=1ωn) = 1∼. Here ωn implies that the PFDb≈f − nowhere dense set in (ξ, τ, τ̂f ) If PFDfTS (ξ, τ, τ̂f ) is not the PFDb≈f -first category, then PFDfTS (ξ, τ, τ̂f ) is the PFDb≈f − second category space. Proposition 1. Every PFDb≈f −DS (ξ, τ, τ̂f ) is a PFDb≈f -submaximal space. Proof. Consider a b≈f -dense set in ( ξ, τ, τ̂f ), denoted by ω ⊆ 1∼. If ω is not PFDb≈f - OS then ω is a PFDb≈f − CS since (ξ, τ, τ̂f ) is a PFDb ≈ f −DS. Then ω = ω̄ = 1∼ and ω is a PFDb≈f − OS ( PFDb≈f − CS ) . Therefore ( ξ, τ, τ̂f ) is a PFDb≈f -submaximal space. Since not all PFDb≈f −OS are dense, with the exception of 1∼, the theorem’s counterpart need not be true. Proposition 2. PFDb≈f − CS in ( ξ, τ, τ̂f ) with PFDb≈f − I(ω) = 0∼ implies that ω is a PFDb≈f - nowhere dense set in ( ξ, τ, τ̂f ). Proof. If ω is a PFDb≈f − CS in ( ξ, τ, τ̂f ) with PFDb≈f − I(ω) = 0∼, in ( ξ, τ, τ̂f ). Then, PFDb≈f − C(ω) = ω. So PFDb≈f − I (C(ω)) = PFDb≈f − I(ω) = 0∼, in (ξ, τ, τ̂f ). Hence ω is a PFDb≈f - nowhere dense set in ( ξ, τ, τ̂f ). The converse of the theorem does not necessarily hold. Proposition 3. Every PFD subspace of a PFDb≈f −DS (ξ, τ, τ̂f , ψ) is a PFDb≈f −DS. Proof. If ( G, τ, τ̂f , ψG ) is a subspace of a PFD b≈f −DS (ξ, τ, τ̂f , ψ), implies G ⊂ ξ. Let ω ⊂ G, since ( ξ, τ, τ̂f , ψ ) is a PFDb≈f − DS, then ω is either PFDb≈f − OS or PFDb≈f − CS in (ξ, τ, τ̂f , ψ) and hence in ( G, τ, τ̂f , ψG ). Therefore, ( G, τ, τ̂f , ψG ) is a PFDb≈f −DS. N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 9 of 17 Definition 30. A PFD fTS (ξ, τ, τ̂f ) is considered a PFD b≈f D-Baire space when every PFDb≈f -first category set ω in ( ξ, τ, τ̂f ) satisfies the condition PFDb≈f - I(C(ω)) = 0∼; that is, all such sets are PFDb≈f - nowhere dense set in (ξ, τ, τ̂f ). Proposition 4. Every PFDb≈f - Baire space in ( ξ, τ, τ̂f ) is a PFDb≈f D-Baire space. Proof. Suppose that ω is a PFDb≈f - first category set in a PFDb≈f D-Baire space ( ξ, τ, τ̂f ). Then ω = (∪∞ n=1ωn) = 0∼ Here ω and ωn are PFDb≈f -nowhere dense sets in (ξ, τ, τ̂f ). So, PFDb ≈ f − I(C(ω)) = 0∼, and PFDb ≈ f − I(ω) ≤ PFDb≈f − I(C(ω)) gives that PFD b≈f − I(ω ) = 0∼. Hence PFDb≈f − I (∪∞ n=1ωn) = 0∼ where ωn implies PFDb≈f - nowhere dense sets in ( ξ, τ, τ̂f ). Hence ( ξ, τ, τ̂f ) is a PFDb≈f - Baire space. Proposition 5. Suppose that ( ξ, τ, τ̂f ) be a PFDb≈f -first category space. Therefore, ( ξ, τ, τ̂f ) is not a PFDb≈f D-Baire space. Proof. If (ξ, τ, τ̂f ) is a PFDb≈f -first category space. Hence (∪∞ n=1ωn) = 1∼. Here ωn implies PFDb≈f -nowhere dense sets in (ξ, τ, τ̂f ). We obtain PFDb≈f -I (C (1∼)) = 0∼ for the PFDb≈f -first category set 1∼. So, (ξ, τ, τ̂f ) is not a PFD b≈f D-Baire space. Definition 31. If every PFDf has two disjoint points that may be separated by PFDb≈f disjoint open sets, then a PFDfTS (ξ, τ, τ̂f , ψ) is a PFDb≈f -Hausdorff. Lemma 1. Given K = x(p, q) is a PFDfP in (E, τ, τ̂f , ψ), {K} will only be considered a PFDb≈f −OS if it is a PFDf open set. Lemma 2. Consider two PFDb≈f sets F and G in (ξ, τ, τ̂f , ψ). Then F ∩G is a PFDb≈f - OS in (ξ, τ, τ̂f , ψ) if F is a PFDb≈f −OS and F is a PFDf open set. Proposition 6. Only one limit point exists for a PFDb≈f -Hausdorff DS ( ξ, τ, τ̂f ). Proof. Consider two PFDfPs in (ξ, τ, τ̂f ) ,K = x1(p, q) and L = x2(u, v). Since (ξ, τ, τ̂f ) is a PFDb≈f -Hausdorff, two PFDb≈f −OS exist, F and G , such that K ∈ F,L ∈ G and F∩G = 0∼. Since (ξ, τ, τ̂f , ψ) is a PFDb≈f −DS, ω = (F\K)∪L is either PFDb≈f −OS or PFDb≈f − CS. Then, according to Lemma 1, ω ∩ G = {L} is PFDb≈f − OS in the preceding case, and according to Lemma 2, it is PFD b≈f − OS. In the latter instance, ωc is PFDb≈f −OS; hence, ωc ∩F = {K} is PFDb≈f −OS and consequently, PFDf open set. In the second case, ωc is PFD b≈f − OS; thus, ωc ∩ F = {K} is PFDb≈f − OS and hence PFDf open set. At least one of the two PFDfPs in (ξ, τ, τ̂f ) is therefore an isolated PFDfP in both situations, and this is demonstrated by contradiction. Definition 32. APFDfTS (ξ, τ, τ̂f , ψ) is a PFDb≈f −T1/2 space if every PFDfP {k} is either an PFDb≈f −OS or PFDb≈f − CS in (ξ, τ, τ̂f ). Proposition 7. Every PFDfTS (ξ, τ, τ̂f , ψ) is a PFDb≈f − T1/2 space. N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 10 of 17 Proof. Suppose that {K} is a PFDfP in (ξ, τ, τ̂f ). Given that all PFDf -nowhere dense sets are PFDb≈f −CS, {K} is either a PFDb≈f −OS or a PFDf -nowhere dense set. Therefore, {K} is either PFDb≈f −OS or PFDb≈f −CS. Hence (ξ, τ, τ̂f ) is a PFDb≈f −T1/2 space. Proposition 8. Let (ξ, τ, τ̂f , ψ) be a PFDfTS, The statements listed below are logically equivalent. (i) (ξ, τ, τ̂f , ψ) is a PFDf-discrete space. (ii) (ξ, τ, τ̂f , ψ) is a PFDb≈f −DS. Proof. (i) ⇒ (ii) Assume that ( ξ, τ, τ̂f , ψ ) is a PFDf discrete, then every PFDfS is a PFDf open or PFDf closed, and then PFD b≈f −OS or PFDb≈f −CS. Hence (ξ, τ, τ̂f , ψ) is a PFDb≈f −DS. (ii) ⇒ (i) If K = x(p, q) is a PFDfP in (ξ, τ, τ̂f ). Consequently, K is PFDb≈f − CS if {K} is not PFDb≈f − OS and {K} = ( PFDb≈f − C(I(K)) ∩ ( PFDb≈f − I(C(K)). Since PFDb≈f − I(K) is not empty and {K} = ( PFDb≈f − C(K) ) , then {K} = ( PFDb≈f − C(I(K)) ∩ ( PFDb≈f − I(K) ) = PFDb≈f − I({K}). In both cases, {K} is a PFDf open set. Therefore, (ξ, τ, τ̂f , ψ) is a PFDf -discrete space. Definition 33. A PFDb≈f TS (ξ, τ, τ̂f , ψ) is irreducible if every PFDb≈f set in (ξ, τ, τ̂f , ψ) is PFDb≈f - connected, equivalent to every non-void PFDb≈f − OS in (ξ, τ, τ̂f , ψ) being dense. Proposition 9. Every PFDf irreducible submaximal space ( ξ, τ, τ̂f , ψ ) is a PFDb≈f − DS. Proof. Assume that ω be a PFDb≈f set in (ξ, τ, τ̂f , ψ). Suppose ω is PFDb≈f -dense, then because (ξ, τ, τ̂f , ψ) is PFDb≈f -submaximal, ω is PFDb≈f -OS, if ω is not PFDb≈f - dense, so a non-void PFDb≈f -OS ς and ωc can be found. Since (ξ, τ, τ̂f , ψ) is irreducible, ς and ωc are PFD b≈f -dense. In addition, since ( ξ, τ, τ̂f , ψ ) is PFD b≈f -submaximal, ωc is PFDb≈f -OS or equivalently, ω is PFDb≈f − CS. Thus, in any case, ω is either PFDb≈f −OS or PFDb≈f − CS. Therefore, (ξ, τ, τ̂f , ψ) is a PFDb≈f −DS. Definition 34. An PFDfTS (ξ, τ, τ̂f ) is called PFDb ≈ f -extremely disconnected if the PFDb≈f − C of an PFDb≈f -OS is a PFDb≈f −OS. 4. Pythagorean Fuzzy Digital Fine Functions in b≈f -Door Spaces This section explores the notion of PFDb≈f -quasi compactness and discusses its key properties. Definition 35. Let σ : (ξ, ψ) → (G, η) be a PFDf function. Then, σ is PFDf -quasi compact if ω ⊆ 1∼ is PFDb≈f − OS in (ξ, ψ) such that σ−1(σ(ω)) = ω. Then, σ(ω) is PFDb≈f −OS in (G, η). Proposition 10. A PFDb≈f −DS has a topological property. N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 11 of 17 Proof. Let σ : (ξ, ψ) → (G, η) be an PFDb≈f -homeomorphism from a PFDb≈f −DS (ξ, ψ) into another PFDb≈f TS (G, η). Let ω be PFDb≈f S in (G, η ), that is, ω ⊆ G. Then, σ−1(ω ) is PFDb≈f Sin (ξ, ψ), that is, σ−1(ω) ⊆ ξ. Because (ξ, ψ) is an PFDb≈f − DS, σ−1(ω) is a PFDb≈f - OS or a PFDb≈f − CS in (ξ, ψ), because σ (( σ−1(ω) ) = ω . Then, ω is either PFDb≈f − OS or PFDb≈f −CS in (G, η ). Thus, (G, η ) is the PFDb≈f −DS. Proposition 11. A PFDf -quasi-compact image of an PFDb≈f −DS is an PFDb≈f −DS. Proof. Let σ : (ξ, ψ) → (G, η) be a PFDb≈f -quasi compact function from a PFDb≈f − DS (ξ, ψ) into another PFD b≈f TS (G, η). Let ς be PFDb≈f S in (G, η). We have to prove that ς is either PFDb≈f −OS or PFDb≈f −CS in (G, η). Because (ξ, ψ) is an PFDb≈f −DS and σ−1(ξ) = ω, then ω is either PFDb≈f −OS or PFDb≈f −CS in ( ξ, ψ ). Suppose that ω is PFDb≈f −OS, and clearly σ(ω) ⊆ ς. Thus, σ−1(σ(ω)) ⊆ σ−1(ς) = ω ⊆ σ−1(σ(ω)) or equiv- alently, ω = σ−1(σ(ω)). By assumption, σ(ω) = σ (( σ−1(ς) ) = ς ∩ σ(1∼) = ς ∩ 1∼ = ς is PFD b≈f − OS in (G, η ). We now assume that ω is PFDb≈f − CS in ( ξ, ψ ). Then, 1∼\σ−1(ζ) = σ−1(1∼\ζ) is PFDb≈f − OS in ( ξ, ψ ). Hence, (1∼\ζ) ∩ σ(1∼) = 1∼\ζ is PFDb≈f -OS in (G, η); thus, 1∼\(1∼\ζ) = ζ is PFDb≈f -CS (G, η). Therefore, (G, η) is the PFDb≈f −DS. Corollary 1. PFDb≈f -open images as well as PFDb≈f -closed images of a PFDb≈f −DS are PFDb≈f −DS. Proof. Since every PFDb≈f -OS (resp, PFDb≈f -CS) surjective function is PFDb≈f - quasicompact, it is an PFDb≈f −DS. 5. Selection of best menstrual hygiene product for Rural Women Selecting the best menstrual hygiene product for rural women serves multiple vital pur- poses that go beyond basic hygiene. First of all, it improves health by lowering the risk of diseases including recurrent and urinary tract infections, which are frequently brought on by the use of unsanitary substitutes like ash, sand, or old fabric. A suitable product also enhances a woman’s comfort and confidence, helping her manage menstruation with dignity, free from embarrassment or fear of leaks and odor. Importantly, access to proper menstrual hygiene products enables girls to continue attending school and women to par- ticipate fully in work and community life, thereby supporting their educational and eco- nomic growth. Moreover, choosing the right product contributes significantly to women’s empowerment and gender equality by encouraging open conversations about menstruation and helping women take control of their health. Environmentally, sustainable options such as reusable or biodegradable products help reduce plastic waste, which is especially cru- cial in rural areas lacking proper waste disposal systems. Economically, reusable products offer long-term cost savings, reducing reliance on continuous external aid. Additionally, promoting locally made menstrual products creates livelihood opportunities for women through community-based production, further boosting rural economies. Lastly, involving N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 12 of 17 communities in menstrual health initiatives raises awareness, dispels myths, and fosters a more supportive environment for women’s health and well-being. In this regard, the current study explores the possibilities and attempts to obtain a meaningful solution using PFDf collection sets and the PFDfTS. The structural and topological features of this proposed space are leveraged to establish a clear and coherent framework that aligns with theoretical rigor. This application illustrates how the con- struction and conditions of the PFDfTS can facilitate the systematic organization and refinement of the selection process. By anchoring the approach in a robust mathematical foundation, the study highlights the practical relevance and real-world applicability of the theoretical model, particularly in addressing the needs of rural women. Beyond the theoretical framework, the study explores the practical implementation of the PFDfTS and its associated fine collection sets. It demonstrates the critical role this space plays in identifying the most suitable menstrual hygiene product for rural women. By utilizing defined parameters, theoretical structures, and specific conditions, this approach enhances the effectiveness and precision of the selection process, as detailed in the following subsec- tions. 5.1. Algorithm of Selecting the best menstrual hygiene product using Pythagorean fuzzy digital fine topological space Step-1: Let ξ = {a} be a non empty set and τ = {0∼, 1∼, ω} is a PFD topology on ξ. Step-2: Define τ̂f = { 0∼, 1∼, ⋃ α∈J Gα } . Here the operator “ ⋃ ”denotes the union of the set of all fine collections of Gα. Then τ̂f is said to be the PFDfS of ξ. Step-3: Construct the PFDfTS ( ξ, τ, τ̂f ) generated by the topology τ on ξ. Step-4: Assume the weight vector w = (w1,w2,w3, . . . . . . ,wn) T the PFDfNs such that wi(i = 1, 2, 3, . . .n) ∈ [0, 1] and ∑n i=1wi = 1 corresponding to the six criteria afford- ability and accessibility, cultural acceptance and awareness, hygiene and health Impact, product availability and supply consistency, comfort and ease of use, and disposal and waste management. Step-5: Calculate the score function using PFWA operator formula PFWA (I1, I2, I3, . . . Jn) = 〈√ 1− ∏n i=1 ( 1− µ2i )wi , ∏n i=1 ϑ wi i 〉 , PFWG operator formula PFWG ( I1, J2, J3, . . . , In) = 〈∏n i=1 µ wi i , √ 1− ∏n i=1 ( 1− ϑ2i )wi 〉 and select the best menstrual hygiene product. 5.2. Proposed study to identify the best menstrual hygiene product for Rural Women using Pythagorean fuzzy digital fine topological space In order to make this process, consider three different menstrual hygiene products as cloth pads, menstrual cups and sanitary pads say H = {H1,H2,H3} and the six factors C = {C1,C2,C3,C4,C5,C6}. The selected evaluation criteria affordability and accessibility, N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 13 of 17 cultural acceptance and awareness, hygiene and health Impact, product availability and supply consistency, comfort and ease of use, and disposal and waste management serve as comprehensive and practical dimensions for assessing menstrual hygiene products, par- ticularly in rural and under-resourced settings. These parameters collectively address essential aspects such as economic feasibility, sociocultural acceptability, health consider- ations, supply reliability, user experience, and environmental impact. Affordability and accessibility reflect the importance of cost and product reach. Cultural acceptance and awareness consider societal norms and awareness levels that influence product adoption. Hygiene and health impact captures the sanitary and safety implications of use. Product availability and supply consistency ensures the reliability of access over time. Comfort and ease of use examines user-friendliness and physical comfort. Disposal and waste man- agement accounts for environmental concerns and the practical aspects of waste handling. Together, these parameters offer a balanced and realistic framework for decision-making. In this study, both membership and non-membership values are assigned to each prod- uct based on these six criteria. These values are subjectively assigned by the authors based on an analysis of common product attributes, user preferences, and practical challenges en- countered in everyday life. For instance, cloth pads are considered economically affordable and culturally acceptable, which leads to higher membership and lower non-membership values in those categories. However, they score lower in terms of hygiene and availability. Menstrual cups exhibit high membership values in hygiene and long-term affordability but face lower cultural acceptance and usability, resulting in higher non-membership in those areas. Sanitary pads generally show moderate performance across most criteria but receive low scores in waste management due to their environmental impact. These values conform to the PFS conditions and serve as input for evaluating the suitability of each product using PFWA and PFWG operators in the proposed model. The following are the analyses performed to discover the best alternative among the possible ones utilising the PFWA and PFWG operators stated in definition 20. Step 1: Let ξ = {a} be a non empty set and τ = {0∼, 1∼, ω} is a PFD topology on ξ where 0∼ = {(0, 1)}, 1∼ = {(1, 0)} and ω = {(0.21, 0.97)} where ω is a PFD open set on ξ. Step 2: Define τ(ω) = τ̂α = { Gα (̸= 1∼) ;Gα ∩ ω ̸= 0∼, for ω ∈ τ ω ̸= 0∼ for some α ∈ J, J is an index set. • ConsiderG1 = {(0.34, 0.84)},G2 = {(0.38, 0.72)},G3 = {(0.46, 0.77)}, G4 = {(0.71, 0.42)}, G5 = {(0.91, 0.26)},G6 = {(0.28, 0.83)},G7 = { (0.44, 0.85)},G8 = {(0.61, 0.66)}, G9 = {(0.75, 0.72)},G10 = { (0.54, 0.77)},G11 = {(0.49, 0.63)},G12 = {(0.58, 0.61)}, G13 = { (0.67, 0.51)},G14 = {(0.62, 0.75)},G15 = {(0.73, 0.56)} be the PFDS’s. • So the PFDf collections are Gα = {(0.34, 0.84), (0.38,0.72), (0.46,0.77), (0.71,0.42), (0.91,0.26), (0.28,0.83), (0.44,0.85), (0.61,0.66), (0.75,0.72), (0.54,0.77), (0.49,0.63), (0.58, 0.61), (0.67, 0.51), (0.62, 0.75), (0.73, 0.56)} which satisfies the condition Gα ∩ ω ̸= 0∼. N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 14 of 17 • Hence τ̂f = {0∼, 1∼,Gα} is called the PFDf collections of subsets of ξ and ( ξ, τ, τ̂f ) is said to be the PFDfTS generated by the topology τ on ξ. Step 3: All the PFDf subsets in the PFDfT on ξ can be represented using the MS and NMS values corresponding to the parameters C1,C2,C3,C4,C5, and C6. These parame- ters denote affordability and accessibility, cultural acceptance and awareness, hygiene and health Impact, product availability and supply consistency, comfort and ease of use, and disposal and waste management respectively. The values for C1 to C6 for the products are assigned randomly, based on the characteristics described by these parameters. For instance, in terms of affordability, menstrual cups are generally less affordable than cloth pads for women in rural areas. Therefore, the MS value for the affordability of men- strual cups is taken as 0.44, which is lower than that of cloth pads. Similarly, for cultural awareness, the MS value of cloth pads is higher than that of sanitary pads, which in turn is higher than that of menstrual cups. Following this rationale, the PFDf subsets are assigned values as illustrated in the Table 2. Table 2: Criteria wise membership and non-membership values for menstrual hygiene products C1 C2 C3 C4 C5 C6 H1 (Cloth pads) (0.91,0.26) (0.71,0.42) (0.38,0.72) (0.21,0.97) (0.46,0.77) (0.34,0.84) H2 (Menstrual cups) (0.44,0.85) (0.28,0.83) (1, 0) (0.61,0.66) (0.75,0.72) (0.54,0.77) H3 (Sanitary pads) (0.49,0.63) (0.58,0.61) (0.67,0.51) (0.73,0.56) (0.62,0.75) (0, 1) Step 4: The weight vector corresponding to these six criteria C = {C1,C2,C3,C4,C5,C6} is w = {0.17, 0.24, 0.3, 0.04, 0.15, 0.1}T where wi ∈ [0, 1] and ∑n i=1wi = 1. Step 5: To determine the most suitable product among cloth pads, menstrual cups, and sanitary pads, the scoring function values must be calculated using the formula pro- vided in Equation 2.20. PFWA calculation: PFwA(J1, J2, J3,..., Jn) = 〈√ 1− ∏n i=1 ( 1− µ2i )wi , ∏n i=1 ϑi wi 〉 The Score values corresponding to them are SC (H1) = −0.136, SC (H2) = 1.000 and SC (H3) = 0.009. Since SC (H2) > SC (H3) > SC (H1), we have H2 > H3 > H1. Hence H2 (Menstrual cups) is the greatest product for rural women’s menstruation hygiene. PFWG calculation: PFWG(I1, I2, I3, . . . . . . .., In) = 〈∏n i=1 µ wi i , √ 1− ∏n i=1 ( 1− ϑ2i )wi 〉 The Score values corresponding to them are SC (H1) = −0.627, SC (H2) = −0.399 and SC (H3) = -1.000. Since SC (H2) > SC (H1) > SC (H3), we have H2 > H1 > H3. Hence H2 (Menstrual cups) is the is the greatest product for rural women’s menstruation hygiene. The fuzzy topological framework adopted in this study offers a structured and action- able basis for real world decision making in the domain of menstrual hygiene management. N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 15 of 17 By utilizing the score outputs derived from PFWA and PFWG operators, policymakers can make informed decisions regarding resource allocation and program design. For in- stance, if the model identifies menstrual cups as the most suitable product based on the considered criteria, interventions can focus on subsidizing these products and organizing targeted distribution and training programs to ensure cost effectiveness and community acceptance. Furthermore, the inclusion of sociocultural factors such as cultural accep- tance and awareness within the model enables local health educators to pinpoint barriers to adoption. This facilitates the development of tailored educational campaigns to pro- mote lesser known but more beneficial products. Additionally, the model’s adaptability allows for regular updates using real time feedback from the field, making it a dynamic tool for continuous evaluation and policy refinement. In this way, the proposed fuzzy topological framework serves not only as an evaluative mechanism but also as a practical decision support system for stakeholders working to improve menstrual health outcomes. 6. Validation In [16], the authors indicate that menstruation cups surpass sanitary pads in terms of comfort, leakage protection, capacity, and discretion. Menstrual cups and sanitary pads are comparable in terms of accessibility and usability. Higher scores were observed for menstrual cups regarding skin rash prevention and reduced frequency of change compared to sanitary pads. Overall, menstrual cups demonstrate significant improvement with each menstrual cycle when compared to sanitary pads. However, sanitary pads are still sug- gested during the first few cycles of menstrual cup use. Notably, all performance scores show improvement as users become more accustomed to menstrual cups. Further research is warranted to evaluate cost-effectiveness and effectiveness in specific subgroups, such as individuals with menorrhagia. Promoting awareness and educating the population on the environmental benefits and initial learning curve associated with menstrual cup use is essential. Accordingly, the current study conveys the concept through a topological framework in a clear and simplified way. 7. Conclusion Utilizing the PFDb≈f S as a basis, we proposed the notion of PFDb≈f − DS in this study and studied its features. Furthermore, we provided the ideas of PFDb≈f -Baire space and PFDb≈f -submaximal space. The characteristics of the PFDb≈f − DS have been defined. This relate with various PFDb≈f -spaces was shown, and the results were interpreted appropriately. In addition, it is demonstrated that a quasi compact image of a PFDb≈f −DS is a PFDb≈f −DS and that a PFDb≈f −DS has a topological property. Moreover, the framework and conditions of the PFDfTS, together with its associated fine collection sets, are systematically utilized to determine the most appropriate menstrual hygiene product for women in rural areas. In rural areas, cultural awareness regarding the use of menstrual cups remains relatively low. Many women are unfamiliar with menstrual cups due to limited access to menstrual N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 16 of 17 health education and prevailing cultural taboos surrounding menstruation. Traditional practices, such as the use of cloth pads, are deeply rooted and often preferred, while myths and misconceptions-particularly concerning virginity, insertion, and safety create additional barriers to acceptance. The resistance to change is often reinforced by genera- tional norms and a lack of open dialogue about menstrual health. Consequently, menstrual cups are perceived as unfamiliar and intimidating. To improve awareness and acceptance, targeted education initiatives and awareness programs are essential. These efforts should include hands-on demonstrations, involvement of local health workers, and the support of community leaders to dispel myths and encourage cultural acceptance of menstrual cups as a safe, sustainable, and effective menstrual hygiene option. The present work explains the selection of the most suitable menstrual hygiene product using PFDfT . This topological approach provides a structured framework, and in the future, it can be applied to real-time data collected from specific rural areas to identify the most appropriate product based on local conditions and preferences. Disclosure Statement The author(s) disclosed no potential conflicts of interest. References [1] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. [2] K. M. Ordenshiya and G. K. Revathi. Enhanced air quality index prediction using fuzzy center merge labeling graph based fuzzy inference system model. Cybernetics and Systems, pages 1–43, 2025. [3] Harish Garg. Confidence levels based pythagorean fuzzy aggregation operators and its application to decision-making process. Computational and Mathematical Organi- zation Theory, 23(4):546–571, 2017. [4] R. R. Yager. Pythagorean fuzzy subsets. In Proc Joint IFSA World Congress and NAFIPS Annual Meeting, 2013. [5] A. Rosenfeld. Fuzzy digital topology. Information and Control, 40(1):76–87, 1979. [6] Ankita Shahasane, Richa Singh, and Shruti Ugran. Menstrual cup versus sanitary cup in menstrual hygiene: A prospective, randomised, two-way crossover study. In- ternational Journal of Pharmaceutical and Clinical Research, 15(12):1536–1542, 2023. [7] R. Usha Parameswari and P. Thangavelu. On b#-open sets. International Journal of Mathematics Trends and Technology, 5:202–218, 2014. [8] S. Anjalmose and G. Thangaraj. Inter relations between fuzzy door space and some fuzzy topological spaces. International Journal of Mathematics And its Applications, 4(4):129–136, 2016. [9] Dimitrije Andrijević. On b-open sets. Matematički Vesnik, (205):59–64, 1996. [10] N. Preethi and G. K. Revathi. A conceptual view on pfd functions and its properties. Test Engineering and Management, 83:20050–20056, 2020. [11] Abdul Gawad A. Q. Al-Qubati and Mohamed El Sayed. Door spaces in intuitionistic N. Preethi, G. K. Revathi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6598 17 of 17 fuzzy topological spaces. International Journal of Fuzzy Logic and Intelligent Systems, 22(3):296–302, 2022. [12] Murat Olgun, Mehmet Unver, and Seyhmus Yardımcı. Pythagorean fuzzy topological spaces. Complex & Intelligent Systems, 5(2):177–183, 2019. [13] T. Yung Kong and A. Rosenfeld. Digital topology: Introduction and survey. Computer Vision, Graphics, and Image Processing, 48(3):357–393, 1989. [14] A. Rosenfeld. Digital topology. The American Mathematical Monthly, 86(8):621–630, 1979. [15] L. Vidyarani and A. G. Rose Venish. Decomposition of intuitionistic fine closed sets in intuitionistic fine topological spaces. Advances and Applications in Mathematical Sciences, 22:2243–2254, 2023. [16] K. T. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1):87–96, 1986.