Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 87 https://internationalpubls.com Novel Picture of Topological Spaces in the Frame of Fuzzy Soft Sequential Sets *K.Japhia Tino Mercy1, K.Bageerathi2 1 Research Scholar (Registration Number: 22122022092006), PG and Research Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur, Affiliated to Manonmaniam Sundaranar University, Abishekapatti,Tirunelveli-627 012. 2 Assistant Professor, PG and Research Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur. E-mail Id: 1jaffy306@gmail.com,2bageerathi14@gmail.com Article History: Received: 20-04-2024 Revised: 10-06-2024 Accepted: 24-06-2024 Abstract: The goal of this paper is to introduce fuzzy soft sequential topological spaces over the universal set via fuzzy soft sequential set by merging the concept of fuzzy soft set and sequential set. We provide a comprehensive study of their properties and explore a relationship between fuzzy soft topology and fuzzy soft sequential topology. We offers a new outlook on the theory of fuzzy soft sequential neighborhood of a fuzzy soft sequential set and we investigate the quasi-coincident relation. Keywords: fuzzy soft set, fuzzy soft sequential set, fuzzy soft sequential topological spaces, fuzzy soft sequential neighborhood system, fuzzy soft sequential quasi-coincident 1. Introduction The concept of fuzzy sets by defining them in terms of mappings from a set into the unit interval on the real line was proposed by L.A.Zadeh [18] in 1965, which is dealing with uncertainty and representing vague concepts. In 1968, C. L. Chang [4] initiated fuzzy topological spaces as an extension to classical topological spaces. In 1999, Molodtsov [11] introduced the concept of soft set theory, which provides a new mathematical theory for dealing with uncertainty. In 2003, Maji et al [10] defined and studied the theory of soft sets which is used to construct new soft sets from the given soft sets. In 2011, Shabir and Naz [14] introduced soft topological spaces which are defined over an initial universe with a fixed set of parameters. Further in the same year, Hussain and Ahmad [6] extended the soft topology and strengthen the foundations of the theory of soft topological spaces. Also Cagman, Enginoglu and Citak [3] investigated soft topology and its properties. In recent years, the researchers have contributed a lot towards fuzzification of soft set theory. In 2001, Maji et al [9] proposed the concept of fuzzy soft set which is a new mathematical approach to vagueness by involving the ideas of both fuzzy sets and soft sets. Fuzzy soft sets was further revised and improved by Ahmad and Kharal [1] in the year 2009. In 2011, Tanay and Kandemir [16] introduced the topological structure of fuzzy soft sets and studied some of its structural properties. Based on this theory many researchers like Roy and Samanta [12] in the year 2012, Atmaca and Zorlutuna [13] in the year 2013, Mahanta and Das [8] in the year 2018 studied and investigated many properties in fuzzy soft topology. In 2002, Bose and Indrajit Lahiri [2] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 88 https://internationalpubls.com introduced the concept of sequential topological spaces. He defined any sequence of subsets of a non void set is called a sequential set. To go along this line of research, we introduce sequence of fuzzy soft set and named as fuzzy soft sequential set. In this paper, we extend the concept of topological space namely fuzzy soft sequential topological space and analyze their properties. 2. Preliminaries Definition 2.1 [18] A fuzzy set 𝑋 over a universal set π‘ˆ is a set defined by a function πœ‡π‘₯ performing a mapping πœ‡π‘₯ ∢ π‘ˆ β†’ [0,1] , here this πœ‡π‘₯ is the membership function of 𝑋, and the value πœ‡π‘₯(𝑒) will be the grade of membership of u ∈ π‘ˆ. Definition 2.2 [11] Let π‘ˆ be an initial universe set and 𝐸 be a set of parameters. A pair (𝐹, 𝐸) is called a soft set over π‘ˆ if and only if 𝐹 is a mapping of 𝐸 into the set of all subsets of the set π‘ˆ. Definition 2.3 [9] Let π‘ˆ be a common universe, 𝐸 be a set of parameters and 𝐴 βŠ† 𝐸. A pair (𝐹, 𝐴) is called a fuzzy soft set over π‘ˆ, where 𝐹 is a mapping given by 𝐹: 𝐴 β†’ πœ‡π‘₯(π‘ˆ) , where πœ‡π‘₯(π‘ˆ) denote the set of all fuzzy subsets of π‘ˆ. Definition 2.4 [16] Let π‘ˆ be a common universe, 𝐸 be a set of parameters and 𝐴, 𝐡 βŠ† 𝐸. Let (𝐹, 𝐴) , (𝐺, 𝐡) be an element of 𝐹𝑆(π‘ˆ, 𝐸), (Briefly, family of all fuzzy soft subsets) and οΏ½ΜƒοΏ½ be a subfamily of 𝐹𝑆(π‘ˆ, 𝐸). Then οΏ½ΜƒοΏ½ is called a fuzzy soft topology on π‘ˆ if the following conditions are satisfied: (i) βˆ…Μƒ , π‘ˆ ∈ οΏ½ΜƒοΏ½ (ii) (𝐹, 𝐴) , (𝐺, 𝐡) ∈ οΏ½ΜƒοΏ½ β‡’ (𝐹, 𝐴) βˆ©Μƒ (𝐺, 𝐡) ∈ οΏ½ΜƒοΏ½ (iii){(𝐹, 𝐴)π‘˜|π‘˜ ∈ 𝐾} βŠ‚ οΏ½ΜƒοΏ½ β‡’ β‹ƒΜƒπ‘˜βˆˆπΎ(𝐹, 𝐴)π‘˜ ∈ οΏ½ΜƒοΏ½. The pair (π‘ˆ, οΏ½ΜƒοΏ½) is called a fuzzy soft topological space. Definition 2.5 [13] Let (𝐹, 𝐴) , (𝐺, 𝐡) ∈ 𝐹𝑆(π‘ˆ, 𝐸). (𝐹, 𝐴) is said to be fuzzy soft quasi-coincident with (𝐺, 𝐡), denoted by (𝐹, 𝐴) π‘ž(𝐺, 𝐡), if there exists an 𝑒 ∈ 𝐸, π‘₯ ∈ 𝑋 such that 𝐹(𝑒)(π‘₯) + 𝐺(𝑒)(π‘₯) > 1. If (𝐹, 𝐴) is not soft quasi-coincident with (𝐺, 𝐡), then we write (𝐹, 𝐴) οΏ½Μ…οΏ½(𝐺, 𝐡). Definition 2.6 [18] 𝐴 is contained in 𝐡 (or, equivalently, 𝐴 is a subset of 𝐡, or 𝐴 is smaller than or equal to 𝐡) if and only if πœ‡π΄ ≦ πœ‡π΅. In symbols, π΄βŠ‚π΅ ⇔ πœ‡π΄ ≦ πœ‡π΅. Definition 2.7 [2] Any sequence of subsets of a non void set 𝑋 is called a sequential set in 𝑋. That is, 𝐴(𝑠) = {𝐴𝑛}𝑛=1 ∞ , where each 𝐴𝑛 is a subset of 𝑋, is a sequential set in 𝑋. The subsets 𝐴𝑛, 𝑛 ∈ β„• are called the components of 𝐴(𝑠). Definition 2.8[7] A sequence of fuzzy soft sets is a mapping from β„• to the family of all fuzzy soft sets and is denoted by {(𝐹, 𝐴)𝑛} or {(𝐹, 𝐴)𝑛; 𝑛 = 1,2, … }.That is, {(𝐹, 𝐴)𝑛, 𝑛 ∈ β„•} where (𝐹, 𝐴)𝑛 for each 𝑛 ∈ β„• represents components of fuzzy soft set in {(𝐹, 𝐴)𝑛} and 𝑛 ∈ β„•, the set of all natural numbers. A sequence of fuzzy soft sets is called fuzzy soft sequential set. 3. Main Results Definition 3.1. A family οΏ½ΜƒοΏ½ of fuzzy soft sequential sets on π‘ˆ satisfying the properties: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 89 https://internationalpubls.com (i) βˆ…Μƒ, οΏ½ΜƒοΏ½ ∈ οΏ½ΜƒοΏ½ (ii) {(𝐹, 𝐴)𝑛}, {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½ β‡’ {(𝐹, 𝐴)𝑛} βˆ©Μƒ {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½ and (iii) For any family {{(πΉπœ†, 𝐴)𝑛} , πœ† ∈ Ξ›} ∈ οΏ½ΜƒοΏ½ β‡’ {(πΉπœ†, 𝐴)𝑛} ∈ οΏ½ΜƒοΏ½πœ†βˆˆΞ› βˆͺΜƒ is called a fuzzy soft sequential topology on π‘ˆ and the triplet (π‘ˆ, οΏ½ΜƒοΏ½, 𝐸) is called a fuzzy soft sequential topological space over π‘ˆ (briefly, 𝐹𝑆𝑆𝑇𝑆). Definition 3.2. Let (π‘ˆ, οΏ½ΜƒοΏ½, 𝐸) be a fuzzy soft sequential topology on π‘ˆ. Then the members of οΏ½ΜƒοΏ½ are called fuzzy soft sequential open sets (𝐹𝑆𝑆-open sets). The Complement of a fuzzy soft sequential open set is called fuzzy soft sequential closed set (𝐹𝑆𝑆 -closed set). Example 3.3. Let us consider the universe π‘ˆ = {π‘₯1, π‘₯2} and 𝐴 = {𝑒1, 𝑒2} βŠ‚Μƒ 𝐸 where {𝑒1, 𝑒2} be a collection of sets of parameters. Let us take a fuzzy soft sequential topological space (π‘ˆ, οΏ½ΜƒοΏ½, 𝐸) where οΏ½ΜƒοΏ½ = {βˆ…Μƒ, οΏ½ΜƒοΏ½, {(𝐹, 𝐴)𝑛}}. The nth component of the fuzzy soft sequential set {(𝐹, 𝐴)𝑛} is defined as (𝐹, 𝐴)𝑛 = { (𝑒1, { π‘₯1 (1 3𝑛)⁄ , π‘₯2 (𝑛 𝑛+2)⁄ }) , (𝑒2, { π‘₯1 (2 5𝑛)⁄ , π‘₯2 (1 𝑛)⁄ })} . Then οΏ½ΜƒοΏ½ is a fuzzy soft sequential topology on π‘ˆ. Definition 3.4. Let π‘ˆ be the universal set and 𝐸 be a set of parameters and οΏ½ΜƒοΏ½ = {βˆ…Μƒ, οΏ½ΜƒοΏ½}. Then οΏ½ΜƒοΏ½ is called the fuzzy soft sequential indiscrete topology on π‘ˆ. Definition 3.5. Let π‘ˆ be the universal set and E be a set of parameters and let οΏ½ΜƒοΏ½ be the collection of all fuzzy soft sequential sets which can be defined over π‘ˆ. Then οΏ½ΜƒοΏ½ is called the fuzzy soft sequential discrete topology on π‘ˆ. Definition 3.6. {(𝐹, 𝐴)𝑛} is contained in {(𝐺, 𝐡)𝑛}, symbolically {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐺, 𝐡)𝑛}, if and only if for each 𝑒𝑖 ∈ 𝐴, 𝐡 and for all π‘₯𝑗 ∈ π‘ˆ, 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) ≀̃ 𝐺𝑛(𝑒𝑖)(π‘₯𝑗) for all 𝑛 ∈ β„• . Definition 3.7. {(𝐹, 𝐴)𝑛} is weakly contained in {(𝐺, 𝐡)𝑛}, symbolically {(𝐹, 𝐴)𝑛} βŠ‚Μƒπ‘€ {(𝐺, 𝐡)𝑛}, if and only if for each 𝑒𝑖 ∈ 𝐴, 𝐡 and for all π‘₯𝑗 ∈ π‘ˆ, there exists 𝑛 ∈ β„• such that 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) ≀̃ 𝐺𝑛(𝑒𝑖)(π‘₯𝑗). Definition 3.8. Let (π‘ˆ, οΏ½ΜƒοΏ½1, 𝐸) and (π‘ˆ, οΏ½ΜƒοΏ½2, 𝐸) be two fuzzy soft sequential topological spaces. Then the following holds: (i) If οΏ½ΜƒοΏ½1 βŠ‚Μƒ οΏ½ΜƒοΏ½2 , then οΏ½ΜƒοΏ½2 is fuzzy soft sequential finer than οΏ½ΜƒοΏ½1 or οΏ½ΜƒοΏ½1 is fuzzy soft sequential coarser than οΏ½ΜƒοΏ½2. (ii) If either οΏ½ΜƒοΏ½1 βŠ‚Μƒ οΏ½ΜƒοΏ½2 or οΏ½ΜƒοΏ½2 βŠ‚Μƒ οΏ½ΜƒοΏ½1, then οΏ½ΜƒοΏ½1 is comparable with οΏ½ΜƒοΏ½2. Theorem 3.9 Let (π‘ˆ, οΏ½ΜƒοΏ½1, 𝐸) and (π‘ˆ, οΏ½ΜƒοΏ½2, 𝐸) be two fuzzy soft sequential topological spaces over π‘ˆ, then (π‘ˆ, οΏ½ΜƒοΏ½1 βˆ©Μƒ οΏ½ΜƒοΏ½2, 𝐸) is a fuzzy soft sequential topological space overπ‘ˆ. Proof. (i) βˆ…Μƒ, οΏ½ΜƒοΏ½ ∈ οΏ½ΜƒοΏ½1 βˆ©Μƒ οΏ½ΜƒοΏ½2. (ii) Let the two fuzzy soft sequential sets {(𝐹, 𝐴)𝑛}, {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½1 βˆ©Μƒ οΏ½ΜƒοΏ½2. Then {(𝐹, 𝐴)𝑛}, {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½1 and {(𝐹, 𝐴)𝑛}, {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½2. Therefore {(𝐹, 𝐴)𝑛} βˆ©Μƒ {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½1 and {(𝐹, 𝐴)𝑛} βˆ©Μƒ {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½2. Hence {(𝐹, 𝐴)𝑛} βˆ©Μƒ {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½1 βˆ©Μƒ οΏ½ΜƒοΏ½2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 90 https://internationalpubls.com (iii) Let {{(πΉπœ†, 𝐴)𝑛} , πœ† ∈ Ξ›} be a family of fuzzy soft sequential sets in οΏ½ΜƒοΏ½1 βˆ©Μƒ οΏ½ΜƒοΏ½2. Then {(πΉπœ†, 𝐴)𝑛} ∈ οΏ½ΜƒοΏ½1 and {(πΉπœ†, 𝐴)𝑛} ∈ οΏ½ΜƒοΏ½2 βˆ€πœ† ∈ Ξ›. Therefore {(πΉπœ†, 𝐴)𝑛}πœ†βˆˆΞ› βˆͺΜƒ ∈ οΏ½ΜƒοΏ½1 and {(πΉπœ†, 𝐴)𝑛}πœ†βˆˆΞ› βˆͺΜƒ ∈ οΏ½ΜƒοΏ½2. Hence {(πΉπœ†, 𝐴)𝑛}πœ†βˆˆΞ› βˆͺΜƒ ∈ οΏ½ΜƒοΏ½1 βˆ©Μƒ οΏ½ΜƒοΏ½2. Remark 3.10. Let (π‘ˆ, οΏ½ΜƒοΏ½1, 𝐸) and (π‘ˆ, οΏ½ΜƒοΏ½2, 𝐸) be two fuzzy soft sequential topological spaces over π‘ˆ, then (π‘ˆ, οΏ½ΜƒοΏ½1 βˆͺΜƒ οΏ½ΜƒοΏ½2, 𝐸) need not be a fuzzy soft sequential topological space over π‘ˆ. Proposition 3.11. If οΏ½ΜƒοΏ½ is a fuzzy soft topology on π‘ˆ, then οΏ½ΜƒοΏ½β„• forms a fuzzy soft sequential topology on π‘ˆ. Proof. Let οΏ½ΜƒοΏ½ be a fuzzy soft topology on π‘ˆ. (i) βˆ…Μƒ, οΏ½ΜƒοΏ½ ∈ οΏ½ΜƒοΏ½β„•, Since βˆ…Μƒ, π‘ˆ ∈ οΏ½ΜƒοΏ½. (ii) Let {(𝐹, 𝐴)𝑛}, {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½β„•. Then (𝐹, 𝐴)𝑛,(𝐺, 𝐡)𝑛 ∈ οΏ½ΜƒοΏ½ for all 𝑛 ∈ β„•. Thus (𝐹, 𝐴)𝑛 βˆ©Μƒ (𝐺, 𝐡)𝑛 ∈ οΏ½ΜƒοΏ½ for all 𝑛 ∈ β„•. Thus {(𝐹, 𝐴)𝑛} βˆ©Μƒ {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½β„•. (iii)Let {{(πΉπœ†, 𝐴)𝑛} ∈ οΏ½ΜƒοΏ½ , πœ† ∈ Ξ›} be a family of fuzzy soft sequential sets in οΏ½ΜƒοΏ½β„•. Then for each 𝑛 ∈ β„•, (𝐹, 𝐴)πœ† ∈ οΏ½ΜƒοΏ½ π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ πœ† ∈ Ξ›. Thus πœ†βˆˆΞ› βˆͺΜƒ (𝐹, 𝐴)πœ† ∈ οΏ½ΜƒοΏ½ . Hence {(πΉπœ†, 𝐴)𝑛}πœ†βˆˆΞ› βˆͺΜƒ ∈ οΏ½ΜƒοΏ½β„•. Proposition 3.12. Any fuzzy soft sequential topological space induces fuzzy soft topological space. Proof. Let οΏ½ΜƒοΏ½ be a fuzzy soft sequential topology, οΏ½ΜƒοΏ½1 be a fuzzy soft topology and 𝑛 ∈ β„•, the set of all natural numbers. Then (i) βˆ…Μƒ, π‘ˆ ∈ οΏ½ΜƒοΏ½1, since βˆ…Μƒ, οΏ½ΜƒοΏ½ ∈ οΏ½ΜƒοΏ½. (ii) Let (𝐹, 𝐴), (𝐺, 𝐡) ∈ οΏ½ΜƒοΏ½1. Then there exists fuzzy soft sequential sets {(𝐹, 𝐴)𝑛},{(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½ such that (𝐹, 𝐴)𝑛 = (𝐹, 𝐴) and (𝐺, 𝐡)𝑛 = (𝐺, 𝐡). Since {(𝐹, 𝐴)𝑛} βˆ©Μƒ {(𝐺, 𝐡)𝑛} ∈ οΏ½ΜƒοΏ½ implies (𝐹, 𝐴) βˆ©Μƒ (𝐺, 𝐡) ∈ οΏ½ΜƒοΏ½1. (iii)Let {(𝐹, 𝐴)πœ† βˆ• πœ† ∈ Ξ›} be a family of fuzzy soft open sets. Then for each πœ† ∈ Ξ›, there exists a fuzzy soft sequential set {{(πΉπœ†, 𝐴)𝑛} βˆ• πœ† ∈ Ξ›} ∈ οΏ½ΜƒοΏ½ having nth component (𝐹, 𝐴)πœ†. Since {(πΉπœ†, 𝐴)𝑛}πœ†βˆˆΞ› βˆͺΜƒ ∈ οΏ½ΜƒοΏ½, πœ†βˆˆΞ› βˆͺΜƒ (𝐹, 𝐴)πœ† ∈ οΏ½ΜƒοΏ½1. Definition 3.13. (π‘ˆ, οΏ½ΜƒοΏ½) in the Proposition 3.12, is called the nth component fuzzy soft topological space of the fuzzy soft sequential topological space (π‘ˆ, οΏ½ΜƒοΏ½ , 𝐸). Proposition 3.14. Let {(𝐹, 𝐴)𝑛} be a fuzzy soft sequential open (closed) set in fuzzy soft sequential topological space (π‘ˆ, οΏ½ΜƒοΏ½ , 𝐸). Then for each 𝑛 ∈ β„•, (𝐹, 𝐴)𝑛 is fuzzy soft open (closed) set in (π‘ˆ, οΏ½ΜƒοΏ½). Proof. Let {(𝐹, 𝐴)𝑛} be a fuzzy soft sequential open set in fuzzy soft sequential topological space(π‘ˆ, οΏ½ΜƒοΏ½ , 𝐸) and (𝐹, 𝐴)𝑛 be a fuzzy soft open set in (π‘ˆ, οΏ½ΜƒοΏ½). From the Proposition 3.12, fuzzy soft topologies are induced by fuzzy soft sequential topologies and any fuzzy soft topology οΏ½ΜƒοΏ½ on (π‘ˆ, οΏ½ΜƒοΏ½) can be considered as a component of fuzzy soft sequential topology οΏ½ΜƒοΏ½ on π‘ˆ. Hence we get, for every 𝑛 ∈ β„•, (𝐹, 𝐴)𝑛 is fuzzy soft open set in (π‘ˆ, οΏ½ΜƒοΏ½). The following example shows that, the converse of the above proposition need not be true. Example 3.15. Example 3.3 shows that, {(𝐹, 𝐴)𝑛} is a fuzzy soft sequential open set in (π‘ˆ, οΏ½ΜƒοΏ½ , 𝐸) implies (𝐹, 𝐴)1, (𝐹, 𝐴)2 ,….. (𝐹, 𝐴)𝑛,….. is fuzzy soft open set in (π‘ˆ, οΏ½ΜƒοΏ½). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 91 https://internationalpubls.com Now, let us take, for each 𝑛 ∈ β„•, (𝐹, 𝐴)𝑛 is fuzzy soft open set in (π‘ˆ, οΏ½ΜƒοΏ½) implies {(𝐹, 𝐴)𝑛} is not a fuzzy soft sequential open set in (π‘ˆ, οΏ½ΜƒοΏ½ , 𝐸). Theorem 3.16. Arbitrary intersection of fuzzy soft sequential closed sets is fuzzy soft sequential closed set. Proof. Let {{(πΉπœ†, 𝐴)𝑛} βˆ• πœ† ∈ Ξ›} be an arbitrary collection of fuzzy soft sequential closed sets. We take {(𝐹, 𝐴)𝑛} = {(πΉπœ†, 𝐴)𝑛}πœ†βˆˆΞ› βˆ©Μƒ . Then π‘ˆ βˆ’ {(𝐹, 𝐴)𝑛} = π‘ˆ βˆ’ {(πΉπœ†, 𝐴)𝑛} =πœ†βˆˆΞ› βˆ©Μƒ (π‘ˆ βˆ’ {(πΉπœ†, 𝐴)𝑛})πœ†βˆˆΞ› βˆͺΜƒ (by De Morgan’s law).Since {(πΉπœ†, 𝐴)𝑛} is fuzzy soft sequential closed set in π‘ˆ for all πœ† ∈ Ξ›, π‘ˆ βˆ’ {(πΉπœ†, 𝐴)𝑛} is fuzzy soft sequential open set in π‘ˆ. Since arbitrary union of fuzzy soft sequential open set is fuzzy soft sequential open, π‘ˆ βˆ’ {(𝐹, 𝐴)𝑛} is fuzzy soft sequential open set. Hence {(𝐹, 𝐴)𝑛} is fuzzy soft sequential closed set. Theorem 3.17. Finite union of fuzzy soft sequential closed sets are fuzzy soft sequential closed. Proof. Let {(𝐹𝑖, 𝐴)𝑛}𝑖=1 π‘˜ be a finite collection of fuzzy soft sequential closed sets. Let {(𝐹, 𝐴)𝑛} = βˆͺ̃𝑖=1 π‘˜ {(𝐹𝑖, 𝐴)𝑛} . Then π‘ˆ βˆ’ {(𝐹, 𝐴)𝑛} = π‘ˆ βˆ’βˆͺ̃𝑖=1 π‘˜ {(𝐹𝑖, 𝐴)𝑛} =βˆ©Μƒπ‘–=1 π‘˜ (π‘ˆ βˆ’ {(𝐹𝑖, 𝐴)𝑛} (by De Morgan’s law). Since {(𝐹𝑖, 𝐴)𝑛} is fuzzy soft sequential closed set in π‘ˆ, π‘ˆ βˆ’ {(𝐹𝑖, 𝐴)𝑛} fuzzy soft sequential open set in π‘ˆ. Since finite intersection of fuzzy soft sequential open set is fuzzy soft sequential open set, π‘ˆ βˆ’ {(𝐹, 𝐴)𝑛} is fuzzy soft sequential open set. Hence {(𝐹, 𝐴)𝑛} is fuzzy soft sequential closed set. Remark 3.18. The set of all fuzzy soft sequential closed sets in fuzzy soft sequential topological space (π‘ˆ, οΏ½ΜƒοΏ½ , 𝐸), form a fuzzy soft sequential topology on π‘ˆ. 4. Fuzzy soft sequential neighborhood system and Properties of fuzzy soft sequential quasi- coincident of a fuzzy soft sequential set Definition 4.1. Let (π‘ˆ, οΏ½ΜƒοΏ½, 𝐸) be a fuzzy soft sequential topological space over π‘ˆ and {(𝐹, 𝐴)𝑛} be a fuzzy soft sequential set over π‘ˆ. A fuzzy soft sequential set {(𝐹, 𝐴)𝑛} is called a fuzzy soft sequential neighborhood (briefly, 𝐹𝑆𝑆-nbhd) of a fuzzy soft sequential set {(𝐺, 𝐡)𝑛} if and only if there exists a fuzzy soft sequential open set {(𝐻, 𝐢)𝑛} such that {(𝐺, 𝐡)𝑛} βŠ‚Μƒ {(𝐻, 𝐢)𝑛} βŠ‚Μƒ {(𝐹, 𝐴)𝑛}. A fuzzy soft sequential neighborhood {(𝐹, 𝐴)𝑛} is said to be an open if and only if {(𝐹, 𝐴)𝑛} is fuzzy soft sequential open set. The collection of all fuzzy soft sequential neighborhood of {(𝐹, 𝐴)𝑛} is called the fuzzy soft sequential neighborhood system of {(𝐹, 𝐴)𝑛} up to topology οΏ½ΜƒοΏ½ and is denoted by 𝒩 {(𝐹,𝐴)𝑛}. Theorem 4.2. A fuzzy soft sequential set {(𝐹, 𝐴)𝑛} over π‘ˆ is a fuzzy soft sequential open set if and only if {(𝐹, 𝐴)𝑛} is a fuzzy soft sequential neighborhood of each fuzzy soft sequential set {(𝐺, 𝐡)𝑛} contained in {(𝐹, 𝐴)𝑛}. Proof. Necessary Part. Let {(𝐹, 𝐴)𝑛} be a fuzzy soft sequential open set and {(𝐺, 𝐡)𝑛} be any fuzzy soft sequential set contained in {(𝐹, 𝐴)𝑛} . Since {(𝐺, 𝐡)𝑛} βŠ‚Μƒ {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐹, 𝐴)𝑛} , {(𝐹, 𝐴)𝑛} is a fuzzy soft sequential neighborhood of each {(𝐺, 𝐡)𝑛} contained in {(𝐹, 𝐴)𝑛}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 92 https://internationalpubls.com Sufficient Part. Let {(𝐹, 𝐴)𝑛} be a fuzzy soft sequential neighborhood for every fuzzy soft sequential set contained it. Since {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐹, 𝐴)𝑛}, there exists a fuzzy soft sequential open set {(𝐺, 𝐡)𝑛} such that {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐺, 𝐡)𝑛} βŠ‚Μƒ {(𝐹, 𝐴)𝑛} .Hence {(𝐹, 𝐴)𝑛} = {(𝐺, 𝐡)𝑛} and {(𝐹, 𝐴)𝑛} is fuzzy soft sequential open set. Theorem 4.3 If 𝒩 {(𝐹,𝐴)𝑛} is the fuzzy soft sequential neighborhood system of fuzzy soft sequential set {(𝐹, 𝐴)𝑛}, then (i) Finite intersection of members of 𝒩 {(𝐹,𝐴)𝑛} belongs to 𝒩 {(𝐹,𝐴)𝑛}. (ii) Each fuzzy soft sequential set which contains a member of 𝒩 {(𝐹,𝐴)𝑛} belongs to 𝒩 {(𝐹,𝐴)𝑛}. Proof. (i) If {(𝐺, 𝐡)𝑛}, {(𝐻, 𝐢)𝑛} ∈ 𝒩 {(𝐹,𝐴)𝑛}. Then there exists {(𝐺′, 𝐡′)𝑛}, {(𝐻′, 𝐢′)𝑛} ∈ οΏ½ΜƒοΏ½ such that {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐺′, 𝐡′)𝑛} βŠ‚Μƒ {(𝐺, 𝐡)𝑛} and {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐻′, 𝐢′)𝑛} βŠ‚Μƒ {(𝐻, 𝐢)𝑛} . Since {(𝐺′, 𝐡′)𝑛} βˆ©Μƒ {(𝐻′, 𝐢′)𝑛} ∈ οΏ½ΜƒοΏ½, {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐺′, 𝐡′)𝑛} βˆ©Μƒ {(𝐻′, 𝐢′)𝑛} βŠ‚Μƒ {(𝐺, 𝐡)𝑛} βˆ©Μƒ {(𝐻, 𝐢)𝑛}. Hence {(𝐺, 𝐡)𝑛} βˆ©Μƒ {(𝐻, 𝐢)𝑛} ∈ 𝒩 {(𝐹,𝐴)𝑛}. (ii) Let {(𝐺, 𝐡)𝑛} ∈ 𝒩 {(𝐹,𝐴)𝑛} and {(𝐻, 𝐢)𝑛} be a fuzzy soft sequential set which contains {(𝐺, 𝐡)𝑛}. Since {(𝐺, 𝐡)𝑛} ∈ 𝒩 {(𝐹,𝐴)𝑛}, there exists a fuzzy soft sequential open set {(𝐺′, 𝐡′)𝑛} such that {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐺′, 𝐡′)𝑛} βŠ‚Μƒ {(𝐺, 𝐡)𝑛}. Since {(𝐺, 𝐡)𝑛} βŠ‚Μƒ {(𝐻, 𝐢)𝑛}, {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐺′, 𝐡′)𝑛} βŠ‚Μƒ {(𝐻, 𝐢)𝑛} . Hence {(𝐻, 𝐢)𝑛} ∈ 𝒩 {(𝐹,𝐴)𝑛}. Definition 4.4. Fuzzy soft sequential sets {(𝐹, 𝐴)𝑛} and {(𝐺, 𝐡)𝑛} are said to be fuzzy soft sequential quasi-coincident, denoted by {(𝐹, 𝐴)𝑛}π‘ž{(𝐺, 𝐡)𝑛} , if and only if there exists 𝑒𝑖 ∈ 𝐴, 𝐡 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹n(𝑒𝑖)(π‘₯𝑗) + 𝐺n(𝑒𝑖)(π‘₯𝑗) >Μƒ 1 (π‘œπ‘Ÿ)𝐹n(𝑒𝑖)(π‘₯𝑗) >Μƒ 𝐺n c(𝑒𝑖)(π‘₯𝑗) for all 𝑛 ∈ β„•. If there exists 𝑒𝑖 ∈ 𝐴, 𝐡 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹n(𝑒𝑖)(π‘₯𝑗) + 𝐺n(𝑒𝑖)(π‘₯𝑗) ≀̃ 1 for all 𝑛 ∈ β„•, then {(𝐹, 𝐴)𝑛} and {(𝐺, 𝐡)𝑛} are not fuzzy soft sequential quasi-coincident and it is denoted by {(𝐹, 𝐴)𝑛}οΏ½Μ…οΏ½{(𝐺, 𝐡)𝑛} . Definition 4.5. Fuzzy soft sequential sets {(𝐹, 𝐴)𝑛} and {(𝐺, 𝐡)𝑛} are said to be fuzzy soft sequential weakly quasi-coincident, denoted by {(𝐹, 𝐴)𝑛}π‘žπ‘€{(𝐺, 𝐡)𝑛}, if and only if there exists 𝑒𝑖 ∈ 𝐴, 𝐡 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹n(𝑒𝑖)(π‘₯𝑗) + 𝐺n(𝑒𝑖)(π‘₯𝑗) >Μƒ 1 (π‘œπ‘Ÿ)𝐹n(𝑒𝑖)(π‘₯𝑗) >Μƒ 𝐺n c(𝑒𝑖)(π‘₯𝑗) for some 𝑛 ∈ β„•. If {(𝐹, 𝐴)𝑛} and {(𝐺, 𝐡)𝑛} are not fuzzy soft sequential weakly quasi-coincident, then it can be written as {(𝐹, 𝐴)𝑛}π‘žπ‘€Μ…Μ…Μ…Μ… {(𝐺, 𝐡)𝑛}. Proposition 4.6 The fuzzy soft sequential sets {(𝐹, 𝐴)𝑛} and {(𝐺, 𝐡)𝑛} are fuzzy soft sequential quasi-coincident, if and only if each pair of non-zero fuzzy soft sets (𝐹, 𝐴)𝑛 and (𝐺, 𝐡)𝑛 are also so. Proof. Let the fuzzy soft sequential sets {(𝐹, 𝐴)𝑛} and {(𝐺, 𝐡)𝑛} are fuzzy soft sequential quasi- coincident ⇔ there exists 𝑒𝑖 ∈ 𝐴, B and π‘₯𝑗 ∈ π‘ˆ such that 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) + 𝐺𝑛(𝑒𝑖)(π‘₯𝑗) >Μƒ 1 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ 𝑛 ∈ β„• Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 93 https://internationalpubls.com ⇔ each pair of non-zero fuzzy soft sets (𝐹, 𝐴)𝑛 and (𝐺, 𝐡)𝑛 are also fuzzy soft quasi-coincident. Proposition 4.7. Let {(𝐹, 𝐴)𝑛} and {(𝐺, 𝐡)𝑛} be a fuzzy soft sequential sets over π‘ˆ. Then the following properties are true. (i) {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐺, 𝐡)𝑛} if and only if {(𝐹, 𝐴)𝑛} οΏ½Μ…οΏ½{(𝐺, 𝐡)𝑛}𝑐 (ii) {(𝐹, 𝐴)𝑛} βŠ‚Μƒπ‘€ {(𝐺, 𝐡)𝑛} if and only if {(𝐹, 𝐴)𝑛} π‘žπ‘€{(𝐺, 𝐡)𝑛}𝑐 (iii) {(𝐹, 𝐴)𝑛} βŠ„Μƒ {(𝐺, 𝐡)𝑛}𝑐 if and only if {(𝐹, 𝐴)𝑛} π‘ž{(𝐺, 𝐡)𝑛} (iv) {(𝐹, 𝐴)𝑛} π‘ž {(𝐺, 𝐡)𝑛} implies {(𝐹, 𝐴)𝑛} βˆ©Μƒ {(𝐺, 𝐡)𝑛} β‰  βˆ…Μƒ (v) {(𝐹, 𝐴)𝑛} οΏ½Μ…οΏ½ {(𝐹, 𝐴)𝑛}𝑐 Proof. (i) {(𝐹, 𝐴)𝑛} βŠ‚Μƒ {(𝐺, 𝐡)𝑛} ⇔ for each 𝑒𝑖 ∈ 𝐴, 𝐡 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) ≀̃ 𝐺𝑛(𝑒𝑖)(π‘₯𝑗) for all 𝑛 ∈ β„•. ⇔ for each 𝑒𝑖 ∈ 𝐴, 𝐡 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) + 1 βˆ’ 𝐺𝑛(𝑒𝑖)(π‘₯𝑗) ≀̃ 1 for all 𝑛 ∈ β„•. ⇔ for each 𝑒𝑖 ∈ 𝐴, 𝐡 and π‘₯𝑗 ∈ π‘ˆ such that {(𝐹, 𝐴)𝑛} + {(𝐺, 𝐡)𝑛}𝑐 ≀̃ 1 for all 𝑛 ∈ β„•. Hence {(𝐹, 𝐴)𝑛} οΏ½Μ…οΏ½{(𝐺, 𝐡)𝑛}𝑐. (ii) The proof is similar to (i) (iii) The proof is similar to (i) (iv) Let {(𝐹, 𝐴)𝑛} π‘ž {(𝐺, 𝐡)𝑛}. Then there exists 𝑒𝑖 ∈ 𝐴, 𝐡 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) + 𝐺𝑛(𝑒𝑖)(π‘₯𝑗) >Μƒ 1 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ 𝑛 ∈ β„• implies {(𝐹, 𝐴)𝑛} β‰  βˆ…Μƒ and {(𝐺, 𝐡)𝑛} β‰  βˆ…Μƒ for all 𝑛 ∈ β„•. Hence {(𝐹, 𝐴)𝑛} βˆ©Μƒ {(𝐺, 𝐡)𝑛} β‰  βˆ…Μƒ. (v) Suppose {(𝐹, 𝐴)𝑛} π‘ž {(𝐹, 𝐴)𝑛}𝑐. Then there exists 𝑒𝑖 ∈ 𝐴 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) + 𝐹𝑛 c(𝑒𝑖)(π‘₯𝑗) >Μƒ 1 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ 𝑛 ∈ β„• implies there exists 𝑒𝑖 ∈ 𝐴 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) + 1 βˆ’ 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) >Μƒ 1 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ 𝑛 ∈ β„•. Therefore, there exists 𝑒𝑖 ∈ 𝐴 and π‘₯𝑗 ∈ π‘ˆ such that 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) >Μƒ 𝐹𝑛(𝑒𝑖)(π‘₯𝑗) π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ 𝑛 ∈ β„•, which is a contradiction. Hence {(𝐹, 𝐴)𝑛} οΏ½Μ…οΏ½ {(𝐹, 𝐴)𝑛}𝑐. 5. Conclusion In this paper, we have introduced fuzzy soft sequential topological spaces via fuzzy soft sequential set. We studied their properties and a relationship between fuzzy soft topology and fuzzy soft sequential topology are explored. We investigated fuzzy soft sequential neighborhood of a fuzzy soft sequential set, fuzzy soft sequential quasi-coincident along with their properties. References [1] Ahmad B, Athar Kharal 2009,’On Fuzzy soft sets’, Advances in fuzzy systems, pp 1-6. [2] Bose M K, Indrajit Lahiri 2002, β€˜Sequential topological spaces and separation axioms’, Bulletin of the Allahabad Mathematical Society, 17, pp 23-37. 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