Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 835 https://internationalpubls.com 𝛉-separation Axioms on Fuzzy Hypersoft Topological Spaces P. Revathi 𝟏 B. Premamalini 𝟐 K. Chitirakala 𝟑 and G. Saravanakumar 𝟒 1 Government Polytechnic College, Kuduveli, Chidambaram - 608 305, India. 1,2Department of Mathematics, Annamalai University, Annamalai Nagar - 608 002, India. 3Department of Mathematics, M.Kumarasamy College of Engineering, Karur - 639 113, India. 4Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology (Deemed to be University), Avadi, Chennai-600062, India. Corresponding authors: K. Chitirakala and B. Premamalini 1revathimathsau@gmail.com, 2premamalinips@gmail.com, 3chitrakalalaksana@gmail.com, 4saravananguru2612@gmail.com Article History: Received: 10-11-2024 Revised:24-12-2024 Accepted:09-01-2025 Abstract: In this article, the concept of fuzzy hypersoft θ (resp. θ semi & θ pre)-separation axioms in fuzzy hypersoft topological spaces are introduced by developing fuzzy hypersoft θ (resp. θ semi & θ pre)-neighbourhood with respect to fuzzy hypersoft points. Also, the properties and relations between fuzzy hypersoft θ (resp. θ semi & θ pre)- Ti-spaces (i = 0,1,2,3,4) are discussed. Keywords: FHS θ (resp. θ semi & θ pre)-neighbourhood, FHS θ (resp. θ semi & θ pre)- separation axioms, FHS θ (resp. θ semi & θ pre)- Ti-spaces (i = 0,1,2,3,4). AMS (2000) subject classification: 03E72, 54A05, 54A40. 1 Introduction The real-world decision-making problems in medical diagnosis, engineering, economics, management computer science, artificial intelligence, social sciences, environmental science and sociology contain more uncertain and inadequate data. Traditional mathematical methods cannot deal with these kinds of problems due to imprecise data. To deal with the problems with uncertainty, Zadeh [29] introduced the fuzzy set in 1965 which contains the membership value in [0,1]. A fuzzy set is a set where each element of the universe belongs to it but with some value or degree of belongingness which lies between 0 and 1 and such values are called the membership value of an element in that set. The topological structure on fuzzy set was undertaken by Chang [9] as fuzzy topological space. Molodstov [12] introduced a new mathematical tool, soft set theory in 1999 to deal with uncertainties in which a soft set is a collection of approximate descriptions of an object. A soft set is a parameterized family of subsets where parameters are the properties, attributes or characteristics of the objects. The soft set theory has several applications in different fields such as decision-making, optimization, forecasting, data analysis etc. Shabir and Naz [23] presented soft topological spaces. Smarandache [24] extended the notion of a soft set to a hypersoft set and then to plithogenic set by replacing a function with a multi-argument function described in the cartesian product with a different set of attributes. This new concept of hypersoft set is more flexible than the soft set and more suitable in decision-making issues involving a different kinds of attributes. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 836 https://internationalpubls.com Saeed et al. [20, 21] studied the fundamentals of hypersoft set theory by introducing aggregate operators, relations, functions, matrices and operations on hypersoft matrices. Abbas et al. [2] defined the basic operations on hypersoft sets and hypersoft point in the fuzzy, intuitionistic and neutrosophic environments. Ajay and Charisma [3] introduced fuzzy hypersoft topology, intuitionistic hypersoft topology and neutrosophic hypersoft topology. Neutrosophic hypersoft topology is the generalized framework which generalizes intuitionistic hypersoft topology and fuzzy hypersoft topology. Saha [22] defined δ-open sets and continuous maps in fuzzy topological spaces. Aranganayagi et al., Revathi et al., Surendra et al. and Vadivel et al. [4, 5, 14, 15, 16, 18, 25, 26, 27] introduced δ-open sets, e-open sets in neutrosophic, neutrosophic soft, fuzzy hypersoft, neutrosophic hypersoft topological spaces and studied its maps, separation axioms and compact spaces. In 2023, Revathi et al. [17] developed contra e-continuous maps in neutrosophic soft topological spaces. In 2019, the separation axioms on neutrosophic soft topological spaces were studied by Aras et al. [6]. The soft b-separation axioms were introduced by Khattak et al. [11] and pre- separation axioms were developed by Acikgoz et al. [1] in neutrosophic soft topological spaces. Gunduz et al. [10] and Ozturk [13] introduced separation axioms in neutrosophic hypersoft and fuzzy hypersoft topological spaces. The class of sets namely, θ open sets are playing more important role in topological spaces, because of their applications in various fields of Mathematics and other real fields. In 1968 Velicko [28] defined θ open set in H-closed Topological Spaces. In [7, 8], Caldas et al. studied various kinds of θ open sets and their properties in topological spaces. Revathi et al. [19] introduced θ open sets and studied its maps in fuzzy hypersoft topological spaces. The goal of this paper is to define the notions of fuzzy hypersoft θ (resp. semi, pre, θ semi & θ pre)-neighbourhood and fuzzy hypersoft θ (resp. semi, pre, θ semi & θ pre)-separation axioms in fuzzy hypersoft topological spaces using fuzzy hypersoft points. In addition, the characteristics of fuzzy hypersoft θ (resp. semi, pre, θ semi & θ pre)- Ti- spaces (i = 0,1,2,3,4) and relations between them are studied. Preliminaries Definition 2.1 [29] Let 𝔐 be an initial universe. A function λ from 𝔐 into the unit interval I is called a fuzzy set in 𝔐. For every 𝔪 ∈ 𝔐, λ(𝔪) ∈ I is called the grade of membership of 𝔪 in λ. Some authors say that λ is a fuzzy subset of 𝔐 instead of saying that λ is a fuzzy set in 𝔐. The class of all fuzzy sets from 𝔐 into the closed unit interval I will be denoted by I𝔐. Definition 2.2 [12] Let 𝔐 be an initial universe, Q be a set of parameters and 𝒫(𝔐) be the power set of 𝔐. A pair (Φ̃,∧) is called the a soft set over 𝔐 where Φ̃ is a mapping Φ̃: Q → 𝒫(𝔐). In other words, the soft set is a parametrized family of subsets of the set 𝔐. Definition 2.3 [24] Let 𝔐 be an initial universe and 𝒫(𝔐) be the power set of 𝔐. Consider 𝔮1, 𝔮2, 𝔮3, . . . , 𝔮n for n ≥ 1, be n distinct attributes, whose corresponding attribute values are respectively the sets Q1, Q2, . . . , Qn with Qi ∩ Qj = ∅, for i ≠ j and i, j ∈ {1,2, . . . , n}. Then the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 837 https://internationalpubls.com pair (Φ̃, Q1 × Q2 ×. . .× Qn) where Φ̃: Q1 × Q2 ×. . .× Qn → 𝒫(𝔐) is called a hypersoft set over 𝔐. Definition 2.4 [2] Let 𝔐 be an initial universal set and Q1, Q2, . . . , Qn be pairwise disjoint sets of parameters. Let 𝒫(𝔐) be the set of all fuzzy sets of 𝔐. Let Ei be the nonempty subset of the pair Qi for each i = 1,2, . . . , n. A fuzzy hypersoft set (briefly, FHySs) over 𝔐 is defined as the pair (Φ̃, E1 × E2 ×. . .× En) where Φ̃: E1 × E2 ×. . .× En → 𝒫(𝔐) and Φ̃(E1 × E2 ×. . .× En) = {(𝔮, 〈𝔪, μΦ̃(𝔮)(𝔪)〉: 𝔪 ∈ 𝔐): 𝔮 ∈ E1 × E2 ×. . .× En ⊆ Q1 × Q2 ×. . .× Qn} where μΦ̃(𝔮)(𝔪) is the membership value such that μΦ̃(𝔮)(𝔪) ∈ [0,1]. Definition 2.5 [2] Let (Φ̃,∧1) and (Ψ̃,∧2) be two FHySs’s over 𝔐. Then (Φ̃,∧1) is the fuzzy hypersoft subset of (Ψ̃,∧2) if μΦ̃(𝔮)(𝔪) ≤ μΨ̃(𝔮)(𝔪). It is denoted by (Φ̃,∧1) ⊆ (Ψ̃,∧2). Definition 2.6 [2] Let (Φ̃,∧1) and (Ψ̃,∧2) be FHySs’s over 𝔐. (Φ̃,∧1) is equal to (Ψ̃,∧2) if μΦ̃(𝔮)(𝔪) = μΨ̃(𝔮)(𝔪). Definition 2.7 [2] A FHySs (Φ̃,∧) over 𝔐 is called null fuzzy hypersoft set if μΦ̃(𝔮)(𝔪) = 0, ∀𝔮 ∈∧ and 𝔪 ∈ 𝔐. It is denoted by 0̃(𝔐,Q). A FHySs (Ψ̃,∧) over 𝔐 is called absolute fuzzy hypersoft set if μΦ̃(𝔮)(𝔪) = 1 ∀𝔮 ∈∧ and 𝔪 ∈ 𝔐. It is denoted by 1̃(𝔐,Q). Clearly, 0̃(𝔐,Q) c = 1̃(𝔐,Q) and 1̃(𝔐,Q) c = 0̃(𝔐,Q). Definition 2.8 [2] Let (Φ̃,∧) be FHySs over 𝔐. (Φ̃,∧)c is the complement of (Φ̃,∧) if μH̃(𝔮) c (𝔪) = 1̃(𝔐,Q) − μH̃(𝔮)(𝔪) where ∀𝔮 ∈∧ and ∀𝔪 ∈ 𝔐. It is clear that ((Φ̃,∧)c)c = (Φ̃,∧). Definition 2.9 [2] Let (Φ̃,∧1) and (Ψ̃,∧2) be FHySs’s over 𝔐. Extended union (Φ̃,∧1) ∪ (Ψ̃,∧2) is defined as Μ ((Φ̃,∧1) ∪ (Ψ̃,∧2)) = { μΦ̃(𝔮)(𝔪) if 𝔮 ∈∧1−∧2 μΨ̃(𝔮)(𝔪) if 𝔮 ∈∧2−∧1 max{μΦ̃(𝔮)(𝔪), μΨ̃(𝔮)(𝔪)} if 𝔮 ∈∧1∩∧2 Definition 2.10 [2, 3] Let (Φ̃,∧1) and (Ψ̃,∧2) be FHSs’s over 𝔐. Extended intersection (Φ̃,∧1) ∩ (Ψ̃,∧2) is defined as μ((Φ̃,∧1) ∩ (Ψ̃,∧2)) = { μΦ̃(𝔮)(𝔪) if 𝔮 ∈∧1−∧2 μΨ̃(𝔮)(𝔪) if 𝔮 ∈∧2−∧1 min{μΦ̃(𝔮)(𝔪), μΨ̃(𝔮)(𝔪)} if 𝔮 ∈∧1∩∧2 Definition 2.11 [3] Let (𝔐, Q) be the family of all FHySs’s over 𝔐 and τ̃ ⊆ FHySs(𝔐, Q). Then τ̃ is said to be a fuzzy hypersoft topology (briefly, FHySt) on 𝔐 if Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 838 https://internationalpubls.com (i). 0̃(𝔐,Q) and 1̃(𝔐,Q) belongs to τ̃ (ii). the union of any number of FHySs’s in τ̃ belongs to τ̃ (iii). the intersection of finite number of FHySs’s in τ̃ belongs to τ̃. Then (𝔐, Q, τ̃) is known as a fuzzy hypersoft toplogical space (briefly, FHySts) over 𝔐. Each member of τ̃ is said to be fuzzy hypersoft open set (briefly, FHySos). A FHySs (Φ̃,∧) is called a fuzzy hypersoft closed set (briefly, FHyScs) if its complement (Φ̃,∧)c is FHySos. Definition 2.12 [3] Let (𝔐, Q, τ̃) be a FHySts over 𝔐 and (Φ̃,∧) be a FHySs in 𝔐. Then, (i). the fuzzy hypersoft interior (briefly, FHSint) of (Φ̃,∧) is defined as FHSint(Φ̃,∧) =∪ {(Ψ̃,∧): (Ψ̃,∧) ⊆ (Φ̃,∧) where (Ψ̃,∧) is FHSos}. (ii). the fuzzy hypersoft closure (briefly, FHScl) of (Φ̃,∧) is defined as FHScl(Φ̃,∧) =∩ {(Ψ̃,∧): (Ψ̃,∧) ⊇ (Φ̃,∧) where (Ψ̃,∧) is FHScs}. Definition 2.13 [2] Let FHS’s (Φ̃,∧) be the family of all FHS’s over 𝔐 and let 𝔪 ∈ 𝔐, 0 ≤ φ ≤ 1, 𝔮 ∈ Q. Then the FHSs 𝔪φ q is called a fuzzy hypersoft point (briefly, FHSp) and is defined as follows: For each 𝔫 ∈ 𝔐, 𝔪φ 𝔮 (𝔮′)(𝔫) = { φif𝔮′ = 𝔮and𝔫 = 𝔪 0if𝔮′ ≠ 𝔮or𝔫 ≠ 𝔪. Definition 2.14 [13] Let 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ be two FHSp’s. For the FHSp’s 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ over a common universe 𝔐, we say that FHSp’s are distinct points, if 𝔪φ 𝔮 ∩ 𝔫φ′ 𝔮′ = 0(𝔐,Q). It is clear that 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ are distinct FHSp’s iff 𝔪 ≠ 𝔫 and 𝔮′ ≠ 𝔮. Definition 2.15 [13] Let (𝔐, Q, τ̃) be FHSts over 𝔐. A FHS’s (Φ̃,∧) in (𝔐, Q, τ̃) is called a fuzzy hypersoft neighbourhood (briefly, FHSnbd) of the FHSp 𝔪φ 𝔮 ∈ (Φ̃,∧), if there exists a FHSos (Ψ̃,∧) such that 𝔪φ 𝔮 ∈ (Ψ̃,∧) ⊆ (Φ̃,∧). Definition 2.16 [13] Let (𝔐, Q, τ̃) be a FHSts over 𝔐. Let (Φ̃,∧) be a FHSs over 𝔐 and 𝔪φ 𝔮 be a FHSp over 𝔐. [(i)] (i). 𝔪φ 𝔮 is a fuzzy hypersoft interior point of (Φ̃,∧), if (Ψ̃,∧) ⊆ (Φ̃,∧) for some (Ψ̃,∧) ∈ FHSnbd of the FHSp 𝔪φ 𝔮 . (ii). 𝔪φ 𝔮 is a fuzzy hypersoft adherent point of (Φ̃,∧), if (Ψ̃,∧) ⋂ (Φ̃,∧) ∉ 0(𝔐,Q) for any (Ψ̃,∧) ∈ FHSnbd of the FHSp 𝔪φ 𝔮 . Theorem 2.1 [13] Let (𝔐, Q, τ̃) be a FHSts over 𝔐 and (Φ̃,∧) be a FHSs over 𝔐. Then 1. FHSint(Φ̃,∧) = ⋃ {𝔪φ 𝔮 : 𝔪φ 𝔮 is a fuzzy hypersoft interior point of (Φ̃,∧)}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 839 https://internationalpubls.com 2. FHScl(Φ̃,∧) = ⋃ {𝔪φ 𝔮 : 𝔪φ 𝔮 is a fuzzy hypersoft adherent point of (Φ̃,∧)}. Definition 2.17 [13] Let (𝔐, Q, τ̃) be a FHSts over 𝔐 and (Φ̃,∧) be an arbitrary FHS’s . Then τ̃(Φ̃,∧) = {(Φ̃,∧) ∩ (Ψ̃,∧): (Ψ̃,∧) ∈ τ̃} is called FHSt on (Φ̃,∧) and ((Φ̃,∧), τ̃(Φ̃,∧), Q) is known as a fuzzy hypersoft topological subspace (briefly, FHStss) of (𝔐, Q, τ̃). 3 Fuzzy hypersoft 𝛉-separation axioms Definition 3.1 Let (𝔐, Q, τ̃) be a FHySts over 𝔐 and (Φ̃,∧) be a FHySs on 𝔐. Then the fuzzy hypersoft (i). θ-interior (briefly, FHSint) of (Φ̃,∧) is defined by FHSθint(Φ̃,∧) = ⋃ {(Ψ̃,∧): (Ψ̃,∧ ) ⊆ (Φ̃,∧) and (Ψ̃,∧) is a FHScs in 𝔐} (ii). θ-closure (briefly, FHScl) of (Φ̃,∧) is defined by FHSθcl(Φ̃,∧) = ⋂ {(Ψ̃,∧): (Ψ̃,∧) ⊇ (Φ̃,∧) and (Ψ̃,∧) is a FHSos in 𝔐} Definition 3.2 Let (𝔐, Q, τ̃) be a FHSts over 𝔐. An FHSs (Φ̃,∧) is said to be a fuzzy hypersoft (i). θ-open set (briefly, FHSθos) if (Φ̃,∧) = FHSθint(Φ̃,∧) (ii). θ-pre open set (briefly, FHSθ𝒫os) if (Φ̃,∧) ⊆ FHSint(FHSθcl(Φ̃,∧)) (iii). θ-semi open set (briefly, FHSθ𝒮os) if (Φ̃,∧) ⊆ FHScl(FHSθint(Φ̃,∧)) The complement of FHSθos (resp. FHSθ𝒫os & FHSθ𝒮os) is called a FHSθ (resp. FHSθ pre & FHSθ semi) closed set (briefly, FHSθcs (resp. FHSθ𝒫cs & FHSθ𝒮cs)) in 𝔐. The family of all FHSθos (resp. FHSθcs, FHSθ𝒫os, FHSθ𝒫cs, FHSθ𝒮os & FHSθ𝒮cs) of 𝔐 is denoted by FHSθOS(𝔐) (resp. FHSθCS(𝔐), FHS𝒫OS(𝔐), FHS𝒫CS(𝔐), FHSθ𝒫OS(𝔐), FHSθ𝒫CS(𝔐), FHSθ𝒮OS(𝔐) & FHSθ𝒮CS(𝔐)). Definition 3.3 Let (𝔐, Q, τ̃) be a FHSts over 𝔐 and (Φ̃,∧) be a FHSs on 𝔐. Then the fuzzy hypersoft (i). θ-pre (resp. θ-semi) interior (briefly, FHSθ𝒫int (resp. FHSθ𝒮int)) of (Φ̃,∧) is defined by FHSθ𝒫int(Φ̃,∧) = ⋃ {(Ψ̃,∧): (Ψ̃,∧) ⊆ (Φ̃,∧) and (Ψ̃,∧) is a FHSθ𝒫os (resp. FHSθ𝒮os) in 𝔐} (ii). θ-pre (resp. θ-semi) closure (briefly, FHSθ𝒫cl (resp. FHSθ𝒮cl)) of (Φ̃,∧) is defined by FHSθ𝒫cl(Φ̃,∧) = ⋂ {(Ψ̃,∧): (Ψ̃,∧) ⊇ (Φ̃,∧) and (Ψ̃,∧) is a FHSθ𝒫cs (resp. FHSθ𝒮cs) in 𝔐} Definition 3.4 Let (𝔐, Q, τ̃) be FHSts over 𝔐. A FHS’s (Φ̃,∧) in (𝔐, Q, τ̃) is called a fuzzy hypersoft θ (resp. θ semi & θ pre)- neighbourhood (briefly, FHSθ(resp. θ semi & θ pre)-nbd) of the FHSp 𝔪φ 𝔮 ∈ (Φ̃,∧), if there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Ψ̃,∧) such that 𝔪φ 𝔮 ∈ (Ψ̃,∧) ⊆ (Φ̃,∧). Theorem 3.1 Let (𝔐, Q, τ̃) be FHSts over 𝔐 and (Φ̃,∧) be a FHS’s on 𝔐. Then (Φ̃,∧) is a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) iff (Φ̃,∧) is a FHSθ- (resp. θ semi & θ pre)nbd of its FHSp’s. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 840 https://internationalpubls.com Proof. Let (Φ̃,∧) be a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) and 𝔪φ 𝔮 ∈ (Φ̃,∧). Then, 𝔪φ 𝔮 ∈ (Φ̃,∧) ⊆ (Φ̃,∧). Thus (Φ̃,∧) is a FHSθ (resp. θ semi & θ pre)-nbd of 𝔪φ 𝔮 . Conversely, let (Φ̃,∧) be a FHSθ (resp. θ semi & θ pre)-nbd of its FHSp’s. Let 𝔪φ 𝔮 ∈ (Φ̃,∧). Since (Φ̃,∧) is a FHSθ (resp. θ semi & θ pre)-nbd of the FHSp 𝔪φ 𝔮 , there exists (Ψ̃,∧) ∈ τ̃ such that 𝔪φ 𝔮 ∈ (Ψ̃,∧) ⊆ (Φ̃,∧). Since (Φ̃,∧) = ⋃ {𝔪φ 𝔮 : 𝔪φ 𝔮 ∈ (Φ̃,∧)}, it follows that (Φ̃,∧) is a union of FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s. Then (Φ̃,∧) is a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os). The FHSθ (resp. θ semi & θ pre)-nbd system of a FHSp 𝔪φ 𝔮 denoted by ⋃ (𝔪φ 𝔮 , Q), is the family of all its FHSθ (resp. δ semi & θ pre)-nbd’s. Theorem 3.2 The FHySθ (resp. θ semi & θ pre)-nbd system ⋃ (𝔪φ 𝔮 , Q) at 𝔪φ 𝔮 in a FHSts (𝔐, Q, τ̃) has the following properties: (i). If (Φ̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q), then 𝔪φ 𝔮 ∈ (Φ̃,∧). (ii). If (Φ̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q) and (Φ̃,∧) ⊆ (Ω̃,∧), then (Ω̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q). (iii). (Φ̃,∧) and (Ψ̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q), then (Φ̃,∧) ∩ (Ψ̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q). (iv). If (Φ̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q), then there exists a (Ψ̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q) such that (Ψ̃,∧) ∈ ⋃ (𝔫φ′ 𝔮′ , Q) for each 𝔫φ′ 𝔮′ ∈ (Ψ̃,∧). Proof. The proofs of (i), (ii) and (iii) directly follow from the Definition 3.4. (iv) Suppose (Φ̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q). Then there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Ψ̃,∧) such that 𝔪φ 𝔮 ∈ (Ψ̃,∧) ⊆ (Φ̃,∧). Then by Theorem 3.1, (Ψ̃,∧) ∈ ⋃ (𝔪φ 𝔮 , Q). So for each 𝔫φ′ 𝔮′ ∈ (Ψ̃,∧), (Ψ̃,∧) ∈ (𝔫φ′ 𝔮′ , Q). Definition 3.5 Let (𝔐, Q, τ̃) be FHSts over 𝔐. Let 𝔪φ 𝔮 and zφ′ 𝔮′ be distinct FHSp’s. If there exist FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s (Φ̃,∧) and (Ψ̃,∧) such that 𝔪φ 𝔮 ∈ (Φ̃,∧) and 𝔪φ 𝔮 ∩ (Ψ̃,∧) = 0(𝔐,Q) or 𝔫φ′ 𝔮′ ∈ (Ψ̃,∧) and 𝔫φ′ 𝔮′ ∩ (Φ̃,∧) = 0(𝔐,Q), then (𝔐, Q, τ̃) is called a fuzzy hypersoft θ (resp. θ semi & θ pre)- T0- space (briefly, FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T0- space). Definition 3.6 Let (𝔐, Q, τ̃) be FHSts over 𝔐. Let 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ be distinct FHSp’s. If there exist FHSθos (resp. FHSθ𝒮os & FHySθ𝒫os)’s (Φ̃,∧) and (Ψ̃,∧) such that 𝔪φ 𝔮 ∈ (Φ̃,∧), 𝔪φ 𝔮 ∩ (Ψ̃,∧) = 0(𝔐,Q) and 𝔫φ′ 𝔮′ ∈ (Ψ̃,∧), 𝔫φ′ 𝔮′ ∩ (Φ̃,∧) = 0(𝔐,Q), then (𝔐, Q, τ̃) is called a fuzzy hypersoft θ (resp. θ semi & θ pre)- T1- space (briefly, FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T1- space). Definition 3.7 Let (𝔐, Q, τ̃) be FHSts over 𝔐. Let 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ be distinct FHSp’s. If there exist FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s (Φ̃,∧) and (Ψ̃,∧) such that 𝔪φ 𝔮 ∈ (Φ̃,∧), 𝔫φ′ 𝔮′ ∈ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 841 https://internationalpubls.com (Ψ̃,∧) and (Φ̃,∧) ∩ (Ψ̃,∧) = 0(𝔐,Q), then (𝔐, Q, τ̃) is called a fuzzy hypersoft θ (resp. θ semi & θ pre)- T2- space (briefly, FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T2- space). Example 3.1 Let 𝔐 = {𝔪1, 𝔪2} be the FHyS initial universe and the attribute be Q = Q1 × Q2. The attribute is given as: Q1 = {a1, a2} & Q2 = {b1} and ∧= {𝔮1 = (a1, b1) & 𝔮2 = (a2, b1)}. Let 𝔪1(0.8) 𝔮1 , 𝔪1(0.7) 𝔮2 , 𝔪2(0.9) 𝔮1 and 𝔪2(0.5) 𝔮2 be FHSp’s. Let (𝔐, Q) be the class of FHyS sets. Let the FHySs’s (Φ̃1,∧), (Φ̃2,∧), (Φ̃3,∧), (Φ̃4,∧), (Φ̃5,∧) over the universe 𝔐 be (Φ̃1,∧) = { 〈(a1, b1), { 𝔪1 0.8 , 𝔪2 0 }〉, 〈(a2, b1), { 𝔪1 0 , 𝔪2 0 }〉 } (Φ̃2,∧) = { 〈(a1, b1), { 𝔪1 0.2 , 𝔪2 0 }〉, 〈(a2, b1), { 𝔪1 0 , 𝔪2 0 }〉 } (Φ̃3,∧) = { 〈(a1, b1), { 𝔪1 0.8 , 𝔪2 0.9 }〉, 〈(a2, b1), { 𝔪1 0.7 , 𝔪2 0.5 }〉 } (Φ̃4,∧) = { 〈(a1, b1), { 𝔪1 0.2 , 𝔪2 0.1 }〉, 〈(a2, b1), { 𝔪1 0.3 , 𝔪2 0.5 }〉 } (Φ̃5,∧) = { 〈(a1, b1), { 𝔪1 0.8 , 𝔪2 0.1 }〉, 〈(a2, b1), { 𝔪1 0.3 , 𝔪2 0.5 }〉 } τ̃ = {0̃(𝔐,Q), 1̃(𝔐,Q), (Φ̃1,∧), (Φ̃2,∧), (Φ̃3,∧), (Φ̃4,∧), (Φ̃5,∧)} is FHySts. Hence, (𝔐, Q, τ̃) is a FHSts over 𝔐. Here, (Φ̃3,∧) and (Φ̃4,∧) are FHS θos’s. Also, (𝔐, Q, τ̃) is a FHSθ- T0- space but not a FHSθ- T1- space because for FHySp’s 𝔪1(0.8) 𝔮1 and 𝔪2(0.5) 𝔮2 , (𝔐, Q, τ̃) is not a FHSθ- T1- space. Example 3.2 Consider a set of natural numbers 𝔐 = N and a parameter set Q = {∧}. Let the FHSp’s be nφn 𝔮 . Now we can take φn appropriate values and the FHSp’s nφn 𝔮 , mφm 𝔮 are distinct FHSp’s iff n ≠ m. It is obvious that there is one-to-one compatibility between the set of natural numbers and the set of FHSp’s N𝔮 = {nφn 𝔮 }. Here we define cofinite topology on this set. Then FHSs’s (Φ̃,∧) is a FHSθos iff the finite FHSp’s are discarded from N𝔮. Hence, (𝔐, Q, τ̃) is a FHSθ- T1- space but not a FHSθ- T2- space. Example 3.3 Let 𝔐 = {𝔪1, 𝔪2} be the FHyS initial universe and the attribute be Q = Q1 × Q2. The attribute is given as: Q1 = {a1, a2} & Q2 = {b1} and ∧= {𝔮1 = (a1, b1) & 𝔮2 = (a2, b1)}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 842 https://internationalpubls.com Let 𝔪1(0.8) 𝔮1 , 𝔪1(0.3) 𝔮2 , 𝔪2(0.9) 𝔮1 and 𝔪2(0.4) 𝔮2 be FHSp’s. Let (𝔐, Q) be the class of FHyS sets. Let the FHySs’s (Φ̃1,∧), (Φ̃2,∧), (Φ̃3,∧), (Φ̃4,∧), (Φ̃5,∧) over the universe 𝔐 be (Φ̃1,∧) = { 〈(a1, b1), { 𝔪1 0.8 , 𝔪2 0 }〉, 〈(a2, b1), { 𝔪1 0 , 𝔪2 0 }〉 } (Φ̃2,∧) = { 〈(a1, b1), { 𝔪1 0.2 , 𝔪2 0 }〉, 〈(a2, b1), { 𝔪1 0 , 𝔪2 0 }〉 } (Φ̃3,∧) = { 〈(a1, b1), { 𝔪1 0.8 , 𝔪2 0.9 }〉, 〈(a2, b1), { 𝔪1 0.7 , 𝔪2 0.6 }〉 } (Φ̃4,∧) = { 〈(a1, b1), { 𝔪1 0.2 , 𝔪2 0.1 }〉, 〈(a2, b1), { 𝔪1 0.3 , 𝔪2 0.4 }〉 } (Φ̃5,∧) = { 〈(a1, b1), { 𝔪1 0.8 , 𝔪2 0.1 }〉, 〈(a2, b1), { 𝔪1 0.3 , 𝔪2 0.4 }〉 } τ̃ = {0̃(𝔐,Q), 1̃(𝔐,Q), (Φ̃1,∧), (Φ̃2,∧), (Φ̃3,∧), (Φ̃4,∧), (Φ̃5,∧)} is FHySts. Hence, (𝔐, Q, τ̃) is a FHSts over 𝔐. Here, (Φ̃3,∧) and (Φ̃4,∧) are FHS θos’s. Also, (𝔐, Q, τ̃) is a FHSθ- T2- space. Theorem 3.3 Let (𝔐, Q, τ̃) be a FHSts over 𝔐. Then (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T1- space iff each FHSp is a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs). Proof. Let (𝔐, Q, τ̃) be a FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T1- space and 𝔪φ 𝔮 be an arbitrary FHSp. Let 𝔫φ′ 𝔮′ ∈ (𝔪φ 𝔮 )c. Then 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ are distinct FHSp’s. Thus 𝔪 ≠ 𝔫 or 𝔮′ ≠ 𝔮. Since (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T1- space, there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Ψ̃,∧) such that 𝔫(α′,β′,γ′) 𝔮′ ∈ (Ψ̃,∧) and 𝔪φ 𝔮 ∩ (Ψ̃,∧) = 0(𝔐,Q). Since 𝔪φ 𝔮 ∩ (Ψ̃,∧ ) = 0(𝔐,Q), we have 𝔫φ′ 𝔮′ ∈ (Ψ̃,∧) ⊆ (𝔪φ 𝔮 )c. Thus (𝔪φ 𝔮 )c is a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os), ie, 𝔪φ 𝔮 is a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs). Conversely, suppose that each FHSp 𝔪φ 𝔮 is a FHSθcs (resp. FHSθ𝒮cs & FHSδ𝒫cs). Then (𝔪φ 𝔮 )c is a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os). Let 𝔪φ 𝔮 ∩ 𝔫φ′ 𝔮′ = 0(𝔐,Q). Thus, 𝔫φ′ 𝔮′ ∈ (𝔪φ 𝔮 )c and 𝔪φ 𝔮 ∩ (𝔪φ 𝔮 )c = 0(𝔐,Q). So (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T1- space on 𝔐 Theorem 3.4 Let (𝔐, Q, τ̃) be a FHSts over 𝔐. Then (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T2- space iff for distinct FHSp’s 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ , there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Φ̃,∧) containing 𝔪φ 𝔮 but not 𝔫φ′ 𝔮′ such that 𝔫φ′ 𝔮′ does not belong to FHScl(Φ̃,∧). Proof. Let 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ be two FHSp’s in FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T2- space (𝔐, Q, τ̃). Then there exist disjoint FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s (Φ̃,∧) and (Ψ̃,∧) such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 843 https://internationalpubls.com 𝔪φ 𝔮 ∈ (Φ̃,∧), 𝔫φ′ 𝔮′ ∈ (Ψ̃,∧). Since 𝔪φ 𝔮 ∩ 𝔫φ′ 𝔮′ = 0(𝔐,Q) and (Φ̃,∧) ∩ (Ψ̃,∧) = 0(𝔐,Q), 𝔫φ′ 𝔮′ does not belong to (Φ̃,∧). It implies that 𝔫φ′ 𝔮′ does not belong to FHScl(Φ̃,∧). Conversely suppose that, for distinct FHSp’s 𝔪φ 𝔮 , 𝔫φ′ 𝔮′ , there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Φ̃,∧) containing 𝔪φ 𝔮 but not 𝔫φ′ 𝔮′ such that 𝔫(α′,β′,γ)′ 𝔮′ does not belong to FHScl(Φ̃,∧). Then 𝔫φ′ 𝔮′ ∈ (FHScl(Φ̃,∧))c, i. e., (Φ̃,∧) and (FHScl(Φ̃,∧))c are disjoint FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s containing 𝔪φ 𝔮 , 𝔫φ′ 𝔮′ respectively Theorem 3.5 Let (𝔐, Q, τ̃) be a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T1- space for every FHSp 𝔪φ 𝔮 ∈ (Φ̃,∧) ∈ τ̃. If there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Ψ̃,∧) such that 𝔪φ 𝔮 ∈ (Ψ̃,∧) ⊆ FHScl(Ψ̃,∧) ⊆ (Φ̃,∧), then (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T2- space. Proof. Suppose that 𝔪φ 𝔮 ∩ 𝔫φ′ 𝔮′ = 0(𝔐,Q). Since (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T1- space, 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ are FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs)’s in τ̃. Then 𝔪φ 𝔮 ∈ (𝔫φ′ 𝔮′ )c ∈ τ̃. Thus there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Ψ̃,∧) in τ̃ such that 𝔪φ 𝔮 ∈ (Ψ̃,∧) ⊆ FHScl(Ψ̃,∧) ⊆ (𝔫φ′ 𝔮′ )c. So, we have 𝔫φ′ 𝔮′ ∈ (FHScl(Ψ̃,∧))c, 𝔪φ 𝔮 ∈ (Ψ̃,∧) and (Ψ̃,∧) ∩ (FHScl(Ψ̃,∧))c = 0(𝔐,Q), i. e., (𝔐, Q, τ̃) is a FHSδ (resp. FHSθ𝒮 & FHSθ𝒫)- T2- space Remark 3.1 Let (𝔐, Q, τ̃) be a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- Ti- space for i = 0,1,2. For each 𝔪 ≠ 𝔫, FHSp’s 𝔪φ and 𝔫φ′ have neighbourhoods satisfying conditions of θ (resp. FHSθ𝒮 & FHSθ𝒫)- Ti- space in FHSts (𝔐, τ̃𝔮) for each 𝔮 ∈ Q because 𝔪φ 𝔮 and 𝔫φ′ 𝔮′ are distinct FHSp ’s. Definition 3.8 Let (𝔐, Q, τ̃) be FHSts over 𝔐. Let (Φ̃,∧) be a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs) and 𝔪φ 𝔮 ∩ (Φ̃,∧) = 0(𝔐,Q). If there exist FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s (Υ̃1,∧) and (Υ̃2,∧) such that 𝔪φ 𝔮 ∈ (Υ̃1,∧), (Φ̃,∧) ⊆ (Υ̃2,∧) and (Υ̃1,∧) ∩ (Υ̃2,∧) = 0(𝔐,Q), then (𝔐, Q, τ̃) is called a fuzzy hypersoft θ (resp. θ semi & θ pre)- regular (briefly, FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)-regular) space. (𝔐, Q, τ̃) is said to be a fuzzy hypersoft θ (resp. θ semi & θ pre)- T3- space (briefly, FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T3- space) if it is both a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)-regular and FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T1-space. Theorem 3.6 Let (𝔐, Q, τ̃) be FHSts over 𝔐. (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T3-space iff for every 𝔪φ 𝔮 ∈ (Φ̃,∧) ∈ τ̃, there exists (Υ̃,∧) ∈ τ̃ such that 𝔪φ 𝔮 ∈ (Υ̃,∧) ⊆ FHScl(Υ̃,∧) ⊆ (Φ̃,∧). Proof. Let (𝔐, Q, τ̃) be a FHSθ- T3-space and 𝔪φ 𝔮 ∈ (Φ̃,∧) ∈ τ̃. Since (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T3- space for the FHSp 𝔪φ 𝔮 and FHSθcs (Φ̃,∧)c, there exist (Υ̃1,∧), (Υ̃2,∧) ∈ τ̃ such that 𝔪φ 𝔮 ∈ (Υ̃1,∧), (Φ̃,∧)c ⊆ (Υ̃2,∧) and (Υ̃1,∧) ∩ (Υ̃2,∧) = 0(𝔐,Q). Then we have 𝔪φ 𝔮 ∈ (Υ̃1,∧) ⊆ (Υ̃2,∧)c ⊆ (Φ̃,∧). Since (Υ̃2,∧)c is a FHSδcs (resp. FHSδ𝒮cs & FHSδ𝒫cs), FHScl(Υ̃1,∧) ⊆ (Υ̃2,∧)c. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 844 https://internationalpubls.com Conversely, let 𝔪φ 𝔮 ∩ (Ω̃,∧) = 0(𝔐,Q) and (Ω̃,∧) be a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs). Then 𝔪φ 𝔮 ∈ (Ω̃,∧)c and from the condition of the theorem, we have 𝔪φ 𝔮 ∈ (Υ̃,∧) ⊆ FHScl(Υ̃,∧ ) ⊆ (Ω̃,∧)c. Thus 𝔪φ 𝔮 ∈ (Υ̃,∧), (Ω̃,∧) ⊆ (FHScl(Υ̃,∧))c and (Υ̃,∧) ∩ (FHScl(Υ̃,∧))c = 0(𝔐,Q). So (𝔐, Q, τ̃) is a FHSδ (resp. FHSθ𝒮 & FHSθ𝒫)- T3-space Definition 3.9 A FHSts (𝔐, Q, τ̃) over 𝔐 is called a FHS θ (resp. θ semi & θ pre)-normal (briefly, FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)-normal) space, if for every pair of disjoint FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs)’s (Φ̃1,∧), (Φ̃2,∧), there exist disjoint FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s (Ω̃1,∧), (Ω̃2,∧) such that (Φ̃1,∧) ⊆ (Ω̃1,∧) and (Φ̃2,∧) ⊆ (Ω̃2,∧). (𝔐, Q, τ̃) is said to be a FHS θ (resp. θ semi & θ pre)- T4-space (briefly, FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T4- space) if it is both a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)-normal and FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T1-space. Theorem 3.7 Let (𝔐, Q, τ̃) be a FHSts over 𝔐. Then (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T4-space iff for each FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs) (Φ̃,∧) and FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Ω̃,∧) with (Φ̃,∧) ⊆ (Ω̃,∧), there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Υ̃,∧) such that (Φ̃,∧) ⊆ (Υ̃,∧) ⊆ FHScl(Υ̃,∧) ⊆ (Ω̃,∧). Proof. Let (𝔐, Q, τ̃) be a FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T4-space. Let (Φ̃,∧) be a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs) and let (Φ̃,∧) ⊆ (Ω̃,∧) ∈ τ̃. Then (Ω̃,∧)c is a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs) and (Φ̃,∧) ∩ (Ω̃,∧)c = 0(𝔐,Q). Since (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T4- space, there exist FHSθos (resp. FHS𝒮os FHS𝒫os, FHSθ𝒮os & FHSθ𝒫os)’s (Υ̃1,∧) and (Υ̃2,∧) such that (Φ̃,∧) ⊆ (Υ̃1,∧), (Ω̃,∧)c ⊆ (Υ̃2,∧) and (Υ̃1,∧) ∩ (Υ̃2,∧) = 0(𝔐,Q). Thus (Φ̃,∧) ⊆ (Υ̃1,∧) ⊆ (Υ̃2,∧)c ⊆ (Ω̃,∧), (Υ̃2,∧)c is a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs) and (Υ̃1,∧) ⊆ (Υ̃2,∧)c. So, (Φ̃,∧) ⊆ (Υ̃1,∧) ⊆ FHScl(Υ̃1,∧) ⊆ (Ω̃,∧). Conversely, let (Φ̃1,∧), (Φ̃2,∧) be two disjoint FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs)’s. Then (Φ̃1,∧) ⊆ (Φ̃2,∧)c. From the condition of theorem, there exists a FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os) (Υ̃,∧) such that (Φ̃1,∧) ⊆ (Υ̃,∧) ⊆ FHScl(Υ̃1,∧) ⊆ (Φ̃2,∧)c. Thus (Υ̃,∧), (FHScl(Υ̃,∧))c are FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s and (Φ̃1,∧) ⊆ (Υ̃,∧), (Φ̃2,∧) ⊆ (FHScl(Υ̃,∧))c and (Υ̃,∧) ∩ (FHScl(Υ̃,∧))c = 0(𝔐,Q). So (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮, & FHSθ𝒫)- T4-space Theorem 3.8 Let (𝔐, Q, τ̃) be a FHSts over 𝔐. If (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- Ti-space, then the FHSts ((Φ̃,∧), τ̃(Φ̃,∧), Q) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- Ti- space for i = 0,1,2,3. Proof. Let 𝔪φ q , 𝔫φ′ q′ ∈ ((Φ̃,∧), τ̃(Φ̃,∧), Q) such that 𝔪φ q ∩ 𝔫φ′ q′ = 0(𝔐,Q). Then there exist FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s (Φ̃1,∧) and (Φ̃2,∧) satisfying the conditions of FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- Ti-space such that 𝔪φ q ∈ (Φ̃1,∧), 𝔫φ′ q′ ∈ (Φ̃2,∧). Thus, 𝔪φ q ∈ (Φ̃1,∧ ) ∩ (Φ̃,∧) and 𝔫φ′ q′ ∈ (Φ̃2,∧) ∩ (Φ̃,∧). Also, the FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s (Φ̃1,∧ ) ∩ (Φ̃,∧), (Φ̃2,∧) ∩ (Φ̃,∧) in τ̃(Φ̃,∧) satisfy the conditions of FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- Ti-space for i = 0,1,2,3. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 845 https://internationalpubls.com Theorem 3.9 Let (𝔐, Q, τ̃) be a FHSts over 𝔐. If (𝔐, Q, τ̃) is a FHSθ (resp.FHySθ𝒮 & FHSθ𝒫)- T4-space and (Ω̃,∧) is a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs) in (𝔐, Q, τ̃), then ((Ω̃,∧ ), τ̃(Ω̃,∧), Q) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T4-space. Proof. Let (𝔐, Q, τ̃) be a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T4-space and (Ω̃,∧) be a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs) in (𝔐, Q, τ̃). Let (Ω̃1,∧) and (Ω̃2,∧) be two FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs)’s in ((Ω̃,∧), τ̃(Ω̃,∧), Q) such that (Ω̃1,∧) ∩ (Ω̃2,∧) = 0(𝔐,Q). When (Ω̃,∧) is a FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs) in (𝔐, Q, τ̃), (Ω̃1,∧) and (Ω̃2,∧) are FHSθcs (resp. FHSθ𝒮cs & FHSθ𝒫cs)’s in (𝔐, Q, τ̃). Since (𝔐, Q, τ̃) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T4- space, there exist FHSθos (resp. FHSθ𝒮os & FHSθ𝒫os)’s (Υ̃1,∧) and (Υ̃2,∧) such that (Ω̃1,∧ ) ⊆ (Υ̃1,∧), (Ω̃2,∧) ⊆ (Υ̃2,∧) and (Υ̃1,∧) ∩ (Υ̃2,∧) = 0(𝔐,Q). Then (Ω̃1,∧) = (Υ̃1,∧) ∩ (Ω̃,∧), (Ω̃2,∧) = (Υ̃2,∧) ∩ (Ω̃,∧) and ((Υ̃1,∧) ∩ (Ω̃,∧)) ∩ ((Υ̃2,∧) ∩ (Ω̃,∧)) = 0(𝔐,Q). Hence ((Ω̃,∧ ), τ̃(Ω̃,∧), Q) is a FHSθ (resp. FHSθ𝒮 & FHSθ𝒫)- T4-space. 4 Conclusion In this paper, FHSθ (resp. θ semi & θ pre)-separation axioms in FHSts are introduced and studied using FHSp’s. The relation and properties between FHSθ (resp. θ semi & θ pre)- Ti- spaces (i = 0,1,2,3,4) are also discussed. The future work can involves the investigation of FHSθ (resp. θ semi & θ pre)- compactness, FHSθ (resp. θ semi & θ pre)- connectedness and FHS contra θ (resp. θ semi & θ pre)- continuous functions. References [1] A. Acikgoz and F. 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