On Fermatean Neutrosophic e-continuous and e-irresolute Maps

University of New Mexico On Fermatean Neutrosophic e-continuous and e-irresolute Maps Vadivel A1∗, Thamilarasi P2, and Priya S3 1,2PG and Research Department of Mathematics, Arignar Anna Government Arts College, Namakkal - 637 002, India; (Vadivel A) a[email protected], (Thamilarasi P) [email protected] 3Department of Mathematics, M.Kumarasamy College of Engineering, Karur - 639 113.; (Priya S) [email protected] 1,2,3Department of Mathematics, Annamalai University, Annamalai Nagar - 608 002, India; ∗Correspondence: (Priya S) [email protected]; Tel.: (+91 ) Abstract.The primary objective of this paper is to introduce and develop the concept of Fermatean neutrosophic e-continuous maps in the framework of Fermatean neutrosophic topological spaces. This new class of mappings extends the idea of continuity by incorporating the higher expressive power of Fermatean neutrosophic sets, which are capable of handling more complex degrees of truth, indeterminacy, and falsity than classical fuzzy or intuitionistic fuzzy settings. In addition to defining and studying the basic formulation of Fermatean neutrosophic e-continuity, we investigate its fundamental properties, structural behavior, and interrelations with other existing types of continuity. Furthermore, we examine the notion of Fermatean neutrosophic e-irresolute maps, which play an important role in the preservation of Fermatean neutrosophic topological structures under mappings. Several characterizations and properties of these maps are provided, establishing their significance in extending the theory of neutrosophic topology. The results presented not only generalize existing concepts from classical and fuzzy topology but also open new avenues for applications of Fermatean neutrosophic theory in decision-making, information systems, and uncertain data analysis. Keywords: Fermatean neutrosophic e-closed sets, fermatean neutrosophic e-continuous maps and Fermatean neutrosophic e-irresolute maps. —————————————————————————————————————————- 1. Introduction In 1965, Zadeh [17] introduced the concept of fuzzy sets to model ambiguity and uncertainty in real-world scenarios. Later, in 1986, Atanassov [1] extended this idea by proposing the concept of intuitionistic fuzzy sets (ifs’s), which incorporated both membership and non-membership functions addressing a limitation in Zadehs model that only considered the membership degree. However, in practical applications, the sum of membership and nonmembership degrees can exceed one, indicating a need for more flexible models. Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 To address these challenges, Yager [16] introduced the Pythagorean fuzzy set (pfs), where the sum of the squares of the membership and non-membership degrees is constrained to be less than or equal to one. Building on this, Senapati and Yager [9] proposed the Fermatean fuzzy set (Ffs), where the cube of the membership and non-membership degrees must sum to less than or equal to one. This extension provides greater flexibility and enhances the capability of managing uncertainty, making Fermatean fuzzy sets more efficient than ifs’s and pfs’s in decision-making and other real-life applications. Smarandache [10] later introduced the concept of neutrosophic sets (NS’s), representing a major advancement in the field of decision theory and beyond. Neutrosophic sets are built on the idea that any concept inherently involves degrees of truth (T), indeterminacy (I), and falsity (F). Unlike intuitionistic fuzzy sets where membership and non-membership are dependent neutrosophic sets allow these three components to be mutually independent, thereby offering greater modeling flexibility for incomplete, inconsistent, and indeterminate information. This independence makes neutrosophic sets a powerful tool for representing uncertainty in a wide range of fields, including engineering, philosophy, economics, and information science. In particular, they support complex analyses involving both dependent and independent variables, contributing significantly to decision-making and strategic planning. To leverage the strengths of both Fermatean fuzzy sets and neutrosophic sets, the Fermatean neutrosophic set was developed. This hybrid framework effectively handles uncertainty, imprecision, and indeterminacy in complex decision making contexts. For example, in evaluating a local restaurant, Fermatean fuzzy theory is applied to assess the importance of attributes such as quality, naturalness, freshness, taste, and presentation which are often subject to human hesitation and subjective judgment. By incorporating neutrosophic principles, the model better captures the vagueness and uncertainty in customer preferences. The domain of local food service was chosen as a practical application due to the high degree of uncertainty in consumer decision making uncertainty that neutrosophic sets can represent more accurately than traditional methods. From a topological perspective, Vadivel et al. [12] introduced the notion of δ-open sets within neutrosophic topological spaces. Prior to this, in 2008, Ekici [4] introduced e-open sets in general topology. Building on this foundation, Seenivasan et al. [8] in 2014 developed the concept of fuzzy e-open sets and their corresponding fuzzy e-continuity. Later, Vadivel et al. [3] extended these ideas into the realm of intuitionistic fuzzy topological spaces. More recently, Vadivel and collaborators [13–15] have made notable contributions by exploring various types of open sets in Fermatean fuzzy topological spaces, advancing the study of fuzzy and neutrosophic topology. Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 411 Research Gap: To the best of our knowledge, no prior investigation has been carried out on the concepts of continuity and irresoluteness with respect to Fermatean neutrosophic e-open sets within the framework of Fermatean fuzzy topological spaces. While various notions of continuity and irresolute mappings have been studied extensively in classical topology, fuzzy topology, and intuitionistic fuzzy topology, the specific treatment of these notions under the richer structure of Fermatean neutrosophic sets remains unexplored in the existing literature. This absence highlights a significant research gap, thereby motivating the present study to initiate and advance the theory of Fermatean neutrosophic e-continuous and e-irresolute maps. In this paper, we put forward the novel concept of Fermatean neutrosophic e-continuity within the framework of Fermatean neutrosophic topological spaces. The study begins with a formal definition of this new type of continuity, followed by an investigation of its essential properties. To enhance clarity and demonstrate the practical significance of the proposed ideas, several illustrative examples are provided. These examples not only validate the theoretical framework but also highlight how Fermatean neutrosophic e-continuity extends and generalizes existing notions of continuity in fuzzy and neutrosophic topological settings. Furthermore, the paper delves into the study of Fermatean neutrosophic e-irresolute maps, presenting their fundamental properties and characterizations in a comprehensive manner. Special attention is given to their structural behavior and their role in preserving Fermatean neutrosophic e-open sets under mappings. By establishing these results, the paper contributes to the enrichment of Fermatean neutrosophic topology and provides a foundation for further theoretical advancements and practical applications in areas involving uncertainty, vagueness, and complex decision-making processes. 2. Preliminaries Definition 2.1. [9] Let Xbe a universe of discourse. A Fermatean fuzzy set (FFs)Fin Xis an object having the form F={< x, αF(x), βF(x)>:x∈X}where αF(x) : X→ [0,1] and βF(x) : X→[0,1], including the condition 0 ≤(αF(x))3+ (βF(x))3≤1, for all x∈X. The numbers αF(x) and βF(x) denote, respectively, the degree of membership and the degree of non-membership of the element xin the set F. For any FFs F and x∈X, πF(x) = 3 √1−[(αF(x))3−(βF(x))3] is identified as the degree of indeterminacy of xto F. In the interest of simplicity, we shall mention the symbol F= (αF, βF) for the FFs F ={< x, αF(x), βF(x)>:x∈X}. Definition 2.2. [7] Let Xbe a non-empty set. A neutrosophic set (briefly, Ns)Lis an object having the form L={⟨x, µL(x), νL(x), σL(x)⟩:x∈X}where µL→[0,1] denote the degree of membership function, νL→[0,1] denote the degree of indeterminacy function and Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 412 σL→[0,1] denote the degree of non-membership function respectively of each element x∈X to the set Land 0 ≤µL(x) + νL(x) + σL(x)≤3 for each x∈X. Definition 2.3. [6] A neutrosophic topology (briefly, Nt) on a non-empty set Xis a family τNof neutrosophic subsets of Xsatisfying (i) 0N, 1N∈τN. (ii) L1∩L2∈τNfor any L1, L2∈τN. (iii) ∪La∈τN,∀La:a∈A⊆τN. Then (X, τN) is called a neutrosophic topological space (briefly, Nts) in X. The τNelements are called neutrosophic open sets (briefly, Nos) in X. A Ns C is called a neutrosophic closed sets (briefly, Ncs) iff its complement Ccis Nos. Definition 2.4. [11] Let Xbe a non-empty set. A Fermatean neutrosophic set (briefly, FNs) Lis an object having the form L={⟨x, µL(x), νL(x), σL(x)⟩:x∈X}where µL→[0,1] denote the degree of membership function, νL→[0,1] denote the degree of indeterminacy function and σL→[0,1] denote the degree of non-membership function respectively of each element x∈Xto the set Lsuch that 0 ≤(µL(x))3+(σL(x))3≤1 and 0 ≤(νL(x))3≤1. Then 0≤(µL(x))3+ (νL(x))3+ (σL(x))3≤2 for all x∈X. Here µL(x) and σL(x) are dependent components and νL(x) is an independent component. The definitions of 1FN and 0FN that will be needed before proceeding to set operations will be given. In [6], possible definitions of 1FN and 0FN neutrosophic sets are given. In this paper, the theory will be constructed by defining 0FN and 1FN Fermatean neutrosophic sets in a single way. 0FN and 1FN are defined as 0FN ={(x, 0,0,1) : x∈X}and 1FN ={(x, 1,1,0) : x∈X} Now, the union, intersection and complement definitions necessary for the definition of the topological space will be given. These definitions are given in several different ways in classical neutrosophic spaces in [2]; to avoid confusion here, only one method will be given for sets with Fermatean structure, and this method is different from the method chosen in [6]. Definition 2.5. [11] Let Xbe a non-empty set & the FNs’s L&Min the form L= {⟨x, µL(x), νL(x), σL(x)⟩:x∈X},M={⟨x, µM(x), νM(x), σM(x)⟩:x∈X}, then (i) 0FN =⟨x, 0,0,1⟩and 1FN =⟨x, 1,1,0⟩, (ii) L⊆Miff µL(x)≤µM(x), νL(x)≤νM(x) & σL(x)≥σM(x) : x∈X, (iii) L=Miff L⊆Mand M⊆L, (iv) 1FN −L={⟨x, σL(x),1FN −νL(x), µL(x)⟩:x∈X}=Lcor C(L), (v) L∪M={⟨x, max(µL(x), µM(x)),max(νL(x), νM(x)),min(σL(x), σM(x))⟩:x∈X}, (vi) L∩M={⟨x, min(µL(x), µM(x)),min(νL(x), νM(x)),max(σL(x), σM(x))⟩:x∈X}. Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 413 Definition 2.6. [5] A Fermatean neutrosophic topology (briefly, FNt) on a non-empty set X is a family τFN of Fermatean neutrosophic subsets of Xsatisfying (i) 0FN, 1FN ∈τFN, (ii) L1∩L2∈τFN for any L1, L2∈τFN, (iii) ∪La∈τFN,∀La:a∈A⊆τFN. Then (X, τFN) is called a Fermatean neutrosophic topological space (briefly, FNts) in X. The τFN elements are called Fermatean neutrosophic open sets (briefly, FNos) in X. A FNs C is called a Fermatean neutrosophic closed sets (briefly, FNcs) iff its complement Ccis FNos. Definition 2.7. [5] Let (X, τFN) be FNts on Xand Lbe an FNson X, then the Fermatean neutrosophic interior of L(briefly, FNint(L)) and the Fermatean neutrosophic closure of L (briefly, FNcl(L)) are defined as FNint(L) = ∪{I:I⊆L&Iis a FNos in X} FNcl(L) = ∩{I:L⊆I&Iis a FNcs in X}. Theorem 2.8. [5] Let Lbe an FNson X. In this case, the following four properties hold: (i) FNcl(L) is a closed Fermatean neutrosophic set, (ii) FNcl(1FN) = 1FN,FNcl(0FN) = 0FN, (iii) FNint(L) is an open Fermatean neutrosophic set. (iv) FNint(1FN) = 1FN,FNint(0FN) = 0FN. Lemma 2.9. [5] For any Fermatean neutrosophic set Ain (X, τFN), we have C(FNint(A)) = FNcl(C(A)) and C(FNcl(A)) = FNint(C(A)).Here C(A) or Adenotes complement of A. 3. Fermatean neutrosophic e-continuous maps In this section we introduce Fermatean neutrosophic e-continuous maps and study some of its properties. Definition 3.1. Let (X, τFN) be an FNts and Abe an FNs. Then Ais said to be an Fermatean neutrosophic (i) regular open set (FNros in short) if A=FNint(FNcl(A)).(ii) regular closed set (FNrcs in short) if A=FNcl(FNint(A)).By Lemma 2.9, it follows that Ais an FNros iff ¯ Ais an FNrcs. Definition 3.2. Let (X, τFN) be an FNts and A={< a, µA(a), νA(a), σA(a)>|a∈X}be an FNsin X. Then the δ-interior and the δ-closure of Aare denoted by FNδint(A) and FNδcl(A) and are defined as follows. FNδint(A) = ∪{G|Gis an FNros and G⊆A},FNδcl(A) = ∩{K|K is an FNrcs and A⊆K}. Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 414 Definition 3.3. Let (X, τFN) be an FNts and A={< a, µA(a), νA(a), σA(a)>|a∈X}be an FNsin X. A set Ais said to be FN (i) δ-open set (briefly, FNδos) if A=FNδint(A), (ii) δ-pre open set (briefly, FNδPos) if A⊆FNint(FNδcl(A)). (iii) δ-semi open set (briefly, FNδSos) if A⊆FNcl(FNδint(A)). (iv) δα open set or a-open set (briefly, FNδαos or FNaos) if A⊆FNint(FNcl(FNδint(A))). (v) δβ open set or e∗-open set (briefly, FNδβos or FNe∗os) if A⊆FNcl(FNint(FNδcl(A))). (vi) eopen set (briefly, FNeos) if A⊆FNcl(FNδint(A)) ∪FNint(FNδcl(A)). (vii) δ(resp. δ-pre, δ-semi, δα,eand δβ) dense if FNδcl(A) (resp. FNδPcl(A), FNδScl(A),FNδαcl(A),FNecl(A) and FNδβcl(A)) = 1FN. The complement of an FNδos (resp. FNδPos, FNδSos, FNδαos,FNeos and FNδβos) is called an FNδ(resp. FNδP,FNδS,FNδα,FNeand FNδβ) closed set (briefly, FNδcs (resp. FNδPcs, FNδScs, FNδαcs,FNecs and FNδβcs)) in X. The family of all FNδos (resp. FNδcs, FNδPos, FNδPcs, FNδSos, FNδScs, FNδαos, FNδαcs, FNeos, FNecs, FNδβos and FNδβcs) of Xis denoted by FNδOS(X),(resp. FNδCS(X),FNδPOS(X),FNδPCS(X),FNδSOS(X),FNδSCS(X), FNδαOS(X),FNδαCS(X),FNeOS(X),FNeCS(X), FNδβOS(X) and FNδβCS(X)). Definition 3.4. Let (X, τFN) be an FNts and A={< a, µA(a), νA(a), σA(a)>|a∈X}be an FNsin X. Then the FNδ(resp. FNδ-pre, FNδ-semi, FNδα,FNeand FNδβ)-interior and the FNδ(resp. FNδ-pre, FNδ-semi, FNδα,FNeand FNδβ)-closure of Aare denoted by FNδint(A) (resp. FNδPint(A), FNδSint(A), FNδαint(A), FNeint(A) and FNδβint(A)) and the FNδcl(A) (resp. FNδPcl(A), FNδScl(A),FNδαcl(A), FNecl(A) and FNδβcl(A)) and are defined as follows: FNδint(A) (resp. FNδPint(A), FNδSint(A),FNδαint(A), FNeint(A) and FNδβint(A) ) = ∪{G|Gin a FNδos (resp. FNδPos, FNδSos, FNδαos, FNeos, and FNδβos) and G⊆A}and FNδcl(A) (resp. FNδPcl(A),FNδScl(A),FNδαcl(A),FNecl(A) and FNδβcl(A) ) = ∩{K|K is an FNδcs (resp. FNδPcs, FNδScs, FNδαcs, FNecs FNδβcs) and A⊆K}. Definition 3.5. A map f: (X, τFN)→(Y, σFN) is called Fermatean neutrosophic (i) continuous (briefly, FNCts) if the inverse image of every FNos in (Y, σFN) is a FNos in (X, τFN), (ii) δ-continuous (briefly, FNδCts) if the inverse image of every FNos in (Y, σFN) is a FNδos in (X, τFN), (iii) δS-continuous (briefly, FNδSCts) if the inverse image of every FNos in (Y, σFN) is a FNδSos in (X, τFN), Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 415 (iv) δP-continuous (briefly, FNδPCts) if the inverse image of every FNos in (Y, σFN) is a FNδPos in (X, τFN), (v) δα-continuous (briefly, FNδαCts) if the inverse image of every FNos in (Y, σFN) is a FNδαos in (X, τFN), (vi) e-continuous (briefly, FNeCts) if the inverse image of every FNos in (Y, σFN) is a FNeos in (X, τFN), (vii) δβ-continuous (briefly, FNδβCts) if the inverse image of every FNos in (Y, σFN) is a FNδβos in (X, τFN). Proposition 3.6. The statements are hold but the converse does not true. (i) Every FNδCts is a FNCts. (ii) Every FNδCts is a FNδSCts. (iii) Every FNδCts is a FNδPCts. (iv) Every FNδPCts is a FNeCts. (v) Every FNδSCts is a FNeCts. (vi) Every FNδPCts is a FNδβCts. (vii) Every FNδSCts is a FNδβCts. (viii) Every FNδαCts is a FNδSCts. (ix) Every FNδαCts is a FNδPCts. Proof. We prove only (iv) and (v), the others are similar. (iv) Let λbe a FNos in Y. Since fis FNδPCts,f−1(λ) is a FNδPos in X. Since every FNδPos is a FNeos,f−1(λ) is a FNeos in X. Hence fis a FNeCts. (v) Let λbe a FNos in Y. Since fis FNδSCts,f−1(λ) is a FNδSos in X. Since every FNδSos is a FNeos,f−1(λ) is a FNeos in X. Hence fis a FNeCts. Example 3.7. Let X=Y={a, b}and the FNs’s A1,A2and A3are defined as µA1(a) = 0.8, νA1(a) = 0.8, σA1(a) = 0.1, µA1(b) = 0.9, νA1(b) = 0.8, σA1(b) = 0.2; µA2(a) = 0.6, νA2(a) = 0.7, σA2(a) = 0.2, µA2(b) = 0.5, νA2(b) = 0.7, σA2(b) = 0.6; µA3(a) = 0.1, νA3(a) = 0.2, σA3(a) = 0.8, µA3(b) = 0.2, νA3(b) = 0.2, σA3(b) = 0.9. Let τFN =σFN ={0FN,1FN, A1, A2, A3}be a FNts on Xand let f: (X, τFN)→(Y, σFN) be an identity mapping, then fis FNCts (resp. FNδPCts,FNδPCts,FNeCts and FNδβCts) but not FNδCts (resp. FNδαCts,FNδCts,FNδSCts and FNδSCts), the set A2is a FNos in Ybut f−1(A2) = A2is not FNδos (resp. FNδαos,FNδos,FNδSos and FNδSos) in X. Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 416 Example 3.8. Let X=Y={a, b}and the FNs’s A1,A2and A3are defined as µA1(a) = 0.2, νA1(a) = 0.5, σA1(a) = 0.8, µA1(b) = 0.3, νA1(b) = 0.5, σA1(b) = 0.7; µA2(a) = 0.1, νA2(a) = 0.5, σA2(a) = 0.9, µA2(b) = 0.1, νA2(b) = 0.5, σA2(b) = 0.9; µA3(a) = 0.2, νA3(a) = 0.5, σA3(a) = 0.8, µA3(b) = 0.4, νA3(b) = 0.5, σA3(b) = 0.6. Let τFN =σFN ={0FN,1FN, A1, A2, A3}be a FNts on Xand let f: (X, τFN)→(Y, σFN) be an identity mapping, then fis FNδSCts but not FNδCts, the set A1is a FNos in Ybut f−1(A1) = A1is not FNδos in X. Example 3.9. Let X=Y={a, b, c, d}and the FNs’s A1,A2,A3,A4and A5are defined as µA1(a) = 1, νA1(a) = 0.5, σA1(a) = 0, µA1(b) = 1, νA1(b) = 0.5, σA1(b) = 0, µA1(c) = 1, νA1(c) = 0.5, σA1(c) = 0, µA1(d) = 1, νA1(d) = 0.5, σA1(d) = 0; µA2(a) = 1, νA2(a) = 0.5, σA2(a) = 0, µA2(b) = 0, νA2(b) = 0.5, σA2(b) = 1, µA2(c) = 0.2, νA2(c) = 0.5, σA2(c) = 0.7, µA2(d) = 0, νA2(d) = 0.5, σA2(d) = 1; µA3(a) = 0, νA3(a) = 0.5, σA3(a) = 1, µA3(b) = 1, νA3(b) = 0.5, σA3(b) = 0, µA3(c) = 0, νA3(c) = 0.5, σA3(c) = 0, µA3(d) = 0, νA3(d) = 0.5, σA3(d) = 1; µA4(a) = 1, νA4(a) = 0.5, σA4(a) = 0, µA4(b) = 1, νA4(b) = 0.5, σA4(b) = 0, µA4(c) = 0.2, νA4(c) = 0.5, σA4(c) = 0.7, µA4(d) = 0, νA4(d) = 0.5, σA4(d) = 0.1; µA5(a) = 0, νA5(a) = 0.5, σA5(a) = 1, µA5(b) = 0, νA5(b) = 0.5, σA5(b) = 1, µA5(c) = 0, νA5(c) = 0.5, σA5(c) = 1, µA5(d) = 0, νA5(d) = 0.5, σA5(d) = 1. Let τFN =σFN ={0FN,1FN, A1, A2, A3, A4, A5}be a FNts on Xand let f: (X, τFN)→ (Y, σFN) be an identity mapping, then fis FNδSCts (resp. FNeCts and FNδβCts) but not FNδαCts (resp. FNδPCts and FNδPCts), the set A2is a FNos in Ybut f−1(A2) = A2is not FNδαos (resp. FNδPos and FNδPos) in X. Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 417 From the above Proposition 3.6, and the following Example, the implications are hold. FNCts FNδCts FNδαCts FNδSCts FNeCts FNδPCts FNδβCts Note: A→Bdenotes Aimplies B, but not conversely. Theorem 3.10. A map f: (X, τFN)→(Y, σFN) is FNeCts iff the inverse image of each FNcs in Yis FNecs in X. Proof. Let λbe a FNcs in Y. This implies λcis FNos in Y. Since fis FNeCts,f−1(λc) is FNeos in X. Since f−1(λc) = (f−1(λ))c,f−1(λ) is a FNecs in X. Conversely, let λbe a FNcs in Y. Then λcis a FNos in Y. By hypothesis f−1(λc) is FNeos in X. Since f−1(λc) = (f−1(λ))c,(f−1(λ))cis a FNeos in X. Therefore f−1(λ) is a FNecs in X. Hence fis FNeCts. Definition 3.11. AFNt(X, τFN)is said to be an Fermatean neutrosophic eU1 2(in short FNeU1 2)-space, if every FNeos in Xis a FNos in X. Theorem 3.12. Let f: (X, τFN)→(Y, σFN) be a FNeCts, then fis a FNCts if Xis a FNeU1 2-space. Proof. Let λbe a FNos in Y. Then f−1(λ) is a FNeos in X, by hypothesis. Since Xis a FNeU1 2-space, f−1(λ) is a FNos in X. Hence fis a FNeCts. Theorem 3.13. Let f: (X, τFN)→(Y, σFN) be a FNeCts map and g: (Y, σFN)→(Z, ρFN) be an FNeCts, then g◦f: (X, τFN)→(Z, ρFN) is a FNeCts. Proof. Let λbe a FNeos in Z. Then g−1(λ) is a FNos in Y, by hypothesis. Since fis a FNeCts map, f−1(g−1(λ))is a FNeos in X. Hence g◦fis a FNeCts map. Theorem 3.14. Let f: (X, τFN)→(Y, σFN) be a FNeCts map. Then the following conditions are hold. (i) f(FNecl(λ)) ⊆FNcl(f(λ)), for all FNcs λ in X. Vadivel A, Thamilarasi P, and Priya S, On Fermatean Neutrosophic e-continuous and e-irresolute Maps Neutrosophic Sets and Systems, Vol. 97, 2026 418