Ratio Mathematica Volume 48, 2023 Fuzzy soft set connected mappings Alpa Singh Rajput* Mahima Thakur† Samajh Singh Thakur‡ Abstract In this paper, the concepts of fuzzy soft connectedness between fuzzy soft sets and fuzzy soft set connected mappings in fuzzy soft topolog- ical spaces has been introduced. It is shown that a fuzzy soft topo- logical space is fuzzy soft connected if and only if it is fuzzy soft connected between every pair of its nonempty fuzzy soft sets. Every fuzzy soft continuous mapping is fuzzy soft set-connected a counter example is given to show the converse may not be true. Several prop- erties of fuzzy soft set-connected mappings in fuzzy soft topological spaces have been studied. Keywords: Fuzzy soft sets; Fuzzy soft connectedness; Fuzzy soft connectedness between fuzzy soft sets and Fuzzy soft set-connected mappings. 2020 AMS subject classifications:54A40, 54D05, 54C08. 1 *Department of Science and Humanities, Vignan Foundation for Science, Technology and Re- search, Guntur, India ; asr sh@vignan.ac.in. †Department of Applied Mathematics, Jabalpur Engineering College Jabalpur, India ; mthakur@jecjabalpur.ac.in. ‡Former Professor Department of Applied Mathematics, Jabalpur Engineering College Ja- balpur, India; ssthakur@jecjabalpur.ac.in. 1Received on April 17, 2023. Accepted on July 1, 2023. Published on August 1,2023. DOI:10.23755/rm.v41i0.1180. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 1 1 Introduction The notion of a fuzzy set was introduced by Zadeh [1965] as a generalization of classical set in the year 1965. Chang [1968] gave the definition of fuzzy topol- ogy and extended some topological concepts to fuzzy sets. In 1999, Molodtsov [1999] introduced the concept of soft sets to deal with uncertainties while mod- elling the problems with incomplete information. In 2011, Shabir and Naz [2011] initiated the study of soft topological spaces as a generalization of topological spaces. The hybrid structure of fuzzy sets and soft set called fuzzy soft set was created by Maji et al. [2001]. To continue the investigation on fuzzy soft sets, Ah- mad and Kharal [2009] presented some more properties of fuzzy soft sets and in- troduced the notion of a mapping on fuzzy soft sets. Tanay and Kandemir [2011] introduced the fuzzy soft topology as an extension of fuzzy topology and soft topology. Fuzzy topological spaces are further investigated by Varol and Aygun [2012] and shown that a fuzzy soft topological space gives a parametrized family of fuzzy topological spaces. Roy and Samanta [2011] and Tridiv et al. [2012] are also investigated various topological concepts in fuzzy soft topological spaces. The study of connectedness in fuzzy soft topology was initiated by Karatas et al. [2015] and further studied by Kandil et al. [2017]. Connectedness between sets and set connected mappings is one of the important topic of research in Topol- ogy. In 2018, Thakur and Rajput [2018] extended and studied these concepts to soft topology. Till the date these concepts are not studied in fuzzy soft topology. Therefore to fill up this gap, the present paper introduces the concept of connect- edness between fuzzy soft sets and studied some of its properties in fuzzy soft topological spaces. Further the concepts of fuzzy soft set-connected mappings are defined and established some theorems related to its characterizations and prop- erties. The results established in this paper generalized many results which are already available in the literature. 2 Preliminaries Throughout this paper, X refers to an initial universe, E is the set of all param- eters for X, I = [0,1] and IX is the set of all fuzzy sets on X. The reader should refer Molodtsov [1999] and Zadeh [1965] for the basic concepts on fuzzy sets and soft sets. Definition 2.1. (Maji et al. [2001]) Let A ⊂ E. A pair (f,A) is called a fuzzy soft set (in short FSS) over X, where f : A → IX defined by , (f,A)(e) = µe(f,A) ,where µe(f,A) = 0̃ if e /∈ A and µ e (f,A) ̸= 0̃ if e ∈ A . The family of all FSSs over (X,E) will be denoted by FS(X, E). For the Fuzzy soft set connected mappings notations, basic operations and properties of FSSs the reader should refer Ahmad and Kharal [2009], Maji et al. [2001]. Definition 2.2. (Tanay and Kandemir [2011]) A subfamily τ of FS(X, E) is called a fuzzy soft topology on X if: (1) ϕ̃, X̃ belong to τ. (2) The union of any number of soft sets in τ belongs to τ. (3) The intersection of any two soft sets in τ belongs to τ. The triplet (X, τ, E) is called a fuzzy soft topological space (in short FSTS). The members of τ are called fuzzy soft open sets in X and their complements called fuzzy soft closed sets in X. The basic fuzzy soft topological concepts can be seen in Kandil et al. [2017], Karatas et al. [2015], Roy and Samanta [2011], Tridiv et al. [2012] and Varol and Aygun [2012]). Lemma 2.1. (Roy and Samanta [2011]) Let (f,A) and (g,B) be two FSSs. Then, (f, A) ⊆ (g, B)c ⇔ (f, A)q̃(g, B). Where (f, A)q̃(g, B) means (f,A) is not quasi- coincident with (g,B). Lemma 2.2. (Tridiv et al. [2012]) Let (Y, τY , E) be a fuzzy soft subspace of a FSTS (X, τ, E) and (F, E) be a fuzzy soft open set in Y. If Ỹ ∈ τ then (F, E) ∈ τ. Lemma 2.3. (Tridiv et al. [2012]) Let (X,τ,E) be a FSTS and (Y, τY ,E) be a fuzzy soft subspace of (X, τ, E), then a fuzzy soft closed set (FY , E) of Y is fuzzy soft closed in X if and only if Ỹ is fuzzy soft closed in X. 3 Connectedness between fuzzy soft sets Throughout this paper fuzzy soft clopen means fuzzy soft closed open. Definition 3.1. (Kandil et al. [2017], Karatas et al. [2015]) A FSTS (X,τ,E) is fuzzy soft connected if and only if there is no nonempty FSS of (X,τ,E) which is both fuzzy soft open and fuzzy soft closed in (X,τ,E). Definition 3.2. A FSTS (X ,τ, E) is said to be fuzzy soft connected between FSSs (f1,E) and (f2,E) if and only if there is no fuzzy soft clopen set (f,E) over X such that (f1,E) ⊂ (f,E) and (f,E) q̃ (f2,E). Theorem 3.1. A FSTS (X,τ,E) is fuzzy soft connected between FSSs (f1,E) and (f2,E) if and only if there is no fuzzy soft clopen set (f,E) over X such that (f1,E) ⊂ (f,E) ⊂ (f2, E)c. Proof. Follows from Definition 3.2. and Lemma 2.1.2 Theorem 3.2. If a FSTS (X,τ,E) is fuzzy soft connected between FSSs (f1,E) and (f2,E) then (f1,E) ̸= ϕ ̸= (f2,E). Proof. If any FSS (f1,E) = ϕ, then (f1,E) is a fuzzy soft clopen set over X such that (f1,E) ⊂ (f2,E) and (f1,E) q̃ (f2,E) and hence (X,τ,E) can not be fuzzy soft connected between FSSs (f1,E) and (f2,E), which is contradiction.2 Theorem 3.3. If a FSTS (X,τ,E) is fuzzy soft connected between FSSs (f1,E) and (f2,E) and if (f1,E) ⊂ (f3,E) and (f2,E) ⊂ (f4,E) then (X,τ,E) is fuzzy soft connected between FSSs (f3,E) and (f4,E). Proof. Suppose FSTS (X,τ,E) is not fuzzy soft connected between FSSs (f3,E) and (f4,E) then there is a fuzzy soft clopen set (f,E) over X such that (f3,E) ⊂ (f,E) and (f,E) q̃ (f4,E) .Clearly (f1,E) ⊂ (f,E). Now we claim that (f,E) q̃ (f2,E) .If (f,E) q (f2,E) then there exists a point x ∈ X such that µe(f,E)(x ) + µ e (f2,E) (x) ≻ 1. Therefore µe(f,E)(x ) + µ e (f4,E) (x) ≻ µe(f,E)(x ) + µ e (f2,E) (x) ≻ 1 and (f,E) q (f4,E), a contradiction. Consequently, (X,τ,E) is not fuzzy soft connected between FSSs (f1,E) and (f2,E).2 Theorem 3.4. A FSTS (X,τ,E) is fuzzy soft connected between FSSs (f1,E) and (f2,E) if and only if (X,τ,E) is fuzzy soft connected between FSSs Cl(f1,E) and Cl(f2,E). Proof. Necessity : Follows from Theorem 3.3. Sufficiency : Suppose FSTS (X,τ,E) is not fuzzy soft connected between FSSs (f1,E) and (f2,E), then there exists fuzzy soft clopen set (f,E) over X such that (f1,E) ⊂ (f,E) and (f,E) q̃ (f2,E). Since (f,E) is fuzzy soft closed, Cl(f1,E) ⊂ Cl(f,E) = (f,E). Clearly, by Lemma 2.1, (f,E) q̃ (f2,E) ⇔ (f,E) ⊂ (f2, E)c. Therefore (f,E) = Int(f,E) ⊂ Int((f2, E)c) = (Cl(f2, E))c. Hence, (f,E) q̃ Cl(f2,E) and (X,τ,E) is not fuzzy soft connected between FSSs Cl(F1,E) and Cl(F2,E).2 Theorem 3.5. If (f1,E) and (f2,E) are two FSSs in FSTS (X,τ,E) and (f1,E) q (f2,E), then (X,τ,E) is fuzzy soft connected between (f1,E) and (f2,E). Proof. If (f,E) is any fuzzy soft clopen set over X such that (f1,E) ⊂ (f,E), then (f1,E) q (f2,E) ⇒ (f,E) q (f2,E). This proves the theorem.2 Remark 3.1. The converse of Theorem 3.5 need not be true . Example 3.1. Let X = {a,b} be universe set and E = {e1 ,e2 } be the set of Fuzzy soft set connected mappings parameters. The FSSs Let (f,E), (f1,E) and (f2,E) over X are defined as follows: f1(e1) = {(a, 0.3), (b, 0.4)} f1(e2) = {(a, 0.2), (b, 0.3)} f2(e1) = {(a, 0.5), (b, 0.4)} f2(e2) = {(a, 0.6), (b, 0.5)} f3(e1) = {(a, 0.3), (b, 0.5)} f3(e2) = {(a, 0.3), (b, 0.2)}. Let τ = {0̃E , 1̃E, (f1, E) } be a fuzzy soft topology on X, then FSTS (X,τ ,E) is fuzzy soft connected between the FSSs (f2,E) and (f3,E), but (f2,E) q̃ (f3,E). Theorem 3.6. If a FSTS (X,τ,E) is neither fuzzy soft connected between (f,E) and (g0,E) nor fuzzy soft connected between (f,E) and (g1,E) then it is not fuzzy soft connected between (f,E) and (g0,E) ∪ (g1,E). Proof. Since a FSTS (X,τ,E) is not fuzzy soft connected between (f ,E) and (g0,E), there is a fuzzy soft clopen set (h0,E) over X such that (f,E) ⊂ (h0,E) and (h0,E) q̃ (g0,E). Also since (X,τ,E) is not fuzzy soft connected between (f ,E) and (g1,E) there exists a fuzzy soft clopen set (h1,E) over X such that (f,E) ⊂ (h1,E) and (g1,E) q̃ (h1,E). Put (h,E) = (h0,E) ∩ (h1,E). Since any intersection of fuzzy soft closed sets is fuzzy soft closed, (h,E) is fuzzy soft closed. Again intersection of finite family of fuzzy soft open sets is fuzzy soft open, (h,E) is fuzzy soft open. Therefore (h,E) is fuzzy soft clopen set over X such that (f,E) ⊂ (h,E) and (h,E) q̃((g0,E) ∪ (g1, E)). If (h,E) q ((g0,E) ∪ (g1, E)) there exists x ∈ X such that µe(h,E) (x) + (µe(g0,E) ∪ µ e (g1,E) )(x) ≻ 1. This implies that (h,E) q (g0,E) or (h,E) q (g1,E) a contradiction. Hence, (X,τ,E) is not fuzzy soft connected between (f,E) and (g0, E) ∪ (g1, E). 2 Theorem 3.7. A FSTS (X,τ,E) is fuzzy soft connected if and only if it is fuzzy soft connected between every pair of its nonempty FSSs. Proof. Let (f,E) and (g,E) be a pair of nonempty FSSs over X. Suppose (X,τ,E) is not fuzzy soft connected between (f,E) and (g,E). Then there is a fuzzy soft clopen set (h,E) over X such that (f,E) ⊂ (h,E) and (g,E) q̃ (h,E). Since (f,E) and (g,E) are nonempty it follows that (h,E) is a nonempty fuzzy soft proper clopen set over X. Hence, (X,τ,E) is not fuzzy soft connected. Conversely, suppose that (X,τ,E) is not fuzzy soft connected. Then there exists a nonempty proper FSS (h,E) over X which is both fuzzy soft open and fuzzy soft closed. Consequently, (X,τ,E) is not fuzzy soft connected between (h,E) and (h, E)c. Thus, (X,τ,E) is not fuzzy soft connected between arbitrary pair of its nonempty FSSs.2 Remark 3.2. If a FSTS (X,τ,E) is fuzzy soft connected between a pair of its FSSs, then it is not necessarily that it is fuzzy soft connected between each pair of its FSSs and so it is not necessarily fuzzy soft connected. Example 3.2. Let X = {a,b} be an universe set, E = {e1, e2} be the set of param- eter and the soft sets (f1, E) , (f2, E) , (f3,E), (f4,E), (f5,E) and (f6,E) over X are defined as follows: f1(e1) = {(a, 0.3), (b, 0.4)} f1(e2) = {(a, 0.2), (b, 0.3)} f2(e1) = {(a, 0.7), (b, 0.6)} f2(e2) = {(a, 0.8), (b, 0.7)} f3(e1) = {(a, 0.3), (b, 0.1)} f3(e2) = {(a, 0.4), (b, 0.2)} f4(e1) = {(a, 0.8), (b, 0.7)} f4(e2) = {(a, 0.9), (b, 0.8)} f5(e1) = {(a, 0.2), (b, 0.1)} f5(e2) = {(a, 0.1), (b, 0.2)} f6(e1) = {(a, 0.4), (b, 0.3)} f6(e2) = {(a, 0.5), (b, 0.4)}. Let τ = {0̃E, 1̃E, ( f1, E), (f2, E)} be a fuzzy soft topology over X. Then the FSTS (X, τ, E) is fuzzy soft connected between the FSSs (f3, E) and (f4, E) but it is not fuzzy soft connected between (f5,E) and (f6,E). Also the FSTS (X, τ, E) is not fuzzy soft connected. Theorem 3.8. Let (Y,τY ,E) be a fuzzy soft subspace of a FSTS (X,τ,E). If (Y,τY ,E) is fuzzy soft connected between the FSSs (f ,E) and (g,E) over Y, then FSTS (X,τ,E) is fuzzy soft connected between (f ,E) and (g,E). Proof. Suppose FSTS (X,τ,E) is not fuzzy soft connected between FSSs (f,E) and (g,E),then there is fuzzy soft clopen set (h,E) over X such that (f,E) ⊂ (h,E) and (h,E) q̃ (g,E). Then Ỹ ∩ (h ,E) is fuzzy soft clopen over Y such that (f,E) ⊂ (h,E) ∩ Ỹ and {(h,E) ∩ Ỹ } q̃ (g ,E). Consequently, (Y,τY ,E) is not fuzzy soft connected between (f,E) and (g,E), a contradiction.2 Theorem 3.9. Let (Y,τY ,E) be a fuzzy soft clopen subspace of a FSTS (X,τ,E) and (f ,E), (g,E) ⊂ Ỹ . If (X,τ,E) is fuzzy soft connected between (f,E) and (g,E) then (Y,τY ,E) is fuzzy soft connected between(f,E) and (g,E). Proof. Suppose (Y,τY ,E) is not fuzzy soft connected between (f,E) and (g,E). Then there is fuzzy soft clopen set (h,E) of (Y,τY ,E) such that (f,E) ⊂ (h,E) and (h,E) q̃ (g,E). Since, (Y,τY ,E) is fuzzy soft clopen in (X,τ,E), by Lemma 2.2 and Lemma 2.3 (h,E) is fuzzy soft clopen set of (X,τ,E) such that (f,E) ⊂ (h,E) and (h,E) q̃ (g,E). Consequently, (X,τ,E) is not fuzzy soft connected between (f,E) and (g,E), a contradiction.2 4 Fuzzy soft set-connected mappings Definition 4.1. A fuzzy soft mapping ϱpu : (X, τ, E) → (Y, ϑ, K) is said to be fuzzy soft set-connected provided, if FSTS(X, τ, E) is fuzzy soft connected be- tween FSSs (f,E) and (g,E) then fuzzy soft subspace (ϱpu(X), ϑϱpu(X), K) is fuzzy Fuzzy soft set connected mappings soft connected between ϱpu(f, E) and ϱpu(g, E) with respect to fuzzy soft relative topology. Theorem 4.1. A fuzzy soft mapping ϱpu : (X, τ, E) → (Y, ϑ, K) is fuzzy soft set- connected mapping if and only if ϱ−1pu (F, K) is a fuzzy soft clopen set over X for any fuzzy soft clopen set (h,K) of (ϱpu(X) ,ϑϱpu(X), K). Proof. Necessity : Let ϱpu be fuzzy soft set-connected mapping and (h,K) be fuzzy soft clopen set in (ϱpu(X), ϑϱpu(X), K). Suppose ϱ −1 pu (h, K) is not fuzzy soft clopen in (X ,τ,E). Then (X ,τ,E) is fuzzy soft-connected between ϱ−1pu (h, K) and (ϱ−1pu (h, K)) c. Therefore, (ϱpu(X), ϑϱpu(X), K) is fuzzy soft-connected between ϱpu(ϱ −1 pu (h, K)) and ϱpu((ϱ −1 pu (h, K)) c) because ϱpu is fuzzy soft set-connected. But, ϱpu(ϱ−1pu (h, K)) = (h, K)∩(ϱpu(X), ϑϱpu(X), K) = (h, K) and ϱpu((ϱ−1pu (h, K))c = (h, K)c imply that (h,K) is not fuzzy soft clopen in (ϱpu(X), ϑϱpu(X), K), a con- tradiction. Hence, ϱ−1pu (h, K) is fuzzy soft clopen in (X ,τ,E). Sufficiency : Let (X ,τ,E) be fuzzy soft-connected between (f,E) and (g,E). If (ϱpu(X), ϑϱpu(X), K) is not fuzzy soft-connected between ϱpu(f, E) and ϱpu(g, E) then there exists a fuzzy soft clopen set (h,K) in (ϱpu(X), ϑϱpu(X), K) such that ϱpu(f, E) ⊂ (h, K) ⊂ (ϱpu(g, E))c. By hypothesis, ϱ−1pu (h, K) is fuzzy soft clopen set over X and (f, E) ⊂ ϱ−1pu (h, K) ⊂ (g, E)c. Therefore, (X ,τ,E) is not fuzzy soft- connected between (f,E) and (g,E). This is a contradiction. Hence, ϱpu is fuzzy soft set-connected.2 Theorem 4.2. Every fuzzy soft continuous mapping ϱpu : (X ,τ,E) → (Y,ϑ,K) is a fuzzy soft set-connected mapping. Proof. It is obvious.2 Remark 4.1. The converse of Theorem 4.2 need not be true. Example 4.1. Let X = {x1, x2} , E = {e1, e2} and Y = {y1, y2} , K = {k1, k2}. The soft sets (f,E) and (g,K) defined as follows: f(e1) = {x1 = 0.3, x2 = 0}, f(e2) = {x1 = 0, x2 = 0.4} g(k1) = {y1 = 0.6, y2 = 0}, g(k2) = {y1 = 0, y2 = 0.5} Let τ = {0̃E,1̃E,(f,E) } and υ = {0̃K , 1̃K , (g ,K) } are fuzzy soft topologies on X and Y respectively. Then the fuzzy soft mapping ϱpu : (X, τ, E) → (Y, υ, K) defined by u(x1) = y1 , u(x2 )= y2 and p(e1 )= k1, p(e2 ) = k2 is fuzzy soft set-connected but it is not fuzzy soft continuous, because fuzzy soft set (g,K) is fuzzy soft open set in Y not fuzzy soft open in X. Theorem 4.3. Every fuzzy soft mapping ϱpu : (X, τ, E) → (Y, ϑ, K) such that (ϱpu(X), ϑϱpu(X), K) is a fuzzy soft connected set is a fuzzy soft set-connected mapping. Proof. Let (ϱpu(X), ϑϱpu(X), K) be fuzzy soft connected. Then by Lemma 3.1, no nonempty proper FSS of (ϱpu(X), ϑϱpu(X), K) which is fuzzy soft clopen. Hence , ϱpu is fuzzy soft set-connected.2 Theorem 4.4. Let ϱpu : (X, τ, E) → (Y, ϑ, K) be a fuzzy soft set-connected mapping. If (X ,τ,E) is fuzzy soft connected set, then (ϱpu(X), ϑϱpu(X), K) is a fuzzy soft connected set of (Y,ϑ,K). Proof. Suppose (ϱpu(X), ϑϱpu(X), K) is not fuzzy soft connected in (Y,ϑ,K), Then by Lemma 3.1, there is a nonempty proper fuzzy soft clopen set (h,K) of (ϱpu(X), ϑϱpu(X), K). Since ϱpu is fuzzy soft set-connected, ϱ −1 pu (h, K) is a nonempty proper fuzzy soft clopen set over X. Consequently,(X ,τ,E) is not fuzzy soft connected.2 Theorem 4.5. Let ϱpu : (X, τ, E) → (Y, ϑ, K) be a fuzzy soft set-connected mapping and (f,E) be a fuzzy soft set over X such that ϱpu(f, E) is fuzzy soft clopen set of (ϱpu(X), ϑϱpu(X), K). Then ϱpu/(f, E) : (f, E) → (Y, ϑ, K) is fuzzy soft set-connected mapping. Proof: Let (f,E) be fuzzy soft connected between (g,E) and (h,E). Then by Theorem 3.8, (X, τ, E) is fuzzy soft connected between (g,E) and (h,E). Since ϱpu is fuzzy soft set-connected, (ϱpu(X), ϑϱpu(X), K) is fuzzy soft connected be- tween ϱpu(g, E) and ϱpu(h, E). Now, since ϱpu(f, E) is fuzzy soft clopen set of (ϱpu(X), ϑϱpu(X), K), it follows by Theorem 3.9 that ϱpu(f, E) is fuzzy soft con- nected between ϱpu(g, E) and ϱpu(h, E). This proves the theorem.2 Theorem 4.6. Let ϱpu : (X, τ, E) → (Y, ϑ, K) be a fuzzy soft set-connected sur- jection. Then for any fuzzy soft clopen set (h,K) of (Y, ϑ, K) is fuzzy soft connected if ϱ−1pu (h, K) is fuzzy soft connected in (X ,τ,E). In particular, if (X ,τ,E) is fuzzy soft connected then (Y,ϑ,K) is fuzzy soft connected. Proof. By Theorem 4.5 ϱpu/ϱ−1pu (h, K) : ϱ −1 pu (h, K) → (Y, ϑ, K) is fuzzy soft set-connected. And, since ϱ−1pu (h, K) is fuzzy soft connected by Theorem 4.4, ϱpu/ϱ −1 pu (h, K)[ϱ −1 pu (h, K)] = (h, K) is fuzzy soft connected.2 Theorem 4.7. Let ϱpu : (X, τ, E) → (Y, ϑ, K) be a fuzzy soft set connected fuzzy soft open surjection and ϱ−1pu ((y α k )K) is fuzzy soft connected for each soft point (yαk )K of Y. Then for every fuzzy soft clopen set (h,K) of Y is fuzzy soft connected if and only if ϱ−1pu ((h, K)) is fuzzy soft connected. Proof. Necessity: Let (h,K) be a fuzzy soft clopen fuzzy soft connected set of Y. Suppose ϱ−1pu (h, K) is not fuzzy soft connected in X. Then there are fuzzy soft open sets (f,E) and (g,E ) of X such that ϱ−1pu (h, K) ∩ ((f, E) ∩ (g, E)) = ϕ, ϱ−1pu (h, K) = ((f, E) ∪ (g, E)) and ϱ−1pu (h, K) ∩ (f, E) ̸= ϕ ̸= ϱ−1pu (h, K) ∩ (g, E). Since, ϱ−1pu ((y α k )K) is fuzzy soft connected either ϱ −1 pu ((y α k )K) ⊂ (f, E) or ϱ−1pu ((y α k )K) ⊂ (g, E) for every fuzzy soft point (y α k )K ∈ (h, K). Therefore (h, K)∩ϱpu(f, E)∩ϱpu(g, E) = ϕ. (h, K) ⊂ ϱpu(f, E)∪ϱpu(g, E) and (h, K)∩ Fuzzy soft set connected mappings ϱpu(f, E) ̸= ϕ ̸= (h, K) ∩ ϱpu(g, E) . Since, ϱpu is fuzzy soft open mapping ϱpu(f, E) and ϱpu(g, E) are fuzzy soft open set over Y. Hence, (h,K) is not fuzzy soft connected. Sufficiency : Follows from Theorem 4.6.2 Theorem 4.8. Let ϱp1u1 : (X ,τ,E) → (Y,ϑ,K) be a surjective fuzzy soft set-connected and σp2u2 :(Y,ϑ,K) → (Z,η,T) a fuzzy soft set-connected mapping. Then (σp2u2oϱp1u1) : (X ,τ,E) → (Z,η,T) is fuzzy soft set-connected. Proof. Let (h,T) be a fuzzy soft clopen set in σp2u2(Y ). Then σ −1 p2u2 (h,T) is fuzzy soft clopen over Y = ϱp1u1 (X) and so ϱ −1 p1u1 (σ−1p2u2 (h,T)) is fuzzy soft clopen in (X ,τ,E). Now (σp2u2oϱp1u1)(X ) = σp2u2 (Y) and (σp2u2oϱp1u1) −1 (h,T) = ϱ−1p1u1 (σ−1p2u2 (h,T)) is fuzzy soft clopen in (X ,τ,E). Hence, (σp2u2oϱp1u1) is fuzzy soft set connected.2 Definition 4.2. A fuzzy soft mapping ϱpu : (X, τ, E) → (Y, ϑ, K) is said to be fuzzy soft weakly continuous if for each fuzzy soft point (xαe )E ∈ X and each fuzzy soft open set (g,K) over Y containing ϱpu((xαe )E), there exists a fuzzy soft open set (f,E) over X containing (xαe )E such that ϱpu(f,E) ⊂ Cl(g,K). Theorem 4.9. A soft mapping ϱpu : (X, τ, E) → (Y, ϑ, K) is fuzzy soft weakly continuous if and only if for each fuzzy soft open set (h,K) over Y, ϱ−1pu (h, K) ⊂ Int(ϱ−1pu (Cl(h, K))). Proof. Necessity : Let (h,K) be a fuzzy soft open set over Y and let (xαe )E ∈ ϱ−1pu (h, K) then ϱpu((x α e )E) ∈ (h, K). Then, there exists a fuzzy soft open set (f,E) over X such that (xαe )E ∈ (f, E) and ϱpu(f, E) ⊂ Cl(h, K). Hence, (xαe )E ∈ (f, E) ⊂ ϱ−1pu (Cl(h, K)) and (xαe )E ∈ Int(ϱ−1pu (Cl(h, K))) since (f,E) is fuzzy soft open. Sufficiency : Let (xαe )E ∈ X and ϱpu((xe)E) ∈ (h, K). Then (xe)E ∈ ϱ−1pu (h, K) ⊂ Int(ϱ−1pu (Cl(h, K))). Let (f, E) = Int(ϱ−1pu (Cl(h, K))) then (f,E) is fuzzy soft open set containing (xαe )E and ϱpu(f, E) = ϱpu(Int(ϱ −1 pu (Cl(h, K)))) ⊂ ϱpu(ϱ −1 pu (Cl(h, K))) ⊂ Cl(h, K). Hence, ϱpu is fuzzy soft weakly continuous.2 Theorem 4.10. If a FSTS space (X ,τ,E) is fuzzy soft connected and ϱpu : (X, τ, E) → (Y, ϑ, K) is a fuzzy soft weakly continuous surjection, then (Y,ϑ,K) is fuzzy soft connected. Proof . Suppose (Y, ϑ, K) is not fuzzy soft connected. Then, there exist nonempty fuzzy soft open sets (h1, K) and (h2, K) in Y such that (h1, K)∩(h2, K) = ϕ and (h1, K) ∪ (h2, K) = X̃. Hence we have ϱ−1pu (h1, K) ∩ ϱ−1pu (h2, K) = ϕ and ϱ−1pu (h1,K) ∪ ϱ−1pu (V2,K) = X̃. Since ϱpu is surjective, ϱ−1pu (Vj,K) ̸= ϕ for j = 1, 2. By Theorem 4.9, we have ϱ−1pu (hj,K) ⊂Int (ϱ−1pu (Cl(hj,K))) because ϱpu is fuzzy soft weakly continuous. Since (hj,K) is fuzzy soft open and also fuzzy soft closed, we have ϱ−1pu (hj,K) ⊂ Int( ϱ−1pu (hj,K. Hence, ϱ−1pu (hj,K) is fuzzy soft open in X for j = 1,2. This implies that X is not fuzzy soft connected. This is contrary to the hypothesis that X is fuzzy soft connected. Hence, (Y,ϑ,K) is fuzzy soft connected.2 Theorem 4.11. A fuzzy soft mapping ϱpu : (X, τ, E) → (Y, ϑ, K) is fuzzy soft weakly continuous, then Cl(ϱ−1pu (h, K)) ⊂ (ϱ−1pu (Cl(h, K)) for each fuzzy soft open set (h,K) over Y. Proof. Suppose there exists a fuzzy soft point (xαe )E ∈ Cl(ϱ−1pu (h, K)) − ϱ−1pu (Cl(h, K)). Then ϱpu((xe)E) /∈ Cl(h, K). Hence there exists a fuzzy soft open set (g ,K) containing ϱpu((xe)E) such that (g, K) ∩ (V, K) = ϕ. Since (h,K) is fuzzy soft open set over Y, we have (h, K) ∩ Cl(g, K) = ϕ. Since ϱpu is fuzzy soft weakly continuous ,there exists a fuzzy soft open set (f,E) over X containing (xαe )E such that ϱpu(f, E) ⊂ Cl(g, K). Thus, we obtain ϱpu(f, E) ∩ (h, K) = ϕ. On the other hand, since (xαe )E ∈ Cl(ϱ−1pu (h, K)), we have (f, E) ∩ ϱ−1pu (h, K) ̸= ϕ and hence, ϱpu(f, E) ∩ (h, K) ̸= ϕ. Thus we have a contradiction. Hence Cl(ϱ−1pu (h, K)) ⊂ (ϱ−1pu (Cl(h, K)).2 Theorem 4.12. If a fuzzy soft surjection ϱpu : (X ,τ,E) → (Y,ϑ,K) is fuzzy soft weakly continuous ,then ϱpu is fuzzy soft set-connected. Proof. Let (h,K) be any fuzzy soft clopen set over Y. Since (h,K) is fuzzy soft closed, We have Cl(h,K) = (h,K). Thus, by Theorem 4.9, we obtain ϱ−1pu (h, K) ⊂ Int(ϱ−1pu (h, K)). This shows that ϱ −1 pu (h, K) is fuzzy soft open set over X. More- over , by Theorem 4.11, we obtain Cl(ϱ−1pu (h, K)) ⊂ ϱ−1pu (h, K). This shows that ϱ−1pu (h, K) is a fuzzy soft closed set over X. Since ϱpu is fuzzy soft surjection, by Theorem 4.1, we observe that ϱpu is a fuzzy soft set-connected mapping.2 Remark 4.2. The converse of Theorem 4.12 is not true. Example 4.2. LetX = {x1, x2}, E = {e1, e2} and Y = {y1, y2} , K = {k1, k2}. The fuzzy soft sets (f,E)and (g,K) are defined as follows : f(e1) = {x1 = 0.3, x2 = 0}, f(e2) = {x1 = 0, x2 = 0.4} g(k1) = {y1 = 0.4, y2 = 0}, g(k2) = {y1 = 0, y2 = 0.5} Let τ = {0̃E,1̃E,(h,E) } and υ = {0̃K , 1̃K , (G ,K) } are fuzzy soft topologies on X and Y respectively. Then fuzzy soft mapping ϱpu : (X, τ, E) → (Y, υ, K) defined by u(x1) = u(x2)=y1 and p(e1 )= k1 , p(e2 ) = k2 is fuzzy soft set-connected but it is not fuzzy soft weakly continuous. Theorem 4.13. Let (Y,ϑ,K) be an fuzzy soft extremally disconnected space .If a fuzzy soft mapping ϱpu : (X, τ, E) → (Y, ϑ, K) is fuzzy soft set-connected, then ϱpu is fuzzy soft weakly continuous. Proof. Let (xαe )E be a fuzzy soft point of X and (g,K) any fuzzy soft open set over Y containing ϱpu((xαe )E). Since (Y,ϑ,K) is fuzzy soft extremally disconnected, Fuzzy soft set connected mappings Cl(g,K) is fuzzy soft clopen set over Y. Thus Cl(g, K)∩ϱpu(X̃) is fuzzy soft clopen set in the fuzzy soft subspace (ϱpu(X), ϑϱpu(X), K). Put ϱ −1(Cl(g, K)∩ϱpu(X̃)) = (f, E). Then, since ϱpu is fuzzy soft set-connected, it follows from Theorem 4.1 that (f,E) is fuzzy soft clopen set over X. Therefore, (f,E) is a fuzzy soft open set containing (xαe )E over X such that ϱpu(f, E) ⊂ Cl(g, K). This implies that ϱpu is fuzzy soft weakly continuous.2 Theorem 4.14. Let (Y, ϑ, K) be a fuzzy soft extremally disconnected space. A fuzzy soft surjection mapping ϱpu : (X, τ, E) → (Y, ϑ, K) is fuzzy soft set- connected if and only if ϱpu is fuzzy soft weakly continuous. Proof. It follows from Theorem 4.12 and Theorem 4.13.2 5 Conclusions Connectedness is an important and major area of topology and it can give many relationships between other scientific areas and mathematical models. The notion of connectedness captures the idea of hanging-togetherness of image el- ements in an object by assigning a strength of connectedness to every possible path between every possible pair of image elements. This paper, introduces the notion of fuzzy soft connectedness between fuzzy soft sets in fuzzy soft topologi- cal spaces. It is shown that a fuzzy soft topological space is fuzzy soft connected if and only if it is fuzzy soft connected between every pair of its nonempty fuzzy soft sets. Further two new classes of fuzzy soft mappings called fuzzy soft set con- nected and soft weakly continuous have been introduced. 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