# Gradation of Continuity for Fuzzy Soft Mappings

Authors: Vildan Çetkin
Journal: Journal of Mathematics
 (2021)

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## Abstract

This paper is devoted to describe the notion of a parameterized degree of continuity for mappings betweenL-fuzzy soft topological spaces, where L is a complete De Morgan algebra. The degrees of openness, closedness, and being a homeomorphism for the fuzzy soft mappings are also presented. The properties and characterizations of the proposed notions are pictured. Besides, the degree of continuity for a fuzzy soft mapping is unified with the degree of compactness and connectedness in a natural way.


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## Body

## 1. Introduction

The theory of fuzzy sets which is a way of modeling real-life problems involves uncertainty, based on the degree of membership of an element to some sets. This idea has impressed many researchers working in diverse areas. Especially, the topology workers applied this idea to the gradation of openness and hence gave birth to the fuzzy topology [1]. The fuzzy-fuzzy case of the topology is the most compatible way of reflecting the gradation of belongingness [2]. So naturally, the notions of the degree of compactness, the degree of connectedness, the degree of separations, and so on have been considered. Later on, the similar argument has been considered for the mappings between fuzzy topological spaces and the degree of continuity, the degree of openness, and the degree of closedness for (fuzzy) mappings have been described [3–5]. The theory of soft sets is one of the other tools to model vague phenomena [6]. Also, the combination of fuzzy sets and soft sets gave birth to the fuzzy soft sets [7–9] and the basic idea of this new kind of sets depends on the parameterized degree of membership of an element to some sets. Nowadays, the studies depending on the soft sets and the fuzzy soft sets are increasing rapidly [10–13].The idea of fuzzy softness (in fact, “parameterized gradation”) is one of the appropriate tools for modeling of environmental and mathematical problems. On the other hand, the mappings play the key role in transforming the characteristics between structured spaces. Especially, the continuous mappings are worth to investigate since they preserve the several properties of the spaces endowed with topology. Motivated from this thinking, we found it reasonable to present a new theory which gives a more accurate and efficient way of transforming the characteristics between the fuzzy soft topological spaces depending on the parameters. Thus, as a continuation of the research studies [14–16], we describe the parameterized gradations of continuity, openness, and closedness for mappings between fuzzy soft topological spaces.The content of this study is organized in the following manner: in Section2, we present the notations and recall the elementary notions which are used throughout the study. In Section 3, we define the parameterized degree of the concepts of continuity, openness, closedness, and being homeomorphism for mappings transformed between fuzzy soft topological spaces. We investigate the parameterized graded extensions of the main properties and results known in general topology, for the proposed concepts. Additionally, we observe several characterizations of the described gradations with the help of the neighborhood systems and closure (interior) operators. At the end, we unify the parameterized graded continuity with the parameterized compactness (connectedness, respectively) degree.


## 2. Preliminaries

Throughout this paper,X refers to a nonempty initial universe, EandK denotes the arbitrary nonempty sets viewed on the sets of parameters, and L=L,∨,∧,′ denotes a complete De Morgan algebra with the smallest element 0L and the largest element 1L. With the underlying lattice L, a mapping A:X⟶L is said to be an L-fuzzy set on X and by LX, we denote the family of all L-fuzzy sets on X.An elementα in L is said to be coprime if α≤β∨γ implies that α≤β or α≤γ. ML denotes the collection of all coprime elements of L. We say α is way below (wedge below) β, in symbols, α≪β,α⊲β, if for every directed (arbitrary) subset D⊆L, ∨D≥β implies α≤γ for some γ∈D. Clearly, if α∈L is coprime, then α≪β if and only if α⊲β. Details for lattices can be found in [17].The binary operation↦ in the complete De Morgan algebra L is given by α↦β=∨γ∈L|α∧γ≤β.For allα,β,γ,δ∈L and αi,βi⊆L, the following properties are satisfied:(1)
α↦β≥γ iff α∧γ≤β(2)
α↦β=1L iff α≤β(3)
α↦∧iβi=∧iα↦βi and ∨iαi↦β=∧iαi↦β(4)
α↦γ∧γ↦β≤α↦β(5)
α≤β⇒γ↦α≤γ↦β and β↦γ≤α↦γ(6)
α↦β∧γ↦δ≤α∧γ↦β∧δDefinition 1 (see [18]).
A mappingf:E⟶LXE is called an L-fuzzy soft set on X. This means that fe:=fe:X⟶L is an L-fuzzy set on X, for each parameter e∈E. Hence, an L-fuzzy soft set can be considered as the parameterized extended version of an L-fuzzy set. Intuitively, by a fuzzy soft set, one can describe the parameterized degree of belongingness.
From now on, we use the symbolLXE to denote the collection of all L-fuzzy soft sets on X.Definition 2.
(see [18, 19]). Let fandg be two L-fuzzy soft sets on X; then, the set operations are defined as follows:(1)
f is an L-fuzzy soft subset of g and written by f⋢g if fe≤ge, for each e∈E. f and g are called equal if f⋢g and g⋢f.(2)
The union off and g is an L-fuzzy soft set h=f⊔g, where he=fe∨ge, for each e∈E.(3)
The intersection off and g is an L-fuzzy soft set h=f⊓g, where he=fe∧ge, for each e∈E.(4)
The complement off is denoted by f′, where f′:E⟶LX is a mapping given by fe′=fe′, for each e∈E. Clearly f′′=f.Definition 3.
(see [18]) (1)
AnL-fuzzy soft set f on X is called a null (or empty) L-fuzzy soft set and denoted by 0˜, if fex=0L, for each e∈E,x∈X.(2)
AnL-fuzzy soft set f on X is called an absolute (or universal) L-fuzzy soft set and denoted by 1˜, if fex=1L, for each e∈E,x∈X. Clearly 1˜′=0˜ and 0˜′=1˜.Definition 4.
(see [20]). Let x∈X and α:E⟶ML be a function. Then, the L-fuzzy soft set defined as follows is called an L-fuzzy soft point and denoted by xα.(1)xeαy=αe,ify=x,0L,otherwise,for alle∈Eandy∈X.
AnL-fuzzy soft point xα is said to belong to an L-fuzzy soft set f and denoted by xα∈f if αe≤fex, for each e∈E.
The set of all nonzero coprime elements ofLXE is denoted by MLXE. It is noted that MLXE is exactly the set of all L-fuzzy soft points.Definition 5.
(see [21]). Let φ:X1⟶X2 and ψ:E1⟶E2 be two functions, where E1 and E2 are parameter sets for the crisp sets X1 and X2, respectively. Then, the pair φ,ψ:=φψ is called an L-fuzzy soft mapping from X1,E1 to X2,E2:(1)
The image off∈LX1E1 under φψ, denoted by φψf, is an L-fuzzy soft set on X2 defined by(2)φψfky=∨φx=y∨ψa=kfax,ifx∈φ−1y,a∈ψ−1k,0L,otherwise.for allk∈E2y∈X2.(2)
The inverse image ofg∈LX2E2 under φψ, denoted by φψ−1g, is an L-fuzzy soft set on X1 defined by(3)φψ−1gex=gψeφx,for alle∈E1,x∈X1.(3)
Ifφ and ψ are both injective (surjective or bijective, respectively), then φψ is said to be injective (surjective or bijective, respectively).(4)
Ifφψ:X1,E1⟶X2,E2 and φψ∗∗:X2,E2⟶X3,E3 are soft mappings, then their composition φψ∗∗°φψ:X1,E1⟶X3,E3 is defined by φψ∗∗°φψ=φ∗°φψ∗°ψ.Definition 6.
(see [14]). A mapping τ:K⟶LLXE which satisfies the following certain axioms is called an L-fuzzy E,K-soft topology on X.(O1)τk0˜=τk1˜=1L, for each k∈K.(O2)τkf⊓g≥τkf∧τkg, for each k∈K and for all f,g∈LXE.(O3)τk⊔i∈Δfi≥∧i∈Δτkfi, for all fii∈Δ⊆LXE and for each k∈K.
Then, the pairX,τ is called an L-fuzzy E,K-soft topological space. The value τkf is interpreted as the degree of openness of an L-fuzzy soft set f with respect to the parameter k∈K. So, the fuzzy soft topology can be thought as the gradation of parameterized degree of openness. Hence, the parameterized degree of closedness of a given L-fuzzy soft set is described by using the complement operator τk∗f=τkf′.
Letτ1 and τ2 be L-fuzzy E,K-soft topologies on X. We say that τ1 is finer than τ2 (τ2 is coarser than τ1), denoted by τ2≤τ1, if τk2f≤τk1f for each k∈Kandf∈LXE.Definition 7.
(see [14]). Let X1,τ1 be an L-fuzzy E1,K1-soft topological space and X2,τ2 be an L-fuzzy E2,K2-soft topological space. Let φ:X1⟶X2,ψ:E1⟶E2 and η:K1⟶K2 be crisp functions. Then, the fuzzy soft mapping φψ,η:X1,τ1⟶X2,τ2 is called continuous if τk1φψ−1h≥τηk2h for all h∈LX2E2 and for all k∈K1.Definition 8.
(see [20]). For a fixed fuzzy soft point xα, let the mappings Qxα:K×LXE⟶L satisfy the following axioms for each k∈K and f,g∈LXE:(FSQ1)Qxαk,1˜=1LandQxαk,0˜=0L(FSQ2) IfQxαk,f≠0L, then xα⋢f′(FSQ3)Qxαk,f⊓g=Qxαk,f∧Qxαk,g(FSQ4)Qxαk,f=∨xα⋢g⊒f′∧yβ⋢gQyβk,g′
Then, the collectionQ=Qxα|xα∈MLXE of maps presented above is called an L-fuzzy E,K-soft quasi-coincident neighborhood (shortly, q-nhood) system on X. The value Qxαk,f represents the parameterized degree of being q-nhood of f to the fuzzy soft point xα.Proposition 1 (see [20]).
Letτ be an L-fuzzy E,K-soft topology on X. Define the mapping Qxατ:K×LXE⟶L as follows: for each k∈K,f∈LXE and xα∈MLXE,(4)Qxατk,f=∨xα⋢g⊒f′τkg′.Then, the set ofQτ=Qxατ|xα∈MLXE is an L-fuzzy E,K-soft q-nhood system on X, called induced L-fuzzy E,K-soft q-nhood system by τ.Definition 9.
For a fixed fuzzy soft pointxα, let the mappings Nxα:K×LXE⟶L satisfy the following axioms for each k∈K and f,g∈LXE.(FSN1)Nxαk,1˜=1LandNxαk,0˜=0L(FSN2) IfNxαk,f≠0L, then xα⋢f(FSN3)Nxαk,f⊓g=Nxαk,f∧Nxαk,g(FSN4)Nxαk,f=∨xα⋢g⋢f′∧yβ⊲gNyβk,g
Then, the collection of mapsN=Nxα|xα∈MLXE is called an L-fuzzy E,K-soft neighborhood (shortly, nhood) system on X.Definition 10.
(see [16]). A mapping cl:K×LXE⟶LMLXE is called an L-fuzzy E,K-soft closure operator on X if it satisfies the following axioms for each k∈K:(C1)clk,fxα=∧β⊲αclk,fxβ, for all xα∈MLXE.(C2)clk,0˜xα=0L for any xα∈MLXE.(C3)clk,fxα=1L for any xα⋢f.(C4)clk,f⊔g=clk,f∨clk,g.(C5)σaclk,⊔σaclk,f⊆σaclk,f, where σaclk,f=xα∈MLXE|clk,fxα≥a.
The valueclk,fxα is interpreted as the degree to which xα belongs to the parameterized closure of the fuzzy soft set f.Example 1.
(see [16]). Let cl:K×LXE⟶LXE be the closure operator given in a parameterized L-soft topological space X,T. In this case, the mapping is Clk,f:MLXE⟶2 defined in such a way that(5)Clk,fxα=1,ifxα⋢clk,f,0,otherwise,satisfies the conditions of Definition 10.Theorem 1 (see [16]).
Letτ be an L-fuzzy E,K-soft topology on X. Then, the mapping C:K×LXE⟶LMLXE defined by(6)Ck,fxα=Qxατk,f′′,is an L-fuzzy E,K-soft closure operator on X, which is called the L-fuzzy E,K-soft closure operator induced by τ.Definition 11.
A mappingInt:K×LXE⟶LMLXE is called an L-fuzzy E,K-soft interior operator on X if it satisfies the following axioms for each k∈K:(I1)
Intk,fxα=∧β⊲αIntk,fxβ, for all xα∈MLXE(I2)
Intk,1˜xα=1L for any xα∈MLXE(I3)
Intk,fxα=0L for any xα⋢f(I4)
Intk,f⊓g=Intk,f∧Intk,g(I5)
δaIntk,f⊆δaIntk,⊔δaIntk,f, where δaIntk,f=xα∈MLXE|Intk,fxα≤a, for all a∈L\1L
The valueIntk,fxα is interpreted as the degree to which xα belongs to the parameterized interior of the fuzzy soft set f.Theorem 2.
Letτ be an L-fuzzy E,K-soft topology on X, and let N=Nxα|xα∈MLXE be the nhood system induced by τ. Define a mapping I:K×LXE⟶LMLXE by(7)Ik,fxα=Nxαk,f.Then, the mappingI is an L-fuzzy E,K-soft interior operator on X, which is called the L-fuzzy E,K-soft interior operator induced by τ.Definition 12.
(see [22]). Let X,τ be an L-fuzzy E,K-soft topological space. Then, identify a mapping comτ:K⟶LLXE in such a way that in order to picture the parameterized compactness degree,(8)comτk,g=∧U⊆LXE∧h∈Uτkh↦∧x∈X∧e∈Ege′x∨∨h∈Uhex↦∨V∈2U∧x∈X∧e∈Ege′x∨∨h∈Vhex.
In this case, the valuecomτk,g is interpreted as the compactness degree of g∈LXE with respect to the parameter k. So, g is said to be compact L-fuzzy soft set with respect to k if comτk,g=1L.Definition 13.
(see [16]). Let X,τ be an L-fuzzy E,K-soft topological space. Then, identify a mapping Con:K×LXE⟶Ł by the following manner in order to describe the connectedness degree in such spaces:(9)Conk,h=∧∨xα⋢fCk,gxα∨∨yβ⋢gCk,fyβ|f,g∈LXE\0˜,h=f⊔g.
In this case, the valueConk,h is said to be the connectedness degree of an L-fuzzy soft set h with respect to k.Theorem 3 (see [16]).
Letτ be an L-fuzzy E,K-soft topology on X. Then, one can characterize the parameterized degree of connectedness of an L-fuzzy soft set g∈LXE in the following way:(10)Conk,g=∧τku′′∨τkv′′|g⊓u≠0˜,g⊓v≠0˜,g⊓u⊓v=0˜andg⋢u⊔v.


## 3. Degree of Continuity for Fuzzy Soft Mappings

In this section, we identify the degrees of continuity, openness, closedness, and being a homeomorphism for a fuzzy soft mapping. Then, we study some of their characterizations by means of the q-nhood, nhood, interior, and closure operators. We also observe the elementary features of the proposed notions.Definition 15.
LetX1,τ1andX2,τ2 be the L-fuzzy E1,K1-soft and E2,K2-soft topological spaces, respectively, and φψ,η:X1,τ1⟶X2,τ2 be a soft mapping. Then, we define the following(1)
The parameterized degree of continuity forφψ,η is as follows: for all k∈K1,(11)Contk,φψ,η=∧g∈LX2E2τηk2g↦τk1φψ−1g.The valueContk,φψ,η represents to which φψ,η is continuous with respect to some parameters. Hence, the degree of continuity for φψ,η is computed by the formula Contφψ,η=∧k∈K1Contk,φψ,η.(2)
The parameterized degree of openness forφψ,η is as follows: for all k∈K1,(12)Openk,φψ,η=∧f∈LX1E1τk1f↦τηk2φψf.The valueOpenk,φψ,η represents to which φψ,η is open with respect to some parameters. Hence, the degree of openness for φψ,η is computed by the formula Openφψ,η=∧k∈K1Openk,φψ,η.(3)
The parameterized degree of closedness forφψ,η is as follows: for all k∈K1,(13)Closk,φψ,η=∧f∈LX1E1τk1f′↦τηk2φψf′.The valueClosk,φψ,η represents to which φψ,η is a closed map with respect to some parameters. Hence, the degree of being a closed map for φψ,η is computed by the formula Closφψ,η=∧k∈K1Closk,φψ,η.Definition 16.
LetX1,τ1andX2,τ2 be the L-fuzzy E1,K1-soft and E2,K2-soft topological spaces, respectively, and φψ,η:X1,τ1⟶X2,τ2 be a bijective soft mapping. Then, the parameterized degree of being a homeomorphism for the soft mapping φψ,η is identified by Homk,φψ,η=Contk,φψ,η∧Openk,φψ,η. Hence, Homφψ,η=Contφψ,η∧Openφψ,η.Remark 1.
(1)
If the valueContφψ,η=1L, then it is seen that τk1φψ,η−1g≥τηk2g for each k∈K1 and for each g∈LX2E2 is satisfied. This gives us the definition of continuity for the mapping φψ,η:X1,τ1⟶X2,τ2. Analogously, if Openφψ,η=1L or Closφψ,η=1L, then (2) and (3) of Definition 15 are just the definitions of openness and closedness for the mapping φψ,η, respectively.(2)
Ifφ=idX,ψ=idE, and η=idK are the corresponding identity mappings, then φψ,η:X,τ⟶X,τ is the identity soft mapping and Contφψ,η=Openφψ,η=Homφψ,η=1L.Theorem 4.
LetXi,τi be the L-fuzzy Ei,Ki-soft topological spaces (where i = 1, 2, 3) and φψ,η:X1,τ1⟶X2,τ2 and δξ,σ:X2,τ2⟶X3,τ3 be the soft mappings. Then, the following is satisfied:(1)
Contk,φψ,η=∧g∈LX2E2τηk2g′↦τk1φψ−1g′(2)
Contδξ,σ°φψ,η≥Contφψ,η∧Contδξ,σ(3)
Openδξ,σ°φψ,η≥Openφψ,η∧Openδξ,σ(4)
Closδξ,σ°φψ,η≥Closφψ,η∧Closδξ,σProof.
(2) If we consider Definition15, then we obtain the following inequality:(14)Contk,φψ,η∧∧Contηk,δξ,σ=∧g∈LX2E2τηk2g↦τk1φψ−1g∧∧h∈LX3E3τσηk3h↦τηk2δξ−1h≤∧h∈LX3E3τηk2δξ−1h↦τk1φψ−1δξ−1h∧∧h∈LX3E3τσηk3h↦τηk2δξ−1h≤∧h∈LX3E3τηk2δξ−1h↦τk1δξ°φψ−1h∧τσ°ηk3h↦τηk2δξ−1h≤∧h∈LX3E3τσ°ηk3h↦τk1δξ°φψ−1h=Contk,δξ,σ°φψ,η,for any k∈K1.
The above fact witnesses the proof.
The other conditions are similarly proved.
As it is well known in general topology, the composition of two homeomorphisms is again a homeomorphism. Then, by the above theorem, we get the following result for the gradation of homeomorphism.Corollary 1.
LetXi,τi be the L-fuzzy Ei,Ki-soft topological spaces (where i = 1, 2, 3) and φψ,η:X1,τ1⟶X2,τ2 and δξ,σ:X2,τ2⟶X3,τ3 be two bijective soft mappings. Then, Homδξ,σ°φψ,η≥Homφψ,η∧Homδξ,σ is satisfied.Theorem 5.
LetXi,τi be the L-fuzzy Ei,Ki-soft topological spaces (where i = 1, 2, 3) and δξ,σ:X2,τ2⟶X3,τ3 be a soft mapping. If the fuzzy soft mapping φψ,η:X1,τ1⟶X2,τ2 is surjective, then we get the following properties:(1)
Openδξ,σ°φψ,η∧Contφψ,η≤Openδξ,σ(2)
Closδξ,σ°φψ,η∧Contφψ,η≤Closδξ,σProof.
(1)δξ°φψφψ−1g=δξg, for each g∈LX2E2, is satisfied by the surjectivity of the soft mapping φψ. Hence, we gain(15)Openk,δξ,σ°φψ,η∧Contk,φψ,η=∧f∈LX1E1τk1f↦τσηk3δξ°φψf∧∧g∈LX2E2τηk2g↦τk1φψ−1g≤∧g∈LX2E2τk1φψ−1g↦τσηk3δξ°φψφψ−1g∧∧g∈LX2E2τηk2g↦τk1φψ−1g=∧g∈LX2E2τk1φψ−1g↦τσηk3δξ°φψφψ−1g∧τηk2g↦τk1φψ−1g≤∧g∈LX2E2τηk2g↦τσηk3δξg=Openηk,δξ,σ,for all k∈K1.
Sincek∈K1 is arbitrary, we obtain the claimed inequality Openδξ,σ°φψ,η∧Contφψ,η≤Openδξ,σ.Theorem 6.
LetXi,τi be the L-fuzzy Ei,Ki-soft topological spaces (where i = 1,2,3) and φψ,η:X1,τ1⟶X2,τ2 be a fuzzy soft mapping. If the fuzzy soft mapping δξ,σ:X2,τ2⟶X3,τ3 is injective, then we get the following properties:(1)
Openδξ,σ°φψ,η∧Contδξ,σ≤Openφψ,η(2)
Closδξ,σ°φψ,η∧Contδξ,σ≤Closφψ,ηProof.
(1) By considering the definitions, we get the following:(16)Openδξ,σ°φψ,η∧Contδξ,σ=∧k∈K1∧f∈LX1E1τk1f↦τσηk3δξ°φψf∧∧k∗∈K2∧h∈LX3E3τσk∗3h↦τk∗2δξ−1h≤∧k∈K1∧f∈LX1E1τk1f↦τσηk3δξφψf∧∧k∈K1∧g∈LX2E2τσηk3δξg↦τηk2g≤∧k∈K1∧f∈LX1E1τk1f↦τσηk3δξφψf∧∧k∈K1∧f∈LX1E1τσηk3δξφψf↦τηk2φψf≤∧k∈K1∧f∈LX1E1τk1f↦τσηk3δξφψf∧τσηk3δξφψf↦τηk2φψf≤∧k∈K1∧f∈LX1E1τk1f↦τηk2φψf=Openφψ,η.Theorem 7.
LetX1,τ1andX2,τ2 be the L-fuzzy E1,K1-soft and L-fuzzy E2,K2-soft topological spaces. If φψ,η:X1,τ1⟶X2,τ2 is a bijective soft mapping, then we have(1)
Contφψ,η=∧k∈K1∧f∈LX1E1τηk2φψf↦τk1f(2)
Openφψ,η=∧k∈K1∧g∈LX2E2τk1φψ−1g↦τηk2g(3)
Contφψ,η−1=Openφψ,η=Closφψ,ηProof.
(1) Since the soft mappingφψ is bijective, then φψ−1φψf=f for each f∈LX1E1 and φψφψ−1g=g for each g∈LX2E2. By considering these facts, we gain the following:(17)∧k∈K1∧f∈LX1E1τηk2φψf↦τk1f=∧k∈K1∧f∈LX1E1τηk2φψf↦τk1φψ−1φψf≥∧k∈K1∧g∈LX2E2τηk2g↦τk1φψ−1g=∧k∈K1∧g∈LX2E2τηk2φψφψ−1g↦τk1φψ−1g≥∧k∈K1∧f∈LX1E1τηk2φψf↦τk1f.
This witnesses the fact that(18)Contφψ,η=∧k∈K1∧f∈LX1E1τηk2φψf↦τk1f.
(2) It is similarly proved to that of (1).
(3) In order to obtain the proof, we will consider the equalitiesφψ−1−1f=φψf and φψf′=φψf′ (by the injectivity property) for each f∈LX1E1. So,(19)Contφψ−1=∧k∗∈K2∧f∈LX1E1τη−1k∗1f↦τk∗2φψ−1−1f=∧k∗∈K2∧f∈LX1E1τη−1k∗1f↦τk∗2φψf=∧k∈K1∧f∈LX1E1τk1f↦τηk2φψf=Openφψ,η.
In addition, we have(20)Openφψ,η=∧k∈K1∧f∈LX1E1τk1f↦τηk2φψf=∧k∈K1∧f∈LX1E1τk1f′↦τηk2φψf′=∧k∈K1∧f∈LX1E1τk1f′↦τηk2φψf′=Closφψ,η.Corollary 2.
LetX1,τ1 and X2,τ2 be an L-fuzzy E1,K1-soft and L-fuzzy E2,K2-soft topological spaces, respectively. If φψ,η:X1,τ1⟶X2,τ2 is bijective, then the following characterizations are valid:(1)
Homφψ,η=Contφψ,η∧Contφψ,η−1=Contφψ,η∧Closφψ,η(2)
Homφψ,η=∧k∈K1∧f∈LX1E1τηk2φψf↔τk1f(3)
Homφψ,η=∧k∈K1∧g∈LX2E2τk1φψ−1g↔τηk2gTheorem 8.
LetX1,τ1 and X2,τ2 be an L-fuzzy E1,K1-soft and L-fuzzy E2,K2-soft topological spaces, respectively. For any fuzzy soft mapping φψ,η:X1,τ1⟶X2,τ2, the following characterizations are satisfied:(1)
Contφψ,η=∧k∈K1∧g∈LX2E2∧xαQφψxα2ηk,g↦Qxα1k,φψ−1g(2)
Contφψ,η=∧k∈K1∧g∈LX2E2∧xαNφψxα2ηk,g↦Nxα1k,φψ−1g(3)
Contφψ,η=∧k∈K1∧g∈LX2E2∧xαInt2ηk,gφψxα↦Int1k,φψ−1gxα(4)
Contφψ,η=∧k∈K1∧g∈LX2E2∧xαCl2ηk,g′φψxα↦Cl1k,φψ−1g′xαProof.
(1) By Proposition1, φψxα⋢h′⊒g′ implies xα⋢φψ−1h′⊒φψ−1g′. Hence, we may obtain(21)∧k∈K1∧g∈LX2E2∧xαQφψxα2ηk,g↦Qxα1k,φψ−1g=∧k∈K1∧g∈LX2E2∧xα∨φψxα⋢h′⊒g′τηk2h↦∨xα⋢w′⊒φψ−1g′τk1w=∧k∈K1∧g∈LX2E2∧xα∨φψxα⋢h′⊒g′τηk2h↦∨xα⋢φψ−1u′⊒φψ−1g′τk1φψ−1u≥∧k∈K1∧g∈LX2E2∧xα∧φψxα⋢h′⊒g′τηk2h↦τk1φψ−1h≥∧k∈K1∧g∈LX2E2τηk2g↦τk1φψ−1g=Contφψ,η.
In order to prove the converse, consider the fact thatτkf=∧xα⊲fQxατk,f, for each f∈LX1E1andk∈K1. Hence, xα⋢φψ−1g′ implies φψxα⋢g′. By these observations, we gain the following:(22)Contφψ,η=∧k∈K1∧g∈LX2E2τηk2g↦τk1φψ−1g=∧k∈K1∧g∈LX2E2∧yβ⊲gQyβ2ηk,g↦∧xα⊲φψ−1gQxα1k,φψ−1g≥∧k∈K1∧g∈LX2E2∧φψxα⊲gQφψxα2ηk,g↦∧xα⊲φψ−1gQxα1k,φψ−1g≥∧k∈K1∧g∈LX2E2∧φψxα⊲gQφψxα2ηk,g↦Qxα1k,φψ−1g≥∧k∈K1∧g∈LX2E2∧xα∈MLX1E1Qφψxα2ηk,g↦Qxα1k,φψ−1g.
This completes the proof.
(2) It is similar to that of (1).
(3) and (4) proofs are obtained by using Theorems1 and 2.Theorem 9.
LetX1,τ1andX2,τ2 be the L-fuzzy E1,K1-soft and L-fuzzy E2,K2-soft topological spaces. For the fuzzy soft mapping φψ,η:X1,τ1⟶X2,τ2, the following properties are satisfied:(1)
Openφψ,η=∧k∈K1∧f∈LX1E1∧xαQxα1k,f↦Qφψxα2ηk,φψf(2)
Openφψ,η=∧k∈K1∧f∈LX1E1∧xαNxα1k,f↦Nφψxα2ηk,φψf(3)
Openφψ,η=∧k∈K1∧f∈LX1E1∧xαInt1k,fxα↦Int2ηk,φψfφψxα(4)
Openφψ,η=∧k∈K1∧f∈LX1E1∧xαCl1k,f′xα↦Cl2ηk,φψf′φψxα(5)
Openφψ,η=∧k∈K1∧g∈LX2E2∧xαQxα1k,φψ−1g↦Qφψxα2ηk,g(6)
Openφψ,η=∧k∈K1∧g∈LX2E2∧xαNxα1k,φψ−1g↦Nφψxα2ηk,g(7)
Openφψ,η=∧k∈K1∧g∈LX2E2∧xαInt1k,φψ−1gxα↦Int2ηk,gφψxα(8)
Openφψ,η=∧k∈K1∧g∈LX2E2∧xαCl1k,φψ−1g′xα↦Cl2ηk,g′φψxαProof.
(5) Let us first consider the following inequality:(23)∧k∈K1∧g∈LX2E2∧xαQxα1k,φψ−1g↦Qφψxα2ηk,g=∧k∈K1∧g∈LX2E2∧xα∨xα⋢h′⊒φψ−1g′τk1h↦∨φψxα⋢d′⊒g′τηk2d≥∧k∈K1∧g∈LX2E2∧xα∨xα⋢h′⊒φψ−1g′τk1h↦∨φψxα⋢φψu′⊒g′τηk2φψu≥∧k∈K1∧g∈LX2E2∧xα∨xα⋢h′⊒φψ−1g′τk1h↦∨xα⋢u′⊒φψ−1g′τηk2φψu≥∧k∈K1∧g∈LX2E2∧xα∧xα⋢h′⊒φψ−1g′τk1h↦τηk2φψh≥∧k∈K1∧f∈LX1E1τk1f↦τηk2φψf=Openφψ,η.
In order to obtain the converse inequality, we will use the following fact:(24)∧yβ⊲φψfQyβ2ηk,φψf=∧φψxα⊲φψfQφψxα2ηk,φψf.
We may also consider(25)∧k∈K1∧f∈LX1E1∧xα⊲fQxα1k,φψ−1φψf↦∧φψxα⊲φψfQφψxα2ηk,φψf≥∧k∈K1∧g∈LX2E2∧xα∈MLX1E1Qxα1k,φψ−1g↦Qφψxα2ηk,g.
Now, let us choose an arbitraryγ∈ML which satisfies(26)γ⊲∧k∈K1∧g∈LX2E2∧xα∈MLX1E1Qxα1k,φψ−1g↦Qφψxα2ηk,g.
Therefore,γ≤Qxα1k,φψ−1g↦Qφψxα2ηk,g is satisfied for all k∈K1,g∈LX2E2 and for all xα∈MLX1E1. By the implication operator properties, we get γ∧Qxα1k,φψ−1g≤Qφψxα2ηk,g. For all f∈LX1E1 with φψxα⊲φψf, we obtain αe≤φψfψefx. Then, there exist z∈X and e∗∈E such that φx=φz and ψe=ψe∗ and also αe∗≤fe∗z. This implies zα⋢f′. From(27)γ∧∧xα⋢f′Qxα1k,φψ−1φψf≤γ∧Qzα1k,φψ−1φψf≤Qφψzα2ηk,φψf=Qφψxα2ηk,φψf,we gain γ∧∧xα⋢f′Qxα1k,φψ−1φψf≤∧φψxα⋢φψf′Qφψxα2ηk,φψf. So we get γ≤∧xα⋢f′Qxα1k,φψ−1φψf↦∧φψxα⋢φψf′Qφψxα2ηk,φψf. Hence, we also have(28)γ≤∧k∈K1∧f∈LX1E1∧xα⋢f′Qxα1k,φψ−1φψf↦∧φψxα⋢φψf′Qφψxα2ηk,φψf.
Since the arbitrariness ofγ, we gain(29)∧k∈K1∧f∈LX1E1∧xα⋢f′Qxα1k,φψ−1φψf↦∧φψxα⋢φψf′Qφψxα2ηk,φψf≥∧k∈K1∧g∈LX2E2∧xα∈MLX1E1Qxα1k,φψ−1g↦Qφψxα2ηk,g.
Sincef⋢φψ−1φψf, for all f∈LX1E1, we obtain the following inequality:(30)Openφψ,η=∧k∈K1∧f∈LX1E1τk1f↦τηk2φψf=∧k∈K1∧f∈LX1E1∧xα⋢f′Qxα1k,f↦∧yβ⋢φψf′Qyβ2ηk,φψf=∧k∈K1∧f∈LX1E1∧xα⋢f′Qxα1k,f↦∧φψxα⋢φψf′Qφψxα2ηk,φψf≥∧k∈K1∧f∈LX1E1∧xα⋢f′Qxα1k,φψ−1φψf↦∧φψxα⋢φψf′Qφψxα2ηk,φψf≥∧k∈K1∧g∈LX2E2∧xα∈MLX1E1Qxα1k,φψ−1g↦Qφψxα2ηk,g.
Hence we obtain the desired result.
By using Theorems1 and 2, and also by considering some similar discussion, one can prove the other claims of the theorem.Theorem 10.
LetX1,τ1 and X2,τ2 be the L-fuzzy E1,K1-soft and L-fuzzy E2,K2-soft topological spaces. For the fuzzy soft mapping φψ,η:X1,τ1⟶X2,τ2, we have the following result:(31)comτ1k,f∧Contk,φψ,η≤comτ2ηk,φψf,for all k∈K1 and f∈LX1E1.Proof.
Let us choose an arbitraryβ∈ML such that β⊲comτ1k,f∧Contk,φψ,η. By the below wedge operation property, we have that(32)β⊲Contk,φψ,η=∧g∈LX2E2τηk2g↦τk1φψ−1g,β⊲comτ1k,f=∧U⊆LX1E1∧g∈Uτk1g∧∧x∈X1∧e∈E1fe′x∨∨g∈Uhex↦∨V∈2U∧x∈X1∧e∈E1fe′x∨∨g∈Vgex.
Hence for anyg∈LX2E2 and for any U⊆LX1E1, we gain(33)β≤τηk2g↦τk1φψ−1g,β≤∧g∈Uτk1g∧∧x∈X1∧e∈E1fe′x∨∨g∈Uhex↦∨V∈2U∧x∈X1∧e∈E1fe′x∨∨g∈Vgex.
By considering the implication properties, we have(34)β∧τηk2g≤τk1φψ−1g,for anyg∈LX2E2,β∧∧g∈Uτk1g∧∧x∈X1∧e∈E1fe′x∨∨g∈Uhex≤∨V∈2U∧x∈X1∧e∈E1fe′x∨∨g∈Vgex.
In order to complete the proof, it is necessary to show that(35)β≤comτ2ηk,φψf=∧W⊆LX2E2∧w∈Wτηk2w∧∧y∈X2∧e∗∈E2φψfe∗y′∨∨w∈Wwe∗y↦∨D∈2W∧y∈X2∧e∗∈E2φψfe∗′y∨∨w∈Dwe∗y.
Letφψ−1W=φψ−1w|w∈W⊆LX1E1. Hence, we have the following facts:(36)β∧∧w∈Wτηk2w∧∧y∈X2∧e∗∈E2φψfe∗′y∨∨w∈Wwe∗y≤β∧∧w∈Wτk1φψ−1w∧∧y∈X2∧e∗∈E2φψfe∗′y∨∨w∈Wwe∗y=β∧∧w∈Wτk1φψ−1w∧∧x∈X1∧e∈E1fe′x∨∨w∈Wφψ−1wex=β∧∧g∈φψ−1Wτk1g∧∧x∈X1∧e∈E1fe′x∨∨g∈φψ−1Wgex≤∨V∈2φψ−1W∧x∈X1∧e∈E1fe′x∨∨g∈Vgex=∨D∈2W∧x∈X1∧e∈E1fe′x∨∨w∈Dφψ−1wex=∨D∈2W∧y∈X2∧e∗∈E2φψfe∗′y∨∨w∈Dwe∗y.
From the implication operator properties, we get(37)β≤∧w∈Wτηk2w∧∧y∈X2∧e∗∈E2φψfe∗y′∨∨w∈Wwe∗y↦∨D∈2W∧y∈X2∧e∗∈E2φψfe∗′y∨∨w∈Dwe∗y.
Therefore, we obtain the following:(38)β≤∧W⊆LX2E2∧w∈Wτηk2w∧∧y∈X2∧e∗∈E2φψfe∗y′∨∨w∈Wwe∗y↦∨D∈2W∧y∈X2∧e∗∈E2φψfe∗′y∨∨w∈Dwe∗y.
This witnesses the desired factβ≤comτ2ηk,φψf.Corollary 3.
LetX1,τ1 and X2,τ2 be the L-fuzzy E1,K1-soft and L-fuzzy E2,K2-soft topological spaces. If the fuzzy soft mapping φψ,η:X1,τ1⟶X2,τ2 is surjective, then we have that comτ1k,1˜X1∧Contk,φψ,η≤comτ2ηk,1˜X2.Theorem 11.
LetX1,τ1 and X2,τ2 be the L-fuzzy E1,K1-soft and L-fuzzy E2,K2-soft topological spaces. For the fuzzy soft mapping φψ,η:X1,τ1⟶X2,τ2, we have(39)Conηk,φψf′∧Contk,φψ,η≤Conk,f′,for all k∈K1 and f∈LX1E1.Proof.
Let us choose an arbitraryβ∈ML such that β⊲Conηk,φψf′∧Contk,φψ,η. By Theorems 3 and 4, the following implications are obtained:(40)β⊲Conηk,φψf′=∨τηk2u′∧τηk2v′|φψf⊓u≠0˜,φψf⊓v≠0˜,φψf⊓u⊓v=0˜,φψf⋢u⊔v,β⊲Contk,φψ,η=∧h∈LX2E2τηk2h′↦τk1φψ−1h′.
Hence, there existsu,v∈LX2E2 which satisfy the conditions φψf⊓u≠0˜,φψf⊓v≠0˜,φψf⊓u⊓v=0˜,andφψf⋢u⊔v such that β≤τηk2u′∧τηk2v′ and also β≤τηk2h′↦τk1φψ−1h′, for each h∈LX2E2. That is, there exists u,v∈LX2E2 satisfying f⊓φψ−1u≠0˜,f⊓φψ−1v≠0˜,f⊓φψ−1u⊓φψ−1v=0˜, and f⋢φψ−1u⊔φψ−1v such that β≤τηk2u′∧τηk2v′. And also, β∧τηk2h′≤τk1φψ−1h′ is satisfied for each h∈LX2E2. We conclude that from these observations,(41)β=β∧τηk2u′∧τηk2v′≤τk1φψ−1u′∧τk1φψ−1v′≤∨τk1g′∧τk1h′|f⊓g≠0˜,f⊓h≠0˜,f⊓g⊓h=0˜,f⋢g⊔h=Conk,f′.
Since the coprime elementβ is arbitrary, we gain the desired inequality for the parameter k, Conηk,φψf′∧Contk,φψ,η≤Conk,f′.


## 4. Conclusion

As it is well known, in real life, nothing is described with the help of the 2-valued logic since there is no only black and white in nature. But the idea of fuzzy thinking, which gives some degrees to the phenomena reflect the facts more correctly. In the topological point of view, fuzzy logic, which is also related with quantum mechanics, was applied to the gradation of belongingness and it gave birth to the fuzzy topology. On the other side, the soft set theory which emphasizes the importance/necessity of the parametrization tool for adequate mathematical modeling of the natural facts is one of the preferred tools by the researchers [13, 23–26]. Besides, the combination of these types of sets, named as fuzzy soft set, is one of the adequate tools for modeling. In this respect, we deal with the gradation of the continuity of mappings between fuzzy soft topological spaces (here both of the sets and the axioms of the structure are all fuzzy soft). Since the mappings play the main role to establish the relations between structured sets, the idea proposed here helps us to obtain more appropriate and compatible results in such spaces. Despite the theoretical benefits of this method, it is not easy to find numerical examples in application. But this could be overcome by taking a unit interval instead of a lattice.In relation with the research in this study, notice that soft continuity seems to be the natural tool to prove results more similar to Weierstrass’s celebrated theorem. For further research, we hope to investigate this idea and try to find reasonable results. Furthermore, we hope to extend the proposed methods to Pythagorean fuzzy uncertain environments [27] as an additional research.



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