Gradation of Continuity for Fuzzy Soft Mappings

This paper is devoted to describe the notion of a parameterized degree of continuity for mappings between L -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.


Introduction
e 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. is 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]. e 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][4][5]. e 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][8][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][11][12][13]. e 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. us, as a continuation of the research studies [14][15][16], we describe the parameterized gradations of continuity, openness, and closedness for mappings between fuzzy soft topological spaces. e content of this study is organized in the following manner: in Section 2, 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.
viewed on the sets of parameters, and L � (L, ∨, ∧, ′ ) denotes a complete De Morgan algebra with the smallest element 0 L and the largest element 1 L . With the underlying lattice L, a mapping A: X ⟶ L is said to be an L-fuzzy set on X and by L X , we denote the family of all L-fuzzy sets on X.
An element α in L is said to be coprime if α ≤ β ∨ c implies that α ≤ β or α ≤ c. M(L) 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 ≤ c for some c ∈ D. Clearly, if α ∈ L is coprime, then α ≪ β if and only if α ⊲ β. Details for lattices can be found in [17]. e binary operation ↦ in the complete De Morgan algebra L is given by For all α, β, c, δ ∈ L and α i , β i ⊆ L, the following properties are satisfied: Definition 1 (see [18]). A mapping f: E ⟶ (L X ) E is called an L-fuzzy soft set on X. is means that f e : � f(e): 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 symbol (L X ) E to denote the collection of all L-fuzzy soft sets on X.
Definition 2 (see [18,19]). Let f and g 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 f e ≤ g e , for each e ∈ E. f and g are called equal if f ⋢ g and g ⋢ f.
(2) e union of f and g is an L-fuzzy soft set h � f⊔g, where h e � f e ∨g e , for each e ∈ E. (3) e intersection of f and g is an L-fuzzy soft set h � f⊓g, where h e � f e ∧g e , for each e ∈ E.
Definition 3 (see [18]) (1) An L-fuzzy soft set f on X is called a null (or empty) L-fuzzy soft set and denoted by 0, if f e (x) � 0 L , for each e ∈ E, x ∈ X. (2) An L-fuzzy soft set f on X is called an absolute (or universal) L-fuzzy soft set and denoted by 1, if f e (x) � 1 L , for each e ∈ E, x ∈ X. Clearly (1) ′ � 0 and 0 ′ � 1.
Definition 4 (see [20]). Let x ∈ X and α: E ⟶ M(L) be a function. en, the L-fuzzy soft set defined as follows is called an L-fuzzy soft point and denoted by x α .
for all e ∈ E and y ∈ X. (1) An L-fuzzy soft point x α is said to belong to an L-fuzzy soft set f and denoted by e set of all nonzero coprime elements of (L X ) E is denoted by M((L X ) E ). It is noted that M((L X ) E ) is exactly the set of all L-fuzzy soft points.
Definition 6 (see [14]). A mapping τ: K ⟶ L (L X ) E which satisfies the following certain axioms is called an L-fuzzy (E, K)-soft topology on X.
, for all f i i∈Δ ⊆(L X ) E and for each k ∈ K. en, the pair (X, τ) is called an L-fuzzy (E, K)-soft topological space. e value τ k (f) 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 τ * k (f) � τ k (f ′ ). Let τ 1 and τ 2 be L-fuzzy (E, K)-soft topologies on X. We say that τ 1 is finer than τ 2 (τ 2 is coarser than τ 1 ), denoted by Definition 7 (see [14]). Let (X 1 , τ 1 ) be an L-fuzzy (E 1 , K 1 )-soft topological space and (X 2 , τ 2 ) be an L-fuzzy (E 2 , K 2 )-soft topological space. Let φ: X 1 ⟶ X 2 , ψ: E 1 ⟶ E 2 and η: K 1 ⟶ K 2 be crisp functions. en, the fuzzy soft mapping φ ψ,η : Definition 8 (see [20]). For a fixed fuzzy soft point x α , let the mappings Q x α : K × (L X ) E ⟶ L satisfy the following axioms for each k ∈ K and f, g ∈ (L X ) E : en, the collection Q � Q x α |x α ∈ M((L X ) E ) of maps presented above is called an L-fuzzy (E, K)-soft quasi-coincident neighborhood (shortly, q-nhood) system on X. e value Q x α (k, f) represents the parameterized degree of being q-nhood of f to the fuzzy soft point x α .
Proposition 1 (see [20]). Let τ be an L-fuzzy (E, K)-soft topology on X. Define the mapping Q τ Definition 9. For a fixed fuzzy soft point x α , let the mappings N x α : K × (L X ) E ⟶ L satisfy the following axioms for each k ∈ K and f, g ∈ (L X ) E .
is called an L-fuzzy (E, K)-soft neighborhood (shortly, nhood) system on X.
Definition 10 (see [16]). A mapping cl: is called an L-fuzzy (E, K)-soft closure operator on X if it satisfies the following axioms for each k ∈ K: is interpreted as the degree to which x α belongs to the parameterized closure of the fuzzy soft set f.
Example 1 (see [16]). Let cl: K × (L X ) E ⟶ (L X ) E be the closure operator given in a parameterized L-soft topological space (X, T). In this case, the mapping is satisfies the conditions of Definition 10.
Theorem 1 (see [16]). Let τ be an L-fuzzy (E, K)-soft topology on X. en, the mapping C: 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.
is called an L-fuzzy (E, K)-soft interior operator on X if it satisfies the following axioms for each k ∈ K: is interpreted as the degree to which x α belongs to the parameterized interior of the fuzzy soft set f. Theorem 2. Let τ be an L-fuzzy (E, K)-soft topology on X, and let N � N x α |x α ∈ M((L X ) E ) be the nhood system induced by τ. Define a mapping I: en, the mapping I 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. en, identify a mapping com τ : in such a way that in order to picture the parameterized compactness degree, In this case, the value com τ (k, g) is interpreted as the compactness degree of g ∈ (L X ) E with respect to the parameter k. So, g is said to be compact L-fuzzy soft set with respect to k if com τ (k, g) � 1 L .
Definition 13 (see [16]). Let (X, τ) be an L-fuzzy (E, K)-soft topological space. en, identify a mapping Con: K × (L X ) E ⟶ Ł by the following manner in order to describe the connectedness degree in such spaces: In this case, the value Con(k, h) is said to be the connectedness degree of an L-fuzzy soft set h with respect to k.
Theorem 3 (see [16]). Let τ be an L-fuzzy (E, K)-soft topology on X. en, one can characterize the parameterized degree of connectedness of an L-fuzzy soft set g ∈ (L X ) E in the following way:

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. en, 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.
is implies , φ ψ (f))). Hence, we also have Since the arbitrariness of c, we gain Since f⋢φ −1 ψ (φ ψ (f)), for all f ∈ (L X 1 ) E 1 , we obtain the following inequality: Hence we obtain the desired result. By using eorems 1 and 2, and also by considering some similar discussion, one can prove the other claims of the theorem. □ Theorem 10. Let (X 1 , τ 1 ) and (X 2 , τ 2 ) be the L-fuzzy (E 1 , K 1 )-soft and L-fuzzy (E 2 , K 2 )-soft topological spaces. For the fuzzy soft mapping φ ψ,η : (X 1 , τ 1 ) ⟶ (X 2 , τ 2 ), we have the following result: Proof. Let us choose an arbitrary β ∈ M(L) such that β⊲(com τ 1 (k, f)∧Cont(k, φ ψ,η )). By the below wedge operation property, we have that Journal of Mathematics 9 Hence for any g ∈ (L X 2 ) E 2 and for any U⊆(L X 1 ) E 1 , we gain By considering the implication properties, we have In order to complete the proof, it is necessary to show that Let φ −1 ψ (W) � φ −1 ψ (w)|w ∈ W ⊆(L X 1 ) E 1 . Hence, we have the following facts: φ ψ (f) e * ′ (y)∨ ∨ w∈D w e * (y) . (36) 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.

Data Availability
e data used to support the findings of this study are cited at relevant places within the text as references and are also available from the corresponding author upon request.

Conflicts of Interest
e author declares no conflicts of interest.