Fuzzy Soft Topology Based on Generalized Intersection and Generalized Union of Fuzzy Soft Sets

In this study, the generalized intersection and union operations of fuzzy soft set (FSS) are established on the basis of traditional FSS operations, which overcome the shortcomings of traditional FSS operations that do not meet De Morgan’s law, and a series of properties of generalized intersection and union operations of FSS are obtained. ,e fuzzy soft topology under generalized intersection and generalized union operation of FSSs is established. Finally, the topological construction of weak FSS and strong FSS is discussed, and the relationship between them and the topological construction of traditional FSS is obtained.


Introduction
In real life, the data of many complex problems often have uncertainty, randomness, and fuzziness. Probability theory, fuzzy sets [1], intuitionistic fuzzy sets [2], rough sets [3], and so on are the main methods to deal with uncertain data, obtained in various fields of engineering, computer science, economics, medical science, etc., but these theories have their own difficulties, mainly reflected in the lack of parameter tools. In 1999, Molodtsov established soft set theory [4], which overcame the deficiency of parameter tools of these methods and was successfully applied to decision analysis, pattern recognition, data mining, and other fields [5][6][7], attracting the attention of scholars in many fields [8][9][10][11][12][13]. In 2021, Guilong Liu outlines a possible relationship between knowledge structures and rough sets [14]. Swarup Kr Ghosh et al. represent a colonogram enhancement approach using intuitionistic fuzzy set [15]. In 2001, Liu et al. combined the fuzzy set with soft set and proposed the concept of fuzzy soft set (FSS) [11]. en, Das et al. redefined the concept of FSS and studied its operation. In 2011, Bekir Tanay and M. Burc Kandemir established the topological structure of fuzzy soft sets based on some basic definitions in the topological space of Munkres point sets and combined with the operational properties of FSSs [13]. In addition, Ayetan and Cemil gave the separation axiom and fuzzy soft connectivity of fuzzy soft topological space. In 2018, Ayten and Cemil established mixed fuzzy soft topological spaces [16]. Finally, Riaz et al. gave a bipolar fuzzy soft topology with decision [17]. In 2020, José Carlos R. Alcantud established soft open bases and a novel construction of soft topologies from bases for topologies [18]. By 2021, Ultami and Haripamyu have systematically studied the application of intuitional fuzzy soft sets in topological structures [19]. e existing intersection and union operations of FSSs do not satisfy De Morgan's law. In the study of fuzzy soft topological space structure and properties, the intersection, union, and complement operations of FSSs need to satisfy De Morgan's law. Considering that De Morgan's law of intersection, union, and complement operations of FSSs is not valid, this paper constructs the generalized intersection and union operations of FSSs on the basis of the existing FSS operations, satisfies De Morgan's law, and further studies the operation properties of the generalized intersection and union operations and constructs the corresponding fuzzy soft topology. Based on the concepts of weak fuzzy soft set and strong fuzzy soft set proposed by F.B.Hua and Y.P.Wang in [20], the corresponding structures and properties of fuzzy soft topology are studied.

Fuzzy Soft Set and Its Operation Properties
Definition 1 (see [11]). Let U be the domain and A be the parameter set; A ⊂ E, F(U) is the whole fuzzy set over U; we define (f, A) a fuzzy soft set (FSS) on U, wheref: [13]).
e union of (g, C) and (h, (1) Definition 7 (see [21]). Let (g, C)and (h, D) be FSSs on U. e intersection of (g, C) and (h, As can be seen from Definition 7, when c ∈ A − B or c ∈ B − A, ∀x ∈ U, and h c (x) has no definition, this will cause De Morgan's law to cease to hold, due to the nature of the algebraic structure of FSS and nature of fuzzy soft topological space structure; setup in a certain level will depend on the virtue of De Morgan's law; therefore, it is very meaningful to improve the intersection and union operation of fuzzy soft sets.

Generalization of Union and Intersection Operation of FSS
Definition 8 Let (f, A) and (g, B) be FSSs on U; the generalized union of (f, A), Remark 2. It is worth noting that because of there is (f, A) ∪ (g, B) � (f, A) ∪ (g, B).
Definition 9 Let (f, A) and (g, B) be FSSs on U; the generalized intersection of (f, A), Remark 3. It is worth noting that because of there It can be seen from Definitions 8 and 9 that the generalized union operation of FSS keeps the original union operation of FSS, while the generalized intersection operation of FSS makes up for the situation, where ∀x ∈ U, h c (x) is undefined when 3 , the parameter set be A � e 1 , e 2 and B � e 2 , e 3 ⊂ E � e 1 , e 2 , e 3 , and the two FSSs on U be (f, Theorem 1. e generalization of the union and intersection operation of FSSs satisfies the commutative law, associative law, and distributive law; that is, let (f, A), (g, B), and (h, C) be FSSs on U; then, there are Proof. (1) and (2) are obviously true according to Definitions 8 and 9, respectively.

Journal of Mathematics
Since 4 Journal of Mathematics erefore, r t (x) � r t ′ (x), that is, (4) is true.
From (5), let Journal of Mathematics erefore, s a (x) � t a (x), that is, (5) is true. e proof of (6) is the same as that of (5 en, Journal of Mathematics erefore, (1) is true.
e proof method of (2) is the same as that of (1). □

Generalization of Remainder Operation of FSS
Example 3. Let the domain be U � h 1 , h 2 , h 3 , the parameter set be A � e 1 , e 2 and B � e 2 , e 3 ⊂ E � e 1 , e 2 , e 3 , and the two FSSs on U be (h, A), (k, B) be FSSs on U; then, there are

Theorem 2. e generalization of the union and intersection operations of fuzzy FSSs satisfies De Morgan's law, that is, let
we get r c a (x) � s c a (x) and e proof of (2) is similar to that of (1).

Topology Construction Based on FSS Generalization under Intersection and Union Operation
Definition 11 (see [13]). LetF(U; E) be the whole family of fuzzy FSSs with parameter set E on U; (c, X) ∈ F(U; E) and P(c, X)are the whole fuzzy soft subset of (c, X). τ is a subfamily of P(c, X); then, τ is a fuzzy soft topology of (c, X). τ) is a fuzzy soft topology space.
Definition 13. Let F(U; E) be the whole family of FSSs with parameter set E on U; (c, X) ∈ F(U; E) and P(c, X) are the whole FSS of (c, X). τ is a subfamily of P(c, X); then, τ is a fuzzy soft topology of (c, X). τ) is a fuzzy soft topology space.
Theorem 3. Let F(U; E) be the whole family of fuzzy soft sets with parameter set E on U; (c, X) ∈ F(U; E) and P(c, X) are the whole FSS of (c, X). τ is a subfamily of P(c, X), if  A k ), erefore, it can be seen that eorem 2 holds. (2) Comparing Definition 10 with Definition 11, it can be seen that the conditions required for topology structure construction based on intersection and union operation in the traditional sense of FSS are relatively weaker than that based on the generalized intersection and union operation of FSS, but the intersection and union operation in the traditional sense of FSS does not satisfy De Morgan's law. is makes it very difficult to further study the structure and properties of fuzzy soft topology. e intersection and union operations in the generalized sense of fuzzy FSS satisfy De Morgan's law, which will bring great convenience for further study the structure of fuzzy soft topology.
It is easy to verify that τ satisfies all the conditions in Definition 4.3 and constitutes a fuzzy soft topology under Definition 4.3.
Definition 14 If (f, A) ⊂ (c, X) is a FSS, for ∀x ∈ U, ∀a ∈ A, and f a (x) � c a (x), then (f, A) is the limit of (c, X) on A, denoted as (f, A) � (c| A , A). Proof we get (f, A) ∈ τ (f,A) .