A Net with Applications for Continuity in a Fuzzy Soft Topological Space

In this paper, the concept of a fuzzy soft point is redefined, and the definition of a fuzzy soft net in a fuzzy soft topological space is given. On this basis, the convergence of a fuzzy soft net is defined by using the Q-neighborhood theory, and the continuity of fuzzy soft mappings is characterized by the net. The obtained results demonstrate that the concepts proposed in this paper are very suitable and will provide powerful research tools for further research in this field.


Introduction
In 1965, Zadeh introduced the concept of fuzzy sets in his classic work [1]. In 1999, Molodtsov [2] introduced the theory of soft sets, which have been applied in several fields, including the smoothness of functions, game theory, Riemann integration, and the theory of probability [3]. In 2001, Maji et al. [4] combined fuzzy sets with soft sets and proposed the concept of fuzzy soft sets. Since then, many researchers have applied fuzzy soft sets to group theory [5], decision making, and medical diagnosis [6,7]. In 2009, Kharal and Ahmad [8] studied the properties of fuzzy soft images and fuzzy soft inverse images of fuzzy soft sets.
In 2011, Tanay and Kandemir [9] proposed the concept of a fuzzy soft set in a fuzzy soft topological space and explained some of its structural properties. ey also claimed that fuzzy soft topological spaces may be used in the theory of information systems. In 2012, Mahanta and Das [10] introduced the definition of a fuzzy soft point and its neighborhood. ey also studied the interior and closure of a fuzzy soft set and investigated the separation axioms and connectedness. Varol and Aygün [11] introduced the fuzzy soft continuity of fuzzy soft mappings. In 2013, Gunduz (Aras) and Bayramov [3] presented fuzzy soft continuous mappings, fuzzy soft open and fuzzy soft closed mappings, and fuzzy soft homeomorphism. In 2014, Ping et al. [12] proposed the sum of fuzzy soft topological spaces. In 2016, Mishra and Srivastava [13] studied compactness in fuzzy soft topological spaces. In 2017, Kandil et al. [14] discussed the connectedness of fuzzy soft sets. Riaz and Hashmi [15] proposed the concept of fuzzifying soft sets, called fuzzy parameterized fuzzy soft sets (FPFS-sets). Mahanta and Das [16] studied fuzzy soft closure and the fuzzy soft interior. In 2018, Abbas et al. [17] explored connectedness in fuzzy soft topological spaces. In 2019, Riaz and Tehrim [18] proved some properties of bipolar fuzzy soft topology (BFS-topology) via the use of the concept of the Q-neighborhood.
In 2012, Roy and Samanta [19] redefined the concept of fuzzy soft topology and obtained some basic results. In a subsequent work [20], they adopted a new definition of a fuzzy soft point, proposed the concepts of quasi-coincidence and Q-neighborhoods, and demonstrated the relationship between the limit point and the closure of a fuzzy soft set.
As pointed out in Example 1 in this paper, the existing concept of fuzzy soft points does not satisfy selectivity, which makes it difficult for a fuzzy soft point to play its expected role. erefore, a more suitable definition of a fuzzy soft point must be given. Moreover, as is commonly known, the net plays an important role in classical topology theory; however, the concept of the net has not been introduced into fuzzy soft topological spaces.
In view of these considerations, this paper first redefines the concept of a fuzzy soft point and introduces the net into fuzzy soft topological spaces. e continuity of a mapping in fuzzy soft topological spaces with the use of a net is then studied. e remainder of this article is organized as follows. In Section 2, some necessary concepts of fuzzy soft sets and fuzzy soft topological spaces are recalled. In Section 3, fuzzy soft points are refined and the notion of a fuzzy soft net consisting of fuzzy soft points is introduced. By using the theory of the Q-neighborhood, the concept of the convergence of a fuzzy soft net is introduced. In Section 4, the net is applied to characterize the continuity of fuzzy soft mappings. Section 5 presents a discussion of convergence for a net of fuzzy soft mappings. In Section 6, an example of the application of the fuzzy soft set theory to medical diagnosis is provided. Finally, the conclusion of this paper is given in Section 7.

Preliminaries
roughout this paper, U refers to an initial universe and E is the set of all parameters of U. In this case, U is also denoted by (U, E). I U is the set of all fuzzy subsets over U, where I � [0, 1]. e elements 0, 1 ∈ I U , respectively, refer to the functions 0(x) � 0 and 1(x) � 1 for all x ∈ U. For an element A ∈ I U , if there exists an x ∈ U such that A(x) � λ > 0 and A(y) � 0, ∀y ∈ U\ x { }, then A is called a fuzzy point over U and is denoted by x λ , and x and λ are called the support and height of x λ , respectively. e set of all fuzzy points over U is denoted by FP(U). e following definitions in this section were obtained from the existing literature [19,20].
e set of all fuzzy soft sets over (U, E) is denoted by FS(U, E). e fuzzy soft set F ϕ ∈ FS(U, E) is called the null fuzzy soft set and is denoted by Φ. Here, F ϕ (e) � 0 for every e ∈ E. For then F E is called the absolute fuzzy soft set and is denoted by E. Let for all e ∈ E, then F A is said to be a fuzzy soft subset of F B and is denoted by Similarly, the union (intersection) of a family of fuzzy soft sets F C α : α ∈ Λ can be defined and is denoted by where Λ is an arbitrary index set.

Remark 2. It is clear that
A fuzzy soft topology τ over (U, E) is a family of fuzzy soft sets over (U, E) satisfying the following properties: If τ is a fuzzy soft topology over (U, E), the triple (U, E, τ) is said to be a fuzzy soft topological space. Each element of τ is called an open set. If F c A is an open set, then F A is called a closed set. is called the interior of F A and is denoted by

Fuzzy Soft Point and Fuzzy Soft Net
Roy and Samanta [20] (2) Also, fuzzy soft sets F A and F B are defined as en, is example demonstrates that a fuzzy soft point that belongs in the union of two fuzzy soft sets does not need to belong in one of these two fuzzy soft sets.
To overcome this shortcoming, a fuzzy soft point is redefined as follows.
In this case, ξ is also denoted by P x λ e , and e is called its parameter support. e set of all fuzzy soft points over It is clear that F e in Example 1 is not a fuzzy soft point in the sense of Definition 5.
en, F e is a fuzzy soft point in the sense of Definition 5 and can be written as P In the remainder of this paper, a fuzzy soft point is always referred to as given by Definition 5 and is called a point for short.
For P However, it is clear that eorem 1(1) does not hold for the union of infinite fuzzy soft sets. For this reason, the concept of quasi-coincidence is subsequently introduced.
Remark 4. In the work by Roy and Samanta [20], a fuzzy soft point F e over (U, E) was said to be quasi-coincident with If F e is a fuzzy soft point in the sense of Definition 5, then the definition of quasi-coincidence in this paper is equivalent to that in the work by Roy and Samanta [20].

Proof
(1) e necessity is evident. To prove the sufficiency, it is supposed that P x λ e ∈ F B for any point P x λ e ∈ F A . If F A ⊆ F B is not true, then there exist an e ∈ E and (3) From Definition 6, it can be seen that P Proof. Suppose that F A qF B ; then, there exist an x ∈ U and e ∈ A∩B such that μ e e converse of eorem 3 does not hold. Indeed, in Example 1, Because then F A qF B . However, F A ∩ F B ≠ Φ. (1) F A is said to be a neighborhood of ξ if there exists an

Theorem 4. Let F A , F B ∈ FS(U, E). If there is a
e set of all Q-neighborhoods of ξ is denoted by U(ξ). In the remainder of this paper, Δ is a directed set with the partial order "≺." In particular, if there exists an F A ∈ FS(U, E) such that S(δ) ∈ F A for any δ ∈ Δ, then S is said to be a fuzzy soft net in F A , or a net for simplicity.

Definition 11. Let F A ∈ FS(U, E) and S � S(δ), δ ∈ Δ
{ } be a net in (U, E). If there exists a δ 0 ∈ Δ such that S(δ)∈ F A whenever δ 0 ≺δ, then S is said to be eventually quasi-coincident with F A . If for each δ ∈ Δ there exists a δ 0 ∈ Δ with δ≺δ 0 such that S(δ 0 )∈ F A , then S is said to be frequently quasi-coincident with F A .

Definition 12. A net S(δ), δ ∈ Δ
{ } in (U, E, τ) is said to be convergent to a point ξ if S is eventually quasi-coincident with each Q-neighborhood of ξ. In this case, ξ is called the limit of S and is denoted by limS(δ).

Fuzzy Soft Continuous Mapping
In this section, the definition of fuzzy soft continuous mapping is first recalled, and the net is then applied to characterize the continuity.
Definition 13 (see the work of Aygünoglu and Aygün [5]). Let FS(U 1 , E 1 ) and FS(U 2 , E 2 ) be the families of all fuzzy soft sets over U 1 and U 2 , respectively. Let φ: U 1 ⟶ U 2 and ψ: E 1 ⟶ E 2 be two functions. en, the pair (φ, ψ) is called a fuzzy soft mapping from U 1 to U 2 and is denoted by (φ, ψ): (1) Let F A ∈ FS(U 1 , E 1 ). en, the image of F A under the fuzzy soft mapping (φ, ψ) is the fuzzy soft set over Mathematical Problems in Engineering (2) Let F B ∈ FS(U 2 , E 2 ). en, the pre-image of F B under the fuzzy soft mapping (φ, ψ) is the fuzzy soft set over If both φ and ψ are injective (surjective), then the fuzzy soft mapping (φ, ψ) is said to be injective (surjective). e composition of two fuzzy soft mappings (φ, ψ) from

Lemma 1 (see the work of Kharal and Ahmad [8]). Let
e following is then obtained: is surjective It is simple to verify the following lemma.

Application of Fuzzy Soft Set Theory to Medical Diagnosis
In a hospital, some doctors usually decide what disease a patient is suffering from by observing the patient's symptoms. However, due to the complexity of symptoms, it is difficult to find the precise relationship between diseases and symptoms. e concept of fuzzy soft sets partially resolves this difficulty. Suppose that the initial universe U � x 1 , x 2 , . . . , x m is the set of all the disease objects that the patient may be infected with, and the set of parameters E � e 1 , e 2 , . . . , e n is all of the patient's symptoms. Generally speaking, from a symptom e ∈ E, one cannot completely determine the corresponding disease x ∈ U; however, one can determine the membership degree in which object x ∈ U holds parameter e ∈ E, which is denoted by μ e F (x); that is, for every e ∈ E, there is a fuzzy subset F(e) of U. Obviously, the mapping F: E ⟶ I U is a fuzzy soft set over (U, E). Let If μ(x i 0 ) � max 1≤i≤m μ(x i ), then it may be claimed that the patient has disease x i 0 .

Conclusions
In this paper, the new concepts of fuzzy soft points and fuzzy soft nets were introduced to fuzzy soft topological spaces. On these bases, the fuzzy soft net was used to accurately describe the convergence, which was used to characterize the continuity. Moreover, the convergence for a net of fuzzy soft mappings was investigated. e obtained results demonstrate that the concepts proposed in this paper are very useful and will provide powerful research tools for further research in this field. Particularly, the convergence of fuzzy soft nets may be used to characterize some important properties of fuzzy soft topological spaces, such as closure, separation, compactness, etc.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.