Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 9098410, 7 pages
https://doi.org/10.1155/2020/9098410

Research Article

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

Rui Gao and Jianrong Wu

School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, China

Correspondence should be addressed to Jianrong Wu; jrwu@mail.usts.edu.cn

Received 24 March 2020; Revised 22 May 2020; Accepted 1 June 2020; Published 30 June 2020

Academic Editor: Suzanne M. Shontz

Copyright © 2020 Rui Gao and Jianrong Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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. They 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. They 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. Therefore, 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. The continuity of a mapping in fuzzy soft topological spaces with the use of a net is then studied. The 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.

2. Preliminaries

Throughout this paper, refers to an initial universe and is the set of all parameters of . In this case, is also denoted by . is the set of all fuzzy subsets over , where . The elements , respectively, refer to the functions and for all . For an element , if there exists an such that and , , then is called a fuzzy point over and is denoted by , and and are called the support and height of , respectively. The set of all fuzzy points over is denoted by .

The following definitions in this section were obtained from the existing literature [19,20].

Definition 1. Let . A mapping , defined by (a fuzzy subset of ), is called a fuzzy soft set over , where if and if .
If , then is shortened to .
The set of all fuzzy soft sets over is denoted by .
The fuzzy soft set is called the null fuzzy soft set and is denoted by . Here, for every .
For , if for all , then is called the absolute fuzzy soft set and is denoted by .
Let , . If for all , then is said to be a fuzzy soft subset of and is denoted by or . In addition, means that is not a fuzzy soft subset of . If and , then and are said to be equivalent, which is denoted by .

Remark 1. If , then .

Definition 2. Let .

(1)The complement of , denoted by , is defined as

(2)The union of and , denoted by , is defined as for all , where .(3)The intersection of and , denoted by , is defined as for all , where .

Similarly, the union (intersection) of a family of fuzzy soft sets can be defined and is denoted by , where is an arbitrary index set.

Remark 2. It is clear that

(1)(2),

Definition 3. A fuzzy soft topology over is a family of fuzzy soft sets over satisfying the following properties:

(1)(2)If , then (3)If for all (any index set), then

If is a fuzzy soft topology over , the triple is said to be a fuzzy soft topological space. Each element of is called an open set. If is an open set, then is called a closed set.

Definition 4. Let be a fuzzy soft topological space, :

(1)The intersection of all closed sets is called the closure of and is denoted by (2)The union of all open subsets of over is called the interior of and is denoted by

Remark 3. Let . It is evident that

(1) is closed and is open(2) is closed if and only if  = (3) is open if and only if  = 

3. Fuzzy Soft Point and Fuzzy Soft Net

Roy and Samanta [20] defined a fuzzy soft point over as a special fuzzy soft set such that if , and if . For , they stated that a fuzzy soft point belongs in , denoted by , if and only if for all .

Example 1. Let and . A fuzzy soft point is defined as

Also, fuzzy soft sets and are defined as

Then,

It is clear that . However, and .
This 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.

Definition 5. A mapping is called a fuzzy soft point over if there is an such that , and when .
In this case, is also denoted by , and is called its parameter support. The set of all fuzzy soft points over is denoted by .
It is clear that in Example 1 is not a fuzzy soft point in the sense of Definition 5.

Example 2. Let and . A mapping is defined as

Then, is a fuzzy soft point in the sense of Definition 5 and can be written as .
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 , , if or or , then it is said that and are different, which is written as .

Theorem 1. Let , , and :

(1)If , then or (2) if and only if

Proof.

(1)Because , then ; that is,     . Therefore, or . Hence, or .(2) is equal to . Equivalently, ; that is, .

However, it is clear that Theorem 1(1) does not hold for the union of infinite fuzzy soft sets. For this reason, the concept of quasi-coincidence is subsequently introduced.

Definition 6. A point is said to be quasi-coincident with , which is denoted by , if .
That is not quasi-coincident with is denoted by .

Remark 4. In the work by Roy and Samanta [20], a fuzzy soft point over was said to be quasi-coincident with if   for some . If 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].

Remark 5. It is evident that if and only if . Additionally, if , then there exists such that .

Theorem 2. Let , , , and :

(1) if and only if for any (2) implies   if and only if and (3) if and only if there exists an such that

Proof.

(1)The necessity is evident. To prove the sufficiency, it is supposed that for any point . If is not true, then there exist an and such that . For with , the point and , which contradicts the assumption. Thus, .(2)If , then . Hence, ; that is, . From Definition 6, it is clear that if and only if . Equivalently, . Thus, and ; that is, and .(3)From Definition 6, it can be seen that if and only if . Equivalently, there exists an such that ; that is, .

Definition 7. Let and . If , then it is said that and are quasi-coincident at .

Remark 6. It is clear that if and only if and are quasi-coincident at a point .

Definition 8. (see the work by Roy and Samanta [20]). A fuzzy soft set is said to be quasi-coincident with , which is denoted by , if for some and .
That is not quasi-coincident with is denoted by .

Theorem 3. Let . If , then .

Proof. Suppose that ; then, there exist an and such that . Set . Then, . Hence, .

Remark 7. The converse of Theorem 3 does not hold. Indeed, in Example 1,

Because

then . However, .

Theorem 4. Let . If there is a with such that , then .

Proof. Because , via Theorem 2(2), and ; that is, . From , . Therefore, ; that is, .

Definition 9. Let and :

(1) is said to be a neighborhood of if there exists an such that (2) is called a Q-neighborhood of if there exists an such that

The set of all Q-neighborhoods of is denoted by .
In the remainder of this paper, is a directed set with the partial order “.”

Remark 8. It is clear that is a directed set with the partial order “.”

Definition 10. The mapping is called a fuzzy soft net in and is denoted by , or for simplicity.
In particular, if there exists an such that for any , then is said to be a fuzzy soft net in , or a net for simplicity.

Definition 11. Let and be a net in . If there exists a such that whenever , then is said to be eventually quasi-coincident with . If for each there exists a with such that , then is said to be frequently quasi-coincident with .

Definition 12. A net in is said to be convergent to a point if is eventually quasi-coincident with each Q-neighborhood of . In this case, is called the limit of and is denoted by .

4. 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 Aygünoğlu and Aygün [5]). Let and be the families of all fuzzy soft sets over and , respectively. Let and be two functions. Then, the pair is called a fuzzy soft mapping from to and is denoted by : :

(1)Let . Then, the image of under the fuzzy soft mapping is the fuzzy soft set over defined by , where

(2)Let . Then, the pre-image of under the fuzzy soft mapping is the fuzzy soft set over defined by , where

If both and are injective (surjective), then the fuzzy soft mapping is said to be injective (surjective).
The composition of two fuzzy soft mappings from to and from to is defined as from to .

Lemma 1. (see the work of Kharal and Ahmad [8]). Let , , and , , . The following is then obtained:

(1)If , then (2)If , then (3); the equality holds if is injective(4); the equality holds if is surjective

It is simple to verify the following lemma.

Lemma 2. Let be a fuzzy soft mapping, and ; then

(1)(2)If , then (3)If , then

Theorem 5. Let , , and : be a fuzzy soft mapping. If , then .

Proof. Suppose that . Then, there are an and such that . Let . Then, and . From Lemma 2, it can be found that , . Thus,

Therefore,

Equivalently,

which implies that .

Definition 14. (see the work by Varol and Aygün [11]). Let and be two fuzzy soft topological spaces. A fuzzy soft mapping : is said to be fuzzy soft continuous if , .
If both : and : are fuzzy soft continuous, then it is clear that is also fuzzy soft continuous. In fact, for a fuzzy soft set on ,

Hence,

Theorem 6. Let and be two fuzzy soft topological spaces and : be a fuzzy soft mapping. The following are then equivalent:

(1): is fuzzy soft continuous(2)For each and each neighborhood of , there exists a neighborhood of such that (3)For each and each Q-neighborhood of , there exists a Q-neighborhood of such that (4)For each net in , if converges to , then is a net in and converges to

Proof.

 : Suppose that and is a neighborhood of . Then, there exists an such that . Because is fuzzy soft continuous, . Then, is a neighborhood of and follows from Lemma 1(4). : Suppose that and is a Q-neighborhood of . Then, there is an open fuzzy soft set such that . From Remark 5 and Lemma 2(1), it can be determined that there exists such that . Thus, is a neighborhood of . Under condition (2), there exists an open neighborhood of such that . Because , . Therefore, is a Q-neighborhood of , and . : Suppose a net in converges to . For any , under condition (3), there exists an such that . From the supposition, there is a such that whenever . Lemma 2 implies that . Therefore, converges to . : This is proven by contradiction. If is not fuzzy soft continuous, then there is an such that is not open. From Remark 3(3), it can be found that there exists a such that , . By Definition 4(2), there is an for any . It follows from Theorem 2(1) that there exists a such that and . Recall from Remark 8 that is a directed set with the partial order “”; thus, a net in is obtained. It is easy to verify that converges to . Under condition (4), the net converges to . Recall that , or ; it is known that is a Q-neighborhood of . Thus, is eventually quasi-coincident with , which conflicts with the fact that , or , for any . Therefore, is fuzzy soft continuous.

5. Convergence for a Net of Fuzzy Soft Mappings

Let and be two fuzzy soft topological spaces. denotes the set of all mappings of into .

Definition 15. A net in is said to be convergent to if, for every point and for every , there exist a and such that for every with .

Theorem 7. A net in converges to if and only if, for every net in which converges to a point , the net converges to the point , where is the product of and .

Proof. (Necessity) Let be a net in that converges to and let arbitrarily. By assumption, there exist an and such that for any with . Because converges to , there is a such that for every with . Then, for every with , it follows from Lemma 2(2) that , which means converges to .
(Sufficiency) Proceeding by contradiction, it is supposed that there are a , such that, for every and for every , there exists a such that . From Theorem 2(1) and Lemma 2(1), a point is obtained such that and . As can be determined from Remark 8, the net converges to . However, the net does not converge to . This conflicts with the condition that converges to .

6. 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. The concept of fuzzy soft sets partially resolves this difficulty.

Suppose that the initial universe is the set of all the disease objects that the patient may be infected with, and the set of parameters is all of the patient’s symptoms. Generally speaking, from a symptom , one cannot completely determine the corresponding disease ; however, one can determine the membership degree in which object holds parameter , which is denoted by ; that is, for every , there is a fuzzy subset of . Obviously, the mapping is a fuzzy soft set over .

Let

If , then it may be claimed that the patient has disease .

7. 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. The 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

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11971343) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX19_2015).

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