Research Article | Open Access

# Some Properties of Fuzzy Soft Proximity Spaces

**Academic Editor:**Feng Feng

#### Abstract

We study the fuzzy soft proximity spaces in Katsaras’s sense. First, we show how a fuzzy soft topology is derived from a fuzzy soft proximity. Also, we define the notion of fuzzy soft *δ*-neighborhood in the fuzzy soft proximity space which offers an alternative approach to the study of fuzzy soft proximity spaces. Later, we obtain the initial fuzzy soft proximity determined by a family of fuzzy soft proximities. Finally, we investigate relationship between fuzzy soft proximities and proximities.

#### 1. Introduction

In 1999, Molodtsov [1] initiated the concept of soft set theory as a new approach for coping with uncertainties and also presented the basic results of the new theory. This new theory does not require the specification of a parameter. We can utilize any parametrization with the aid of words, sentences, real numbers, and so on. This implies that the problem of setting the membership function does not arise. Hence, soft set theory has compelling applications in several diverse fields; most of these applications were shown by Molodtsov [1]. Nowadays, there are a lot of works related to soft set theory and its applications [2–9].

Fuzzy soft set which is a combination of fuzzy set and soft set was introduced by Maji et al. [10]. Roy and Maji [11] gave some results on an application of fuzzy soft sets in decision making problem. Then, Tanay and Kandemir [12] initiated the notion of fuzzy soft topology and gave some fundamental properties of it by following Chang [13]. Also, the fuzzy soft topology in Lowen’s sense [14] was given by Varol and Aygün [15]. Fuzzy soft sets and their applications have been investigated intensively in recent years [16–24].

Proximity structure was introduced by Efremovic in 1951 [25, 26]. It can be considered either as axiomatizations of geometric notions or as suitable tools for an investigation of topology. Moreover, this structure has a very significant role in many problems of topological spaces such as compactification and extension problems. The most comprehensive work on the theory of proximity spaces was done by Naimpally and Warrack [27]. Then, many authors have obtained the concept of fuzzy proximity structure in different approaches. By using fuzzy sets, Katsaras [28] defined fuzzy proximity and studied the relation with fuzzy topology in Chang’s sense. Later, Artico and Moresco [29, 30] proposed a new definition of fuzzy proximity on a set as a map satisfying certain conditions, where is a completely distributive lattice with an order reversing involution. In connection with a fuzzy topology in [31], the different notion of a fuzzy proximity was introduced by Markin and Sostak [32]. In 2005, Ramadan et al. [33] presented fuzzifying proximity structures.

Extensions of proximity structures to the soft sets and also fuzzy soft sets have been studied by some authors. More recently, Hazra et al. [34] defined soft proximity spaces and studied some of their properties. By using fuzzy soft sets, Çetkin et al. [35] introduced soft fuzzy proximity spaces on the base of the axioms suggested by Markin and Sostak [32] and Katsaras [28], respectively. All these works have generalized versions of many of the well-known results on proximity spaces.

Motivated by their works, we continue investigating the properties of fuzzy soft proximity spaces in Katsaras’s sense. We show that each fuzzy soft proximity on induces a fuzzy soft topology on the same set. Also, we define the notion of fuzzy soft neighborhood in a fuzzy soft proximity space and obtain a few results analogous to the ones that hold for -neighborhood in proximity spaces. We prove the existences of initial fuzzy soft proximity structures. Based on this fact, we introduce products of fuzzy soft proximity spaces. The relation between a fuzzy soft proximity and a proximity is also investigated.

#### 2. Preliminaries

In this section, we recall some basic notions regarding fuzzy soft sets which will be used in the sequel. Throughout this work, let be an initial universe, be the set of all fuzzy subsets of and be the set of all parameters for .

*Definition 1 (see [10]). *A fuzzy soft set on the universe with the set of parameters is defined by the set of ordered pairs
where is a mapping given by .

Throughout this paper, the family of all fuzzy soft sets over is denoted by .

*Definition 2 (see [10, 15, 16]). *Let . Then, we have the following.(i)The fuzzy soft set is called null fuzzy soft set, denoted by , if for every .(ii)If , for all , then is called absolute fuzzy soft set, denoted by .(iii) is a fuzzy soft subset of if for each . It is denoted by .(iv) and are equal if and . It is denoted by .(v)The complement of is denoted by , where is a mapping defined by for all . Clearly, .(vi)The union of and is a fuzzy soft set defined by for all . is denoted by .(vii)The intersection of and is a fuzzy soft set defined by for all . is denoted by .

*Definition 3 (see [16]). *Let be an arbitrary index set and let be a family of fuzzy soft sets over . Then,(i)the union of these fuzzy soft sets is the fuzzy soft set defined by for every and this fuzzy soft set is denoted by ;(ii)the intersection of these fuzzy soft sets is the fuzzy soft set defined by for every and this fuzzy soft set is denoted by .

Theorem 4 (see [15]). *Let be an index set and , , , , for all . Then, the following statements are satisfied:*(1)*;*(2)*;*(3)*;*(4)*;*(5)*if , then .*

*Definition 5 (see [18]). *A fuzzy soft set over is said to be a fuzzy soft point if there is an such that is a fuzzy point in (i.e., there exists an such that and for all ) and for every . It will be denoted by .

The fuzzy soft point is said to belong to a fuzzy soft set , denoted by , if .

*Definition 6 (see [20]). *Let and be the families of all fuzzy soft sets over and , respectively. Let and be two mappings. Then, the mapping is called a fuzzy soft mapping from to , denoted by .(1)Let . Then is the fuzzy soft set over defined as follows:
for all and all . is called an image of a fuzzy soft set .(2)Let . Then is the soft set over defined as follows:
for all and all . is called a preimage of a fuzzy soft set .The fuzzy soft mapping is called injective, if and are injective. The fuzzy soft mapping is called surjective, if and are surjective.

Theorem 7 (see [20]). *Let and for all , where is an index set. Then, for a fuzzy soft mapping , the following conditions are satisfied:*(1)*if , then ;*(2)*if , then ;*(3)*;*(4)*;
*(5)*;*(6)*;*(7)*;*(8)*.*

Theorem 8 (see [15, 18]). *Let for all , where is an index set, and let . Then, for a fuzzy soft mapping , the following conditions are satisfied:*(1)*, and the equality holds if is injective;*(2)*, and the equality holds if is surjective;*(3)* if is injective;*(4)* if is surjective.*

*Definition 9 (see [15]). *Let and . The fuzzy soft product is defined by the fuzzy soft set where and for all .

*Definition 10 (see [15]). *Let and and let , and , be the projection mappings in classical meaning. The fuzzy soft mappings and are called fuzzy soft projection mappings from to and from to , respectively, where and .

Theorem 11. *Every parameterized collection of fuzzy subsets in is a fuzzy soft set. Also, every fuzzy soft set is a parameterized collection of fuzzy subsets in some universe.*

*Proof. *Consider any parameterized collection of fuzzy subsets in . Then, defined by is a fuzzy soft set over .

*Definition 12 (see [12]). *Let be the collection of fuzzy soft sets over ; then is said to be a fuzzy soft topology on if belong to ,the union of any number of fuzzy soft sets in belongs to ,the intersection of any two fuzzy soft sets in belongs to .

is called a fuzzy soft topological space. The members of are called fuzzy soft open sets in . A fuzzy soft set over is called a fuzzy soft closed in if .

*Example 13. *Let and . Let us consider the following fuzzy soft sets on with the set of parameters:
Then, is a fuzzy soft topology on .

*Definition 14 (see [12]). *Let be a fuzzy soft topological space and . The fuzzy soft interior of is the fuzzy soft set .

By property for fuzzy soft open sets, is fuzzy soft open. It is the largest fuzzy soft open set contained in .

*Definition 15 (see [15, 17]). *Let be a fuzzy soft topological space and . The fuzzy soft closure of is the fuzzy soft set .

Clearly is the smallest fuzzy soft closed set over which contains .

Theorem 16 (see [15]). *Let us consider an operator associating with each fuzzy soft set on another fuzzy soft set such that the following properties hold:**,**,**,**.**
Then, the family
**
defines a fuzzy soft topology on and, for every , the fuzzy soft set is the fuzzy soft closure of in the fuzzy soft topological space .**This operator is called the fuzzy soft closure operator.*

*Definition 17 (see [15, 18]). *Let and be two fuzzy soft topological spaces. A fuzzy soft mapping is called fuzzy soft continuous if for every .

Theorem 18 (see [15]). *Let be a fuzzy soft topological space, where . Then, the collection for every defines a fuzzy topology on .*

Theorem 19. *Every parameterized collection of fuzzy topological spaces on determines a fuzzy soft topological space over .*

*Proof. *Let be a parameterized family of fuzzy topological spaces. Let us define a fuzzy soft topological space as the following: let be the collection of all the mappings , where such that for each . Then, is a fuzzy soft topology on . Indeed,, because and for each ; similarly, , because and for each ;let be a collection of members in ; then, for all , we have for each ; therefore, is a mapping such that for each ; consequently, ;let ; then, for each and hence is a mapping such that for each ; thus, .

Recall that a binary relation on the power set of a set is called a proximity on if the following axioms are satisfied (see, [27]):;if , then ;if , then ; if and only if or ;if , then there exists a subset of such that and ,

where means negation of .

The pair is called a proximity space; two subsets and of the set are close with respect to if ; otherwise, they are remote with respect to .

*Definition 20 (see [28]). *A binary relation on is called a fuzzy proximity if satisfies the following conditions:;if , then ;if , then ; if and only if or ;if , then there exists a such that and .

A fuzzy proximity space is a pair comprising a set and a fuzzy proximity on the set .

#### 3. Fuzzy Soft Proximities

In this section, we study some elementary properties of fuzzy soft proximity structures in Katsaras’s sense. We induce a fuzzy soft topology from a given fuzzy soft proximity by using the fuzzy soft closure operator. Also, we present an alternative description of the concept of fuzzy soft proximity, which is called fuzzy soft -neighborhood.

*Definition 21 (see [35]). *A mapping is called a Katsaras -soft fuzzy proximity on a set , where and are arbitrary nonempty sets viewed on the sets of parameters, if, for any and , the following conditions are satisfied:;if , then ;if , then ; if and only if or ;if , then there exists an such that and .

The pair is called a Katsaras -soft fuzzy proximity space, where, for every , is a relation on .

The following definition coincides with Definition 21 when the parameter set is a singleton set.

*Definition 22. *A binary relation is called a fuzzy soft proximity on if satisfies the following conditions:;if , then ;if , then ; if and only if or ;if , then there exists an such that and .

The pair is called a fuzzy soft proximity space.

*Example 23. *On any set , let us define if and only if and . This defines a fuzzy soft proximity on .

We have easily the following Lemma.

Lemma 24. *If is a fuzzy soft proximity space, then it satisfies the following properties:*(i)*if and , , then ;*(ii)* for each ;*(iii)* if and only if .*

Let be a fuzzy soft proximity space. For every , we define Then we get the following theorem.

Theorem 25. *Let be a fuzzy soft proximity space. Then, the mapping satisfies the conditions . Therefore, the collection
**
is a fuzzy soft topology on .*

*Proof. *We will show that the mapping has the properties .

Suppose that . Let and . Take any such that . Then, by , . Hence, either or . In both cases, we obtain . Therefore, . Thus, we get

It is enough to show that if and only if . Necessity follows immediately from Lemma 24. For sufficiency, let . Suppose that . Then, by , there is an such that and . Because and , there exist an and an such that . Now, we will choose number , where and let us define . Since , we have . Also, , since, otherwise, we would have which is impossible. By and , . This is a contradiction to the fact that .

It is easy to verify that . Conversely, suppose that there exist an and an such that . Take an satisfying
We may assume (the case is analogous). Then, since , there exists an such that and . By the inequality , we have . Because , there exists a such that and . Now, since and , we obtain . From , it follows that . Also, we get . Thus,
which yields a contradiction.

Because of , we get

Trivially, the fuzzy soft proximity space defined in Example 23 induces the fuzzy soft topological space .

*Definition 26. *Let be a fuzzy soft proximity space. For , the fuzzy soft set is said to be a fuzzy soft -neighborhood of if ; we write this in symbols as .

Theorem 27. *Let be a fuzzy soft proximity space. Then the relation satisfies the following properties:**;** implies ;** implies ;** if and only if and ;** implies ;** implies there is an such that .*

*Proof. * is obvious.

If , then . By , ; that is, .

Let . Then, from , it follows that .

Consider and and .

If , where means negation of , then . Since and , we have . Therefore, , which is a contradiction.

Consider that implies . Then, by , there exists an such that and . Hence, .

Theorem 28. *Let be a fuzzy soft proximity space and . Then, the following statements are satisfied:*(i)* if and only if ;*(ii)*if , then there is a such that ;*(iii)*if , then there exist fuzzy soft sets such that , , and .*

*Proof. *(i) It is clear by the fact that if and only if (see the proof of Theorem 25).

(ii) Let . Then, and this implies that
Set . It is easy to verify that
Thus, we obtain and .

(iii) If , then, from , there is a fuzzy soft set such that and . Because , there is a fuzzy soft set such that and . Thus, there exist fuzzy soft sets and such that , , and .

Theorem 29. *Let be a relation on satisfying . Then, is a fuzzy soft proximity on defined as follows:
**
Also, according to this fuzzy soft proximity, is a fuzzy soft -neighbourhood of if and only if .*

*Proof. *We first need to verify axioms .Let . By , we have and thus .Let . Then, and from it follows that
If , then . By , and hence .Consider and and .Let . Then . Therefore, by , there is a fuzzy soft set such that . Thus, and .

Hence, is a fuzzy soft proximity on . From the definitions of the terms involved, it follows easily that is a fuzzy soft -neighbourhood of if and only if .

Theorem 30. *If is a fuzzy soft proximity space and , then
*

*Proof. *Let us take a fuzzy soft set such that . Therefore, and by we obtain . Hence,
On the other hand, suppose that there are an and an such that . Let . Then there exists an such that
Therefore, there exists a fuzzy soft set such that and . Because , we have . Hence, . Thus,
which leads to a contradiction.

*Definition 31 (see [35]). *Let and be two fuzzy soft proximity spaces. A fuzzy soft mapping is a fuzzy soft proximity mapping if it satisfies
for every .

Using the above definition, we can easily prove the following propositions.

Proposition 32. *Let and be two fuzzy soft proximity spaces. A fuzzy soft mapping is a fuzzy soft proximity mapping if and only if
**
or in another form
**
for every .*

Proposition 33. *The composition of two fuzzy soft proximity mappings is a fuzzy soft proximity mapping.*

Theorem 34. *A fuzzy soft proximity mapping is fuzzy soft continuous with respect to and .*

*Proof. *Let . Now let us take any such that . Since is a fuzzy soft proximity mapping, we obtain . From , it follows that . Then, for every and every , we have
Therefore,
Thus, since , we obtain .

*Definition 35. *If and are two fuzzy soft proximities on , we define
The above is expressed by saying that is finer than , or is coarser than .

#### 4. Initial Fuzzy Soft Proximities

We prove the existences of initial fuzzy soft proximity structure. Based on this fact, we define the product of fuzzy soft proximity spaces.

*Definition 36. *Let be a set and a family of fuzzy soft proximity spaces, and, for each , let be a fuzzy soft mapping. The initial structure is the coarsest fuzzy soft proximity on for which all mappings are fuzzy soft proximity mapping.

Theorem 37 (existence of initial structures). *Let be a set a family of fuzzy soft proximity spaces, and, for each , let be a fuzzy soft mapping. For any , define if and only if, for every finite families and , where and , there exist an and a such that
**
Then is the coarsest fuzzy soft proximity on for which all mappings are fuzzy soft proximity mapping.*

*Proof. *We first prove that is a fuzzy soft proximity on .is obvious.We will show that if , then . Let . Then, there exist finite covers and of and , respectively, such that for some , where and . Since each is a fuzzy soft proximity, . From this, it follows that
Thus, we have .Since each is a fuzzy soft proximity, it is clear that implies .It is easy to verify that if , then for each . Therefore, or implies . Conversely, assume that and . Then, there exist finite covers and of and , respectively, such that for some , where and . In the same way, there are finite covers and of and , respectively, such that for some , where and . Now, and are finite covers of and , respectively. Hence, from the fact that for or , it follows that .Let us define the set of all pairs such that and we have either or for each . The validity of will follow from the fact that is empty. Suppose, on the contrary, that . Then, for each . Indeed, let and . If , then . Because , we have . Similarly, if , then . Hence, since is a fuzzy soft proximity on , we obtain . Also, we observe that for each there are positive integers and covers and such that, for every pair , there exists an satisfying . Let . It easy to see that . Then, for each , let us choose such an integer . But is not uniquely determined by . Let be the set of all integers corresponding to members of and let be the smallest member of . Take a such that is the integer corresponding to it. Then, there are covers and such that and for every pair and there exists an satisfying . One of the , is greater than . Consider and let . In this case, one of the following conditions should be true:(i)for every , either or ;(ii)for every , either or .

In fact, suppose that neither (i) nor (ii) holds. Then, there are such that and . Letting , we obtain and , contradicting the fact that .

Suppose that (i) holds. Because and , this means that . Hence, by (i), we have . But this is now a contradiction since , contrary to the choice of . If (ii) holds, we get a contradiction in a similar way. Therefore, the set is empty. Thus, is a fuzzy soft proximity on .

It is easy to see that all mappings are fuzzy soft proximity mapping. Let be another fuzzy soft proximity on making each of the mappings fuzzy soft proximity mapping. We will show that , which will complete the proof. Let and consider any covers and of and , respectively. Since , by , there is an such that . In the same way, since , by , there is a such that . From the fact that all mappings are fuzzy soft proximity mapping, it follows that for each . Thus, we get .

Theorem 38. *A fuzzy soft mapping is a fuzzy soft proximity mapping if and only if is a fuzzy soft proximity mapping for every .*

*Proof. *The necessity is easy. We prove the sufficiency. Suppose that is a fuzzy soft proximity mapping for every . Let and let and . Then, we have
Since , by , there exist such that . Because
it follows from our hypothesis that for every . This shows that .

*Definition 39. *Let be a family of fuzzy soft proximity spaces and let and be product sets. An initial fuzzy soft proximity structure on with respect to all the fuzzy soft projection mappings , where and , is called the product fuzzy soft proximity structure.

is said to be a product fuzzy soft proximity space.

From Theorems 37 and 38, we obtain the following corollary.

Corollary 40. *Consider be a family of fuzzy soft proximity spaces. Let and be sets and for each let be a fuzzy soft mapping. For any , define if and only if, for every finite families and , where and , there exist an and a such that for each . Then,*(i)* is the coarsest fuzzy soft proximity on for which all mappings are fuzzy soft proximity mapping;*(ii)*a fuzzy soft mapping is a fuzzy soft proximity mapping if and only if is a fuzzy soft proximity mapping for every .*

#### 5. Fuzzy Soft Proximities Induced by Proximities

Our task in this section is to study the connection between fuzzy soft proximity spaces and proximity spaces.

*Definition 41 (see [17]). *Let be a set and let be a subset of . Then, a mapping is a fuzzy soft set on defined as the following:
where is a characteristic function of .

*Example 42. *Let , , and . Then,
is a fuzzy soft set on .

Lemma 43. *For any subsets of ,*(i)*;*(ii)*;*(iii)*.*

*Proof. *Straightforward.

Theorem 44. *Let be a proximity space. By letting, for ,** if and only if there exist subsets of such that , , and , one defines a fuzzy soft proximity on .*

*Proof. *We will show that satisfies axioms .From , , and , it follows that .Let . Then, there are subsets and of such that , , and . By , we have , so that . Thus, we get .It is clear because implies .It is easy to see that if , then and . Conversely, suppose that