Some Properties of Fuzzy Soft Proximity Spaces

Hindawi Publishing Corporation e Scientific World Journal Volume 2015, Article ID 752634, 10 pages http://dx.doi.org/10.1155/2015/752634

Research Article Some Properties of Fuzzy Soft Proximity Spaces Ezzettin Demir1 and Oya Bedre ÖzbakJr2 1

Department of Mathematics, Duzce University, 81620 Duzce, Turkey Department of Mathematics, Ege University, 35100 Izmir, Turkey

2

¨ Correspondence should be addressed to Oya Bedre Ozbakır; [email protected] Received 13 May 2014; Revised 4 August 2014; Accepted 12 August 2014 Academic Editor: Feng Feng ¨ Copyright © 2015 ˙I. Demir and O. B. Ozbakır. 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. 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 𝑋 as a map 𝛿 : 𝐿𝑋 × 𝐿𝑋 → {0, 1} satisfying certain conditions, where 𝐿 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, C ¸ etkin 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 𝛿 on 𝑋 induces a fuzzy

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soft topology 𝜏(𝛿) on the same set. Also, we define the notion of fuzzy soft 𝛿-neighborhood in a fuzzy soft proximity space and obtain a few results analogous to the ones that hold for 𝛿neighborhood 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.

(1) 𝑓 ⊓ (⊔𝑖∈𝐽 𝑔𝑖 ) = ⊔𝑖∈𝐽 (𝑓 ⊓ 𝑔𝑖 ); (2) 𝑓 ⊔ (⊓𝑖∈𝐽 𝑔𝑖 ) = ⊓𝑖∈𝐽 (𝑓 ⊔ 𝑔𝑖 ); (3) (⊓𝑖∈𝐽 𝑓𝑖 )𝑐 = ⊔𝑖∈𝐽 𝑓𝑖𝑐 ;

(4) (⊔𝑖∈𝐽 𝑓𝑖 )𝑐 = ⊓𝑖∈𝐽 𝑓𝑖𝑐 ;

(5) if 𝑓 ⊑ 𝑔, then 𝑔𝑐 ⊑ 𝑓𝑐 .

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 𝑋 be an initial universe, 𝐼𝑋 be the set of all fuzzy subsets of 𝑋 and 𝐸 be the set of all parameters for 𝑋. Definition 1 (see [10]). A fuzzy soft set 𝑓 on the universe 𝑋 with the set 𝐸 of parameters is defined by the set of ordered pairs 𝑓 = {(𝑒, 𝑓 (𝑒)) : 𝑒 ∈ 𝐸, 𝑓 (𝑒) ∈ 𝐼X } ,

Theorem 4 (see [15]). Let 𝐽 be an index set and 𝑓, 𝑔, 𝑓𝑖 , 𝑔𝑖 ∈ 𝐹𝑆(𝑋, 𝐸), for all 𝑖 ∈ 𝐽. Then, the following statements are satisfied:

(1)

where 𝑓 is a mapping given by 𝑓 : 𝐸 → 𝐼𝑋 . Throughout this paper, the family of all fuzzy soft sets over 𝑋 is denoted by 𝐹𝑆(𝑋, 𝐸). Definition 2 (see [10, 15, 16]). Let 𝑓, 𝑔 ∈ 𝐹𝑆(𝑋, 𝐸). Then, we have the following. (i) The fuzzy soft set 𝑓 is called null fuzzy soft set, ̃ if 𝑓(𝑒) = 0𝑋 for every 𝑒 ∈ 𝐸. denoted by 0, (ii) If 𝑓(𝑒) = 1𝑋 , for all 𝑒 ∈ 𝐸, then 𝑓 is called absolute ̃ fuzzy soft set, denoted by 𝑋. (iii) 𝑓 is a fuzzy soft subset of 𝑔 if 𝑓(𝑒) ≤ 𝑔(𝑒) for each 𝑒 ∈ 𝐸. It is denoted by 𝑓 ⊑ 𝑔. (iv) 𝑓 and 𝑔 are equal if 𝑓 ⊑ 𝑔 and 𝑔 ⊑ 𝑓. It is denoted by 𝑓 = 𝑔. (v) The complement of 𝑓 is denoted by 𝑓𝑐 , where 𝑓𝑐 : 𝐸 → 𝐼𝑋 is a mapping defined by 𝑓𝑐 (𝑒) = 1𝑋 − 𝑓(𝑒) for all 𝑒 ∈ 𝐸. Clearly, (𝑓𝑐 )𝑐 = 𝑓. (vi) The union of 𝑓 and 𝑔 is a fuzzy soft set ℎ defined by ℎ(𝑒) = 𝑓(𝑒) ∨ 𝑔(𝑒) for all 𝑒 ∈ 𝐸. ℎ is denoted by 𝑓 ⊔ 𝑔. (vii) The intersection of 𝑓 and 𝑔 is a fuzzy soft set ℎ defined by ℎ(𝑒) = 𝑓(𝑒) ∧ 𝑔(𝑒) for all 𝑒 ∈ 𝐸. ℎ is denoted by 𝑓 ⊓ 𝑔. Definition 3 (see [16]). Let 𝐽 be an arbitrary index set and let {𝑓𝑖 }𝑖∈𝐽 be a family of fuzzy soft sets over 𝑋. Then, (i) the union of these fuzzy soft sets is the fuzzy soft set ℎ defined by ℎ(𝑒) = ∨𝑖∈𝐽 𝑓𝑖 (𝑒) for every 𝑒 ∈ 𝐸 and this fuzzy soft set is denoted by ⊔𝑖∈𝐽 𝑓𝑖 ; (ii) the intersection of these fuzzy soft sets is the fuzzy soft set ℎ defined by ℎ(𝑒) = ∧𝑖∈𝐽 𝑓𝑖 (𝑒) for every 𝑒 ∈ 𝐸 and this fuzzy soft set is denoted by ⊓𝑖∈𝐽 𝑓𝑖 .

Definition 5 (see [18]). A fuzzy soft set 𝑓 over 𝑋 is said to be a fuzzy soft point if there is an 𝑒 ∈ 𝐸 such that 𝑓(𝑒) is a fuzzy point in 𝑋 (i.e., there exists an 𝑥 ∈ 𝑋 such that 𝑓(𝑒)(𝑥) = 𝛼 ∈ (0, 1] and 𝑓(𝑒)(𝑥󸀠 ) = 0 for all 𝑥󸀠 ∈ 𝑋 − {𝑥}) and 𝑓(𝑒󸀠 ) = 0𝑋 for every 𝑒󸀠 ∈ 𝐸 \ {𝑒}. It will be denoted by 𝑒𝑥𝛼 . The fuzzy soft point 𝑒𝑥𝛼 is said to belong to a fuzzy soft set 𝑓, denoted by 𝑒𝑥𝛼 ∈̃𝑓, if 𝛼 ≤ 𝑓(𝑒)(𝑥). Definition 6 (see [20]). Let 𝐹𝑆(𝑋, 𝐸) and 𝐹𝑆(𝑌, 𝐾) be the families of all fuzzy soft sets over 𝑋 and 𝑌, respectively. Let 𝜑 : 𝑋 → 𝑌 and 𝜓 : 𝐸 → 𝐾 be two mappings. Then, the mapping 𝜑𝜓 is called a fuzzy soft mapping from 𝑋 to 𝑌, denoted by 𝜑𝜓 : 𝐹𝑆(𝑋, 𝐸) → 𝐹𝑆(𝑌, 𝐾). (1) Let 𝑓 ∈ 𝐹𝑆(𝑋, 𝐸). Then 𝜑𝜓 (𝑓) is the fuzzy soft set over 𝑌 defined as follows: 𝜑𝜓 (𝑓) (𝑘) (𝑦) { { { ⋁ ( ⋁ 𝑓 (𝑒)) (𝑥) , if 𝜓−1 (𝑘) ≠ 0, 𝜑−1 (𝑦) ≠ 0; = {𝑥∈𝜑−1 (𝑦) 𝑒∈𝜓−1 (𝑘) { { otherwise, {0, (2) for all 𝑘 ∈ 𝐾 and all 𝑦 ∈ 𝑌. 𝜑𝜓 (𝑓) is called an image of a fuzzy soft set 𝑓. (2) Let 𝑔 ∈ 𝐹𝑆(𝑌, 𝐾). Then 𝜑𝜓−1 (𝑔) is the soft set over 𝑋 defined as follows: 𝜑𝜓−1 (𝑔) (𝑒) (𝑥) = 𝑔 (𝜓 (𝑒)) (𝜑 (𝑥))

(3)

for all 𝑒 ∈ 𝐸 and all 𝑥 ∈ 𝑋. 𝜑𝜓−1 (𝑔) is called a preimage of a fuzzy soft set 𝑔. The fuzzy soft mapping 𝜑𝜓 is called injective, if 𝜑 and 𝜓 are injective. The fuzzy soft mapping 𝜑𝜓 is called surjective, if 𝜑 and 𝜓 are surjective. Theorem 7 (see [20]). Let 𝑓𝑖 ∈ 𝐹𝑆(𝑋, 𝐸) and 𝑔𝑖 ∈ 𝐹𝑆(𝑌, 𝐾) for all 𝑖 ∈ 𝐽, where 𝐽 is an index set. Then, for a fuzzy soft mapping 𝜑𝜓 : 𝐹𝑆(𝑋, 𝐸) → 𝐹𝑆(𝑌, 𝐾), the following conditions are satisfied: (1) if 𝑓1 ⊑ 𝑓2 , then 𝜑𝜓 (𝑓1 ) ⊑ 𝜑𝜓 (𝑓2 ); (2) if 𝑔1 ⊑ 𝑔2 , then 𝜑𝜓−1 (𝑔1 ) ⊑ 𝜑𝜓−1 (𝑔2 );

The Scientific World Journal (3) 𝜑𝜓 (⊔𝑖∈𝐽 𝑓𝑖 ) = ⊔𝑖∈𝐽 𝜑𝜓 (𝑓𝑖 ); (4) 𝜑𝜓 (⊓𝑖∈𝐽 𝑓𝑖 ) ⊑ ⊓𝑖∈𝐽 𝜑𝜓 (𝑓𝑖 );

3 Example 13. Let 𝑋 = 𝐼 and 𝐸 = (0, 1). Let us consider the following fuzzy soft sets on 𝑋 with the set 𝐸 of parameters:

(5) 𝜑𝜓−1 (⊔𝑖∈𝐽 𝑔𝑖 ) = ⊔𝑖∈𝐽 𝜑𝜓−1 (𝑔𝑖 );

𝑓 (𝑒) (𝑥) = {

(6) 𝜑𝜓−1 (⊓𝑖∈𝐽 𝑔𝑖 ) = ⊓𝑖∈𝐽 𝜑𝜓−1 (𝑔𝑖 ); (7)

̃ 𝜑𝜓−1 (𝑌)

=

̃ 𝜑−1 (0) ̃ 𝑋, 𝜓

∀𝑒 ∈ 𝐸,

̃ = 0;

𝜑𝜓−1 (𝜑𝜓 (𝑓)),

(1) 𝑓 ⊑ injective;

and the equality holds if 𝜑𝜓 is

(2) 𝜑𝜓 (𝜑𝜓−1 (𝑔)) ⊑ 𝑔, and the equality holds if 𝜑𝜓 is surjective; (3) 𝜑𝜓 (⊓𝑖∈𝐽 𝑓𝑖 ) = ⊓𝑖∈𝐽 𝜑𝜓 (𝑓𝑖 ) if 𝜑𝜓 is injective; ̃ =𝑌 ̃ if 𝜑𝜓 is surjective. (4) 𝜑𝜓 (𝑋) Definition 9 (see [15]). Let 𝑓 ∈ 𝐹𝑆(𝑋, 𝐸) and 𝑔 ∈ 𝐹𝑆(𝑌, 𝐾). The fuzzy soft product 𝑓 × 𝑔 is defined by the fuzzy soft set ℎ where ℎ : 𝐸 × 𝐾 → 𝐼𝑋×𝑌 and ℎ(𝑒, 𝑘) = 𝑓(𝑒) × 𝑔(𝑘) for all (𝑒, 𝑘) ∈ 𝐸 × 𝐾. Definition 10 (see [15]). Let 𝑓 ∈ 𝐹𝑆(𝑋, 𝐸) and 𝑔 ∈ 𝐹𝑆(𝑌, 𝐾) and let 𝑝𝑋 : 𝑋 × 𝑌 → 𝑋, 𝑞𝐸 : 𝐸 × 𝐾 → 𝐸 and 𝑝𝑌 : 𝑋 × 𝑌 → 𝑌, 𝑞𝐾 : 𝐸 × 𝐾 → 𝐾 be the projection mappings in classical meaning. The fuzzy soft mappings (𝑝𝑋 )𝑞𝐸 and (𝑝𝑌 )𝑞𝐾 are called fuzzy soft projection mappings from 𝑋×𝑌 to 𝑋 and from 𝑋 × 𝑌 to 𝑌, respectively, where (𝑝𝑋 )𝑞𝐸 (𝑓 × 𝑔) = 𝑓 and (𝑝𝑌 )𝑞𝐾 (𝑓 × 𝑔) = 𝑔. Theorem 11. Every parameterized collection of fuzzy subsets in 𝑋 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 {𝜇𝛼 : 𝛼 ∈ Δ} of fuzzy subsets in 𝑋. Then, 𝑓 : Δ → 𝐼𝑋 defined by 𝑓(𝛼) = 𝜇𝛼 is a fuzzy soft set over 𝑋. Definition 12 (see [12]). Let 𝜏 be the collection of fuzzy soft sets over 𝑋; then 𝜏 is said to be a fuzzy soft topology on 𝑋 if ̃𝑋 ̃ belong to 𝜏, (𝑓𝑠𝑡1 ) 0, (𝑓𝑠𝑡2 ) the union of any number of fuzzy soft sets in 𝜏 belongs to 𝜏,

if 0 ≤ 𝑥 ≤ 𝑒; if 𝑒 ≤ 𝑥 ≤ 1 all 𝑥 ∈ 𝑋,

𝑒 − 𝑥, if 0 ≤ 𝑥 ≤ 𝑒; 𝑔 (𝑒) (𝑥) = { 0, if 𝑒 ≤ 𝑥 ≤ 1

̃ = 0. ̃ (8) 𝜑𝜓 (0) Theorem 8 (see [15, 18]). Let 𝑓, 𝑓𝑖 ∈ 𝐹𝑆(𝑋, 𝐸) for all 𝑖 ∈ 𝐽, where 𝐽 is an index set, and let 𝑔 ∈ 𝐹𝑆(𝑌, 𝐾). Then, for a fuzzy soft mapping 𝜑𝜓 : 𝐹𝑆(𝑋, 𝐸) → 𝐹𝑆(𝑌, 𝐾), the following conditions are satisfied:

0, 𝑥 − 𝑒,

∀𝑒 ∈ 𝐸,

(4)

all 𝑥 ∈ 𝑋.

̃ 𝑋, ̃ 𝑓, 𝑔, 𝑓 ⊔ 𝑔} is a fuzzy soft topology on 𝑋. Then, 𝜏 = {0, Definition 14 (see [12]). Let (𝑋, 𝜏) be a fuzzy soft topological space and 𝑓 ∈ 𝐹𝑆(𝑋, 𝐸). The fuzzy soft interior of 𝑓 is the fuzzy soft set 𝑓𝑜 = ⊔{𝑔 : 𝑔 is a fuzzy soft open set and 𝑔 ⊑ 𝑓}. By property (𝑓𝑠𝑡2 ) for fuzzy soft open sets, 𝑓𝑜 is fuzzy soft open. It is the largest fuzzy soft open set contained in 𝑓. Definition 15 (see [15, 17]). Let (𝑋, 𝜏) be a fuzzy soft topological space and 𝑓 ∈ 𝐹𝑆(𝑋, 𝐸). The fuzzy soft closure of 𝑓 is the fuzzy soft set 𝑓 = ⊓{𝑔 : 𝑔 is a fuzzy soft closed set and 𝑓 ⊑ 𝑔}. Clearly 𝑓 is the smallest fuzzy soft closed set over 𝑋 which contains 𝑓. Theorem 16 (see [15]). Let us consider an operator associating with each fuzzy soft set 𝑓 on 𝑋 another fuzzy soft set 𝑓 such that the following properties hold: (𝑓𝑜1 ) 𝑓 ⊑ 𝑓, (𝑓𝑜2 ) 𝑓 = 𝑓, (𝑓𝑜3 ) 𝑓 ∨ 𝑔 = 𝑓 ∨ 𝑔, ̃ (𝑓𝑜4 ) 0̃ = 0. Then, the family 𝜏 = {𝑓 ∈ 𝐹𝑆 (𝑋, 𝐸) : 𝑓𝑐 = 𝑓𝑐 }

(5)

defines a fuzzy soft topology on 𝑋 and, for every 𝑓 ∈ 𝐹𝑆(𝑋, 𝐸), the fuzzy soft set 𝑓 is the fuzzy soft closure of 𝑓 in the fuzzy soft topological space (𝑋, 𝜏). This operator is called the fuzzy soft closure operator. Definition 17 (see [15, 18]). Let (𝑋, 𝜏1 ) and (𝑌, 𝜏2 ) be two fuzzy soft topological spaces. A fuzzy soft mapping 𝜑𝜓 : (𝑋, 𝜏1 ) → (𝑌, 𝜏2 ) is called fuzzy soft continuous if 𝜑𝜓−1 (𝑔) ∈ 𝜏1 for every 𝑔 ∈ 𝜏2 .

(𝑓𝑠𝑡3 ) the intersection of any two fuzzy soft sets in 𝜏 belongs to 𝜏.

Theorem 18 (see [15]). Let (𝑋, 𝜏) be a fuzzy soft topological space, where 𝜏 = {𝑓𝛼 : 𝛼 ∈ Δ}. Then, the collection 𝜏𝑒 = {𝑓𝛼 (𝑒) | 𝛼 ∈ Δ} for every 𝑒 ∈ 𝐸 defines a fuzzy topology on 𝑋.

(𝑋, 𝜏) is called a fuzzy soft topological space. The members of 𝜏 are called fuzzy soft open sets in 𝑋. A fuzzy soft set 𝑓 over 𝑋 is called a fuzzy soft closed in 𝑋 if 𝑓𝑐 ∈ 𝜏.

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

4


Proof. Let {(𝑋, 𝜏𝑒 ) : 𝑒 ∈ 𝐸} be a parameterized family of fuzzy topological spaces. Let us define a fuzzy soft topological space (𝑋, 𝜏) as the following: let 𝜏 be the collection of all the mappings 𝑓, where 𝑓 : 𝐸 → 𝐼𝑋 such that 𝑓(𝑒) ∈ 𝜏𝑒 for each 𝑒 ∈ 𝐸. Then, 𝜏 is a fuzzy soft topology on 𝑋. Indeed, ̃ = 0𝑋 ∈ 𝜏𝑒 for (𝑓𝑠𝑡1 ) 0̃ ∈ 𝜏, because 0̃ : 𝐸 → 𝐼𝑋 and 0(𝑒) ̃ ∈ 𝜏, because 𝑋 ̃ : 𝐸 → 𝐼𝑋 each 𝑒 ∈ 𝐸; similarly, 𝑋 ̃ = 1𝑋 ∈ 𝜏𝑒 for each 𝑒 ∈ 𝐸; and 𝑋(𝑒) (𝑓𝑠𝑡2 ) let {𝑓𝑖 | 𝑖 ∈ 𝐽} be a collection of members in 𝜏; then, for all 𝑖 ∈ 𝐽, we have 𝑓𝑖 (𝑒) ∈ 𝜏𝑒 for each 𝑒 ∈ 𝐸; therefore, ⊔𝑖∈𝐽 𝑓𝑖 is a mapping ⊔𝑖∈𝐽 𝑓𝑖 : 𝐸 → 𝐼𝑋 such that (⊔𝑖∈𝐽 𝑓𝑖 )(𝑒) = ∨𝑖∈𝐽 𝑓𝑖 (𝑒) ∈ 𝜏𝑒 for each 𝑒 ∈ 𝐸; consequently, ⊔𝑖∈𝐽 𝑓𝑖 ∈ 𝜏; (𝑓𝑠𝑡3 ) let 𝑓, 𝑔 ∈ 𝜏; then, 𝑓(𝑒), 𝑔(𝑒) ∈ 𝜏𝑒 for each 𝑒 ∈ 𝐸 and hence 𝑓 ⊓ 𝑔 is a mapping 𝑓 ⊓ 𝑔 : 𝐸 → 𝐼𝑋 such that (𝑓 ⊓ 𝑔)(𝑒) = 𝑓(𝑒) ∧ 𝑔(𝑒) ∈ 𝜏𝑒 for each 𝑒 ∈ 𝐸; thus, 𝑓 ⊓ 𝑔 ∈ 𝜏. Recall that a binary relation 𝛿 on the power set of a set 𝑋 is called a proximity on 𝑋 if the following axioms are satisfied (see, [27]): (𝑝1 ) 0𝛿𝐴; (𝑝2 ) if 𝐴 ∩ 𝐵 ≠ 0, then 𝐴𝛿𝐵; (𝑝3 ) if 𝐴𝛿𝐵, then 𝐵𝛿𝐴; (𝑝4 ) 𝐴𝛿(𝐵 ∪ 𝐶) if and only if 𝐴𝛿𝐵 or 𝐴𝛿𝐶; (𝑝5 ) if 𝐴𝛿𝐵, then there exists a subset 𝐶 of 𝑋 such that 𝐴𝛿𝐶 and 𝐵𝛿(𝑋 − 𝐶), where 𝛿 means negation of 𝛿. The pair (𝑋, 𝛿) is called a proximity space; two subsets 𝐴 and 𝐵 of the set 𝑋 are close with respect to 𝛿 if 𝐴𝛿𝐵; otherwise, they are remote with respect to 𝛿. Definition 20 (see [28]). A binary relation 𝛿 on 𝐼𝑋 is called a fuzzy proximity if 𝛿 satisfies the following conditions: (𝑓𝑝1 ) 0𝑋 𝛿𝜇; (𝑓𝑝2 ) if 𝜇 ∧ 𝜌 ≠ 0𝑋 , then 𝜇𝛿𝜌; (𝑓𝑝3 ) if 𝜇𝛿𝜌, then 𝜌𝛿𝜇; (𝑓𝑝4 ) 𝜇𝛿(𝜌 ∨ 𝜎) if and only if 𝜇𝛿𝜌 or 𝜇𝛿𝜎; (𝑓𝑝5 ) if 𝜇𝛿𝜌, then there exists a 𝜎 ∈ 𝐼𝑋 such that 𝜇𝛿𝜎 and 𝜌𝛿(1𝑋 − 𝜎). A fuzzy proximity space is a pair (𝑋, 𝛿) comprising a set 𝑋 and a fuzzy proximity 𝛿 on the set 𝑋.

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 𝛿-neighborhood.

Definition 21 (see [35]). A mapping 𝛿 : 𝐾 → 2𝐹𝑆(𝑋,𝐸)×𝐹𝑆(𝑋,𝐸) is called a Katsaras (𝐸, 𝐾)-soft fuzzy proximity on a set 𝑋, where 𝐸 and 𝐾 are arbitrary nonempty sets viewed on the sets of parameters, if, for any 𝑓, 𝑔, ℎ ∈ 𝐹𝑆(𝑋, 𝐸) and 𝑘 ∈ 𝐾, the following conditions are satisfied: ̃ 𝑘 𝑓; (𝑠𝑓𝑝1 ) 0𝛿 ̃ − 𝑔); (𝑠𝑓𝑝2 ) if 𝑓𝛿𝑘 𝑔, then 𝑓 ⊑ (𝑋 (𝑠𝑓𝑝3 ) if 𝑓𝛿𝑘 𝑔, then 𝑔𝛿𝑘 𝑓; (𝑠𝑓𝑝4 ) 𝑓𝛿𝑘 (𝑔 ⊔ ℎ) if and only if 𝑓𝛿𝑘 𝑔 or 𝑓𝛿𝑘 ℎ; (𝑠𝑓𝑝5 ) if 𝑓𝛿𝑘 𝑔, then there exists an ℎ ∈ 𝐹𝑆(𝑋, 𝐸) such that ̃ − ℎ). 𝑓𝛿𝑘 ℎ and 𝑔𝛿𝑘 (𝑋 The pair (𝑋, 𝛿) is called a Katsaras (𝐸, 𝐾)-soft fuzzy proximity space, where, for every 𝑘 ∈ 𝐾, 𝛿𝑘 ⊂ 𝐹𝑆(𝑋, 𝐸) × 𝐹𝑆(𝑋, 𝐸) is a relation on 𝐹𝑆(𝑋, 𝐸). The following definition coincides with Definition 21 when the parameter set 𝐾 is a singleton set. Definition 22. A binary relation 𝛿 ⊂ 𝐹𝑆(𝑋, 𝐸) × 𝐹𝑆(𝑋, 𝐸) is called a fuzzy soft proximity on 𝑋 if 𝛿 satisfies the following conditions: ̃ (𝑓𝑠𝑝1 ) 0𝛿𝑓; ̃ then 𝑓𝛿𝑔; (𝑓𝑠𝑝2 ) if 𝑓 ⊓ 𝑔 ≠ 0, (𝑓𝑠𝑝3 ) if 𝑓𝛿𝑔, then 𝑔𝛿𝑓; (𝑓𝑠𝑝4 ) 𝑓𝛿(𝑔 ⊔ ℎ) if and only if 𝑓𝛿𝑔 or 𝑓𝛿ℎ;

(𝑓𝑠𝑝5 ) if 𝑓𝛿𝑔, then there exists an ℎ ∈ 𝐹𝑆(𝑋, 𝐸) such that ̃ − ℎ). 𝑓𝛿ℎ and 𝑔𝛿(𝑋 The pair (𝑋, 𝛿) is called a fuzzy soft proximity space. Example 23. On any set 𝑋, let us define 𝑓𝛿𝑔 if and only if ̃ This defines a fuzzy soft proximity on 𝑋. 𝑓 ≠ 0̃ and 𝑔 ≠ 0. We have easily the following Lemma. Lemma 24. If (𝑋, 𝛿) is a fuzzy soft proximity space, then it satisfies the following properties: (i) if 𝑓𝛿𝑔 and 𝑘 ⊒ 𝑓, ℎ ⊒ 𝑔, then 𝑘𝛿ℎ; ̃ (ii) 𝑓𝛿𝑓 for each 𝑓 ≠ 0; ̃ if and only if 𝑓 ≠ 0. ̃ (iii) 𝑓𝛿𝑋 Let (𝑋, 𝛿) be a fuzzy soft proximity space. For every 𝑓 ∈ 𝐹𝑆(𝑋, 𝐸), we define ̃ − ⊔ {𝑔 ∈ 𝐹𝑆 (𝑋, 𝐸) : 𝑓𝛿𝑔} . 𝑓=𝑋

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Then we get the following theorem. Theorem 25. Let (𝑋, 𝛿) be a fuzzy soft proximity space. Then, the mapping 𝑓 → 𝑓 satisfies the conditions (𝑓𝑜1 )–(𝑓𝑜4 ). Therefore, the collection 𝜏 (𝛿) = {𝑓 ∈ 𝐹𝑆 (𝑋, 𝐸) : 𝑓𝑐 = 𝑓𝑐 } is a fuzzy soft topology on 𝑋.

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5

Proof. We will show that the mapping 𝑓 → 𝑓 has the properties (𝑓𝑜1 )–(𝑓𝑜4 ). ̃ Let 𝑒 ∈ 𝐸 and 𝑥 ∈ 𝑋. Take any (𝑓𝑜1 ) Suppose that 𝑓 ≠ 0. ̃ Hence, 𝑔 ∈ 𝐹𝑆(𝑋, 𝐸) such that 𝑔𝛿𝑓. Then, by (𝑓𝑠𝑝2 ), 𝑔⊓𝑓 = 0. either 𝑔(𝑒)(𝑥) = 0 or 𝑓(𝑒)(𝑥) = 0. In both cases, we obtain 𝑔(𝑒)(𝑥) ≤ 1 − 𝑓(𝑒)(𝑥). Therefore, ∨𝑔𝛿𝑓 𝑔(𝑒)(𝑥) ≤ 1 − 𝑓(𝑒)(𝑥). Thus, we get 𝑓 (𝑒) (𝑥) ≤ 1 − ⋁ 𝑔 (𝑒) (𝑥) = 𝑓 (𝑒) (𝑥) . 𝑔𝛿𝑓

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(𝑓𝑜2 ) It is enough to show that 𝑔𝛿𝑓 if and only if 𝑔𝛿𝑓. Necessity follows immediately from Lemma 24. For sufficiency, let 𝑔𝛿𝑓. Suppose that 𝑔𝛿𝑓. Then, by (𝑓𝑠𝑝5 ), there ̃ − ℎ). Because is an ℎ ∈ 𝐹𝑆(𝑋, 𝐸) such that 𝑔𝛿ℎ and 𝑓𝛿(𝑋 𝑔𝛿ℎ and 𝑔𝛿𝑓, there exist an 𝑒 ∈ 𝐸 and an 𝑥 ∈ 𝑋 such that ℎ(𝑒)(𝑥) < 𝑓(𝑒)(𝑥). Now, we will choose number 𝑎, where ℎ(𝑒)(𝑥) < 𝑎 < 𝑓(𝑒)(𝑥) and let us define 𝑒𝑥1−𝑎 ∈ 𝐹𝑆(𝑋, 𝐸). ̃ − ℎ. Also, 𝑒𝑥1−𝑎 𝛿𝑓, Since 1 − 𝑎 ≤ 1 − ℎ(𝑒)(𝑥), we have 𝑒𝑥1−𝑎 ⊑ 𝑋 since, otherwise, we would have 𝑓(𝑒)(𝑥) ≤ 1 − (1 − 𝑎) = 𝑎 ̃ − ℎ), (𝑋 ̃ − ℎ)𝛿𝑓. which is impossible. By 𝑒𝑥1−𝑎 𝛿𝑓 and 𝑒𝑥1−𝑎 ⊑ (𝑋 ̃ − ℎ). This is a contradiction to the fact that 𝑓𝛿(𝑋 (𝑓𝑜3 ) It is easy to verify that 𝑓 ⊔ 𝑔 ⊒ 𝑓 ⊔ 𝑔. Conversely, suppose that there exist an 𝑒 ∈ 𝐸 and an 𝑥 ∈ 𝑋 such that 𝑓 ⊔ 𝑔(𝑒)(𝑥) > 𝑓(𝑒)(𝑥) ∨ 𝑔(𝑒)(𝑥). Take an 𝜖 > 0 satisfying 𝑎 = 𝑓 ⊔ 𝑔 (𝑒) (𝑥) > 𝑓 (𝑒) (𝑥) ∨ 𝑔 (𝑒) (𝑥) + 𝜖.

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We may assume 𝑓(𝑒)(𝑥) ≥ 𝑔(𝑒)(𝑥) (the case 𝑔(𝑒)(𝑥) > 𝑓(𝑒)(𝑥) is analogous). Then, since 𝑓(𝑒)(𝑥) = 1 − ∨{ℎ(𝑒)(𝑥) : ℎ𝛿𝑓} < 𝑎 − 𝜖, there exists an ℎ ∈ 𝐹𝑆(𝑋, 𝐸) such that ℎ𝛿𝑓 and 1 − ℎ(𝑒)(𝑥) < 𝑎 − 𝜖. By the inequality 1 − ℎ(𝑒)(𝑥) ≥ 𝑓(𝑒)(𝑥) ≥ 𝑔(𝑒)(𝑥) > 𝑔(𝑒)(𝑥) − 𝜖/2, we have 1 − ℎ(𝑒)(𝑥) + 𝜖/2 > 𝑔(𝑒)(𝑥). Because 𝑔(𝑒)(𝑥) = 1 − ∨{𝑘(𝑒)(𝑥) : 𝑘𝛿𝑔}, there exists a 𝑘 ∈ 𝐹𝑆(𝑋, 𝐸) such that 𝑘𝛿𝑔 and ℎ(𝑒)(𝑥) − 𝜖/2 < 𝑘(𝑒)(𝑥). Now, since (ℎ⊓𝑘)𝛿𝑓 and (ℎ⊓𝑘)𝛿𝑔, we obtain (ℎ⊓𝑘)𝛿(𝑓 ⊔ 𝑔). From (𝑓𝑠𝑝2 ), it follows that 𝑓 ⊔ 𝑔(𝑒)(𝑥) ≤ 1 − (ℎ ⊓ 𝑘)(𝑒)(𝑥). Also, we get ℎ(𝑒)(𝑥) − 𝜖/2 < (𝑘 ⊓ ℎ)(𝑒)(𝑥). Thus, 𝑎 = 𝑓 ⊔ 𝑔 (𝑒) (𝑥) ≤ 1 − (ℎ ⊓ 𝑘) (𝑒) (𝑥) ≤ 1 − ℎ (𝑒) (𝑥) +

𝜖 𝜖 𝜖 𝑓(𝑒)(𝑥). Let ∧{𝑔(𝑒)(𝑥) : 𝑓 ⋐ 𝑔} = 𝑎. Then there exists an 𝜖 > 0 such that 𝑓 (𝑒) (𝑥) = 1 − ∨ {ℎ (𝑒) (𝑥) : 𝑓𝛿ℎ} < 𝑎 − 𝜖.

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Therefore, there exists a fuzzy soft set 𝑘 such that 𝑓𝛿𝑘 and ̃ − 𝑘). Hence, 1 − 𝑘(𝑒)(𝑥) < 𝑎 − 𝜖. Because 𝑓𝛿𝑘, we have 𝑓 ⋐ (𝑋 ̃ ⊓{𝑔 : 𝑓 ⋐ 𝑔} ⊑ (𝑋 − 𝑘). Thus, 𝑎 = ∧ {𝑔 (𝑒) (𝑥) : 𝑓 ⋐ 𝑔} ≤ 1 − 𝑘 (𝑒) (𝑥) < 𝑎 − 𝜖,

ℎ𝛿2 𝑘 󳨐⇒ 𝜑𝜓−1 (ℎ) 𝛿1 𝜑𝜓−1 (𝑘) ,

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or in another form ℎ⋐2 𝑘 󳨐⇒ 𝜑𝜓−1 (ℎ) ⋐1 𝜑𝜓−1 (𝑘) ,

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for every ℎ, 𝑘 ∈ 𝐹𝑆(𝑌, 𝐾). Proposition 33. The composition of two fuzzy soft proximity mappings is a fuzzy soft proximity mapping. Theorem 34. A fuzzy soft proximity mapping 𝜑𝜓 : (𝑋, 𝛿1 ) → (𝑌, 𝛿2 ) is fuzzy soft continuous with respect to 𝜏(𝛿1 ) and 𝜏(𝛿2 ). Proof. Let 𝑓 ∈ 𝜏(𝛿2 ). Now let us take any ℎ ∈ 𝐹𝑆(𝑌, 𝐾) such ̃ − 𝑓). Since 𝜑𝜓 is a fuzzy soft proximity mapping, that ℎ𝛿2 (𝑌 ̃ − 𝜑−1 (𝑓)). From (𝑓𝑠𝑝2 ), it follows that we obtain 𝜑𝜓−1 (ℎ)𝛿1 (𝑋 𝜓 ̃ − 𝜑−1 (𝑓) ⊑ 𝑋 ̃ − 𝜑−1 (ℎ). Then, for every 𝑒 ∈ 𝐸 and every 𝑋 𝜓 𝜓 𝑥 ∈ 𝑋, we have ̃ − 𝜑−1 (𝑓)) (𝑒) (𝑥) ≤ (𝑋 ̃ − 𝜑−1 (ℎ)) (𝑒) (𝑥) (𝑋 𝜓 𝜓 = 1 − ℎ (𝜓 (𝑒)) (𝜑 (𝑥))

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Proof. Let us take a fuzzy soft set 𝑔 such that 𝑓 ⋐ 𝑔. Therefore, 𝑓 ⋐ 𝑔 and by (𝑓𝑠𝑝𝑛3 ) we obtain 𝑓 ⊑ 𝑔. Hence, 𝑓 ⊑ ⊓ {𝑔 : 𝑓 ⋐ 𝑔} .

Proposition 32. Let (𝑋, 𝛿1 ) and (𝑌, 𝛿2 ) be two fuzzy soft proximity spaces. A fuzzy soft mapping 𝜑𝜓 : (𝑋, 𝛿1 ) → (𝑌, 𝛿2 ) is a fuzzy soft proximity mapping if and only if

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̃ − ℎ) (𝜓 (𝑒)) (𝜑 (𝑥)) . = (𝑌 Therefore, ̃ − 𝜑−1 (𝑓)) (𝑒) (𝑥) (𝑋 𝜓 ̃ − ℎ)) (𝜓 (𝑒)) (𝜑 (𝑥)) (𝑌

≤( ⨅ ̃ ℎ𝛿2 (𝑌−𝑓)

̃ − 𝑓) (𝜓 (𝑒)) (𝜑 (𝑥)) = (𝑌

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̃ − 𝑓) (𝜓 (𝑒)) (𝜑 (𝑥)) = (𝑌 = 1 − 𝜑𝜓−1 (𝑓) (𝑒) (𝑥) ̃ − 𝜑−1 (𝑓)) (𝑒) (𝑥) . = (𝑋 𝜓

which leads to a contradiction. Definition 31 (see [35]). Let (𝑋, 𝛿1 ) and (𝑌, 𝛿2 ) be two fuzzy soft proximity spaces. A fuzzy soft mapping 𝜑𝜓 : (𝑋, 𝛿1 ) → (𝑌, 𝛿2 ) is a fuzzy soft proximity mapping if it satisfies 𝑓𝛿1 𝑔 󳨐⇒ 𝜑𝜓 (𝑓) 𝛿2 𝜑𝜓 (𝑔)

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for every 𝑓, 𝑔 ∈ 𝐹𝑆(𝑋, 𝐸). Using the above definition, we can easily prove the following propositions.

̃ − 𝜑−1 (𝑓) ⊑ 𝑋 ̃ − 𝜑−1 (𝑓), we obtain 𝜑−1 (𝑓) ∈ Thus, since 𝑋 𝜓 𝜓 𝜓 𝜏(𝛿1 ). Definition 35. If 𝛿1 and 𝛿2 are two fuzzy soft proximities on 𝑋, we define 𝛿1 < 𝛿2

iff 𝑓𝛿2 𝑔 implies 𝑓𝛿1 𝑔.

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The above is expressed by saying that 𝛿2 is finer than 𝛿1 , or 𝛿1 is coarser than 𝛿2 .


7

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 𝑋 be a set and {(𝑋𝛼 , 𝛿𝛼 ) : 𝛼 ∈ Δ} a family of fuzzy soft proximity spaces, and, for each 𝛼 ∈ Δ, let (𝜑𝜓 )𝛼 : 𝐹𝑆(𝑋, 𝐸) → (𝑋𝛼 , 𝛿𝛼 ) be a fuzzy soft mapping. The initial structure 𝛿 is the coarsest fuzzy soft proximity on 𝑋 for which all mappings (𝜑𝜓 )𝛼 : (𝑋, 𝛿) → (𝑋𝛼 , 𝛿𝛼 ) (𝛼 ∈ Δ) are fuzzy soft proximity mapping. Theorem 37 (existence of initial structures). Let 𝑋 be a set {(𝑋𝛼 , 𝛿𝛼 ) : 𝛼 ∈ Δ} a family of fuzzy soft proximity spaces, and, for each 𝛼 ∈ Δ, let (𝜑𝜓 )𝛼 : 𝐹𝑆(𝑋, 𝐸) → (𝑋𝛼 , 𝛿𝛼 ) be a fuzzy soft mapping. For any 𝑓, 𝑔 ∈ 𝐹𝑆(𝑋, 𝐸), define 𝑓𝛿𝑔 if and only if, for every finite families {𝑓𝑖 : 𝑖 = 1, . . . , 𝑛} and {𝑔𝑗 : 𝑗 = 1, . . . , 𝑚}, where 𝑓 = ⊔𝑛𝑖=1 𝑓𝑖 and 𝑔 = ⊔𝑚 𝑗=1 𝑔𝑗 , there exist an 𝑓𝑖 and a 𝑔𝑗 such that (𝜑𝜓 )𝛼 (𝑓𝑖 ) 𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 )

for each 𝛼 ∈ Δ.

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Then 𝛿 is the coarsest fuzzy soft proximity on 𝑋 for which all mappings (𝜑𝜓 )𝛼 : (𝑋, 𝛿) → (𝑋𝛼 , 𝛿𝛼 ) (𝛼 ∈ Δ) are fuzzy soft proximity mapping. Proof. We first prove that 𝛿 is a fuzzy soft proximity on 𝑋. (𝑓𝑠𝑝1 ) is obvious. ̃ Let (𝑓𝑠𝑝2 ) We will show that if 𝑓𝛿𝑔, then 𝑓 ⊓ 𝑔 = 0. 𝑓𝛿𝑔. Then, there exist finite covers 𝑓 = ⊔𝑛𝑖=1 𝑓𝑖 and 𝑔 = ⊔𝑚 𝑗=1 𝑔𝑗 of 𝑓 and 𝑔, respectively, such that (𝜑𝜓 )𝛼 (𝑓𝑖 )𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 ) for some 𝛼 = 𝑠𝑖𝑗 ∈ Δ, where 𝑖 = 1, . . . , 𝑛 and 𝑗 = 1, . . . , 𝑚. Since each 𝛿𝛼 is a fuzzy ̃ From this, soft proximity, (𝜑𝜓 )𝛼 (𝑓𝑖 ) ⊓ (𝜑𝜓 )𝛼 (𝑔𝑗 ) = 0. it follows that 𝑛

𝑚

𝑖=1

𝑗=1

(𝜑𝜓 )𝛼 (⨆ 𝑓𝑖 ) ⊓ (𝜑𝜓 )𝛼 (⨆ 𝑔𝑗 )

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̃ = (𝜑𝜓 )𝛼 (𝑓) ⊓ (𝜑𝜓 )𝛼 (𝑔) = 0. ̃ Thus, we have 𝑓 ⊓ 𝑔 = 0. (𝑓𝑠𝑝3 ) Since each 𝛿𝛼 is a fuzzy soft proximity, it is clear that 𝑓𝛿𝑔 implies 𝑔𝛿𝑓. (𝑓𝑠𝑝4 ) It is easy to verify that if 𝑓𝛿𝑔, then 𝑓𝛿ℎ for each ℎ ⊒ 𝑔. Therefore, 𝑓𝛿𝑔 or 𝑓𝛿ℎ implies 𝑓𝛿(𝑔 ⊔ ℎ). Conversely, assume that 𝑓𝛿𝑔 and 𝑓𝛿ℎ. Then, there exist finite covers 𝑓 = ⊔𝑛𝑖=1 𝑓𝑖 and 𝑔 = ⊔𝑚 𝑗=1 𝑔𝑗 of 𝑓 and 𝑔, respectively, such that (𝜑𝜓 )𝛼 (𝑓𝑖 )𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 ) for some 𝛼 = 𝑠𝑖𝑗 ∈ Δ, where 𝑖 = 1, . . . , 𝑛 and 𝑗 = 1, . . . , 𝑚. 𝑞 In the same way, there are finite covers 𝑓 = ⊔𝑝=1 𝑘𝑝 and ℎ = ⊔𝑚+𝑙 𝑗=𝑚+1 𝑔𝑗 of 𝑓 and ℎ, respectively, such

that (𝜑𝜓 )𝛼 (𝑘𝑝 )𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 ) for some 𝛼 = 𝑡𝑝𝑗 ∈ Δ, where 𝑝 = 1, . . . , 𝑞 and 𝑗 = 𝑚 + 1, . . . , 𝑚 + 𝑙. Now,

𝑓 = ⊔{𝑓𝑖 ⊓ 𝑘𝑝 : 𝑖 = 1, . . . , 𝑛; 𝑝 = 1, . . . , 𝑞} and 𝑔 ⊔ ℎ = ⊔{𝑔𝑗 : 𝑗 = 1, . . . , 𝑚 + 𝑙} are finite covers of 𝑓 and 𝑔 ⊔ ℎ, respectively. Hence, from the fact that (𝜑𝜓 )𝛼 (𝑓𝑖 ⊓ 𝑘𝑝 )𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 ) for 𝛼 = 𝑠𝑖𝑗 or 𝛼 = 𝑡𝑝𝑗 , it follows that 𝑓𝛿(𝑔 ⊔ ℎ). (𝑓𝑠𝑝5 ) Let us define the set Ω of all pairs (𝑓, 𝑔) such that ̃ − ℎ) for each 𝑓𝛿𝑔 and we have either 𝑓𝛿ℎ or 𝑔𝛿(𝑋 ℎ ∈ 𝐹𝑆(𝑋, 𝐸). The validity of (𝑓𝑠𝑝5 ) will follow from the fact that Ω is empty. Suppose, on the contrary, that (𝑓, 𝑔) ∈ Ω. Then, (𝜑𝜓 )𝛼 (𝑓)𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔) for each 𝛼 ∈ Δ. Indeed, let ℎ ∈ 𝐹𝑆(𝑋𝛼 , 𝐸𝛼 ) and 𝑘 = (𝜑𝜓 )−1 𝛼 (ℎ). If 𝑓𝛿𝑘, then (𝜑𝜓 )𝛼 (𝑓)𝛿𝛼 (𝜑𝜓 )𝛼 (𝑘). Because (𝜑𝜓 )𝛼 (𝑘) ⊑ ̃ − 𝑘), ℎ, we have (𝜑𝜓 )𝛼 (𝑓)𝛿𝛼 ℎ. Similarly, if 𝑔𝛿(𝑋 ̃ then (𝜑𝜓 )𝛼 (𝑔)𝛿𝛼 (𝑋𝛼 − ℎ). Hence, since 𝛿𝛼 is a fuzzy soft proximity on 𝑋𝛼 , we obtain (𝜑𝜓 )𝛼 (𝑓)𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔). Also, we observe that for each (𝑓, 𝑔) ∈ Ω there are positive integers 𝑛, 𝑚 and covers 𝑓 = ⊔𝑛𝑖=1 𝑓𝑖 and 𝑔 = ⊔𝑚 𝑗=1 𝑔𝑗 such that, for every pair (𝑖, 𝑗) ∈ {1, . . . , 𝑛} × {1, . . . , 𝑚}, there exists an 𝛼 ∈ Δ satisfying (𝜑𝜓 )𝛼 (𝑓𝑖 )𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 ). Let 𝑙 = 𝑛 + 𝑚. It easy to see that 𝑙 > 2. Then, for each (𝑓, 𝑔) ∈ Ω, let us choose such an integer 𝑙. But 𝑙 is not uniquely determined by (𝑓, 𝑔). Let 𝜅 be the set of all integers corresponding to members of Ω and let 𝑙 be the smallest member of 𝜅. Take a (𝑓, 𝑔) ∈ Ω such that 𝑙 is the integer corresponding to it. Then, there are covers 𝑓 = ⊔𝑛𝑖=1 𝑓𝑖 and 𝑔 = ⊔𝑚 𝑗=1 𝑔𝑗 such that 𝑙 = 𝑛 + 𝑚 and for every pair (𝑖, 𝑗) ∈ {1, . . . , 𝑛} × {1, . . . , 𝑚} and there exists an 𝛼 ∈ Δ satisfying (𝜑𝜓 )𝛼 (𝑓𝑖 )𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 ). One of the 𝑛, 𝑚 is greater than 1. Consider 𝑛 > 1 and let 𝑓󸀠 = 𝑓1 ⊔ ⋅ ⋅ ⋅ ⊔ 𝑓𝑛−1 . In this case, one of the following conditions should be true: ̃ (i) for every ℎ ∈ 𝐹𝑆(𝑋, 𝐸), either 𝑓󸀠 𝛿ℎ or 𝑔𝛿(𝑋−ℎ); ̃ (ii) for every ℎ ∈ 𝐹𝑆(𝑋, 𝐸), either 𝑓𝑛 𝛿ℎ or 𝑔𝛿(𝑋−ℎ). In fact, suppose that neither (i) nor (ii) holds. Then, there ̃ − ℎ1 ) and are ℎ1 , ℎ2 ∈ F𝑆(𝑋, 𝐸) such that 𝑓󸀠 𝛿ℎ1 , 𝑔𝛿(𝑋 ̃ 𝑓𝑛 𝛿ℎ2 , 𝑔𝛿(𝑋 − ℎ2 ). Letting ℎ = ℎ1 ⊓ ℎ2 , we obtain 𝑓𝛿ℎ and ̃ − ℎ), contradicting the fact that (𝑓, 𝑔) ∈ Ω. 𝑔𝛿(𝑋 Suppose that (i) holds. Because 𝑓󸀠 ⊑ 𝑓 and 𝑓𝛿𝑔, this means that 𝑓󸀠 𝛿g. Hence, by (i), we have (𝑓󸀠 , 𝑔) ∈ Ω. But this is now a contradiction since (𝑛−1)+𝑚 = 𝑙−1 ∈ 𝜅, contrary to the choice of 𝑙. If (ii) holds, we get a contradiction in a similar way. Therefore, the set Ω is empty. Thus, 𝛿 is a fuzzy soft proximity on 𝑋. It is easy to see that all mappings (𝜑𝜓 )𝛼 : (𝑋, 𝛿) → (𝑋𝛼 , 𝛿𝛼 ) are fuzzy soft proximity mapping. Let 𝛿∗ be another fuzzy soft proximity on 𝑋 making each of the mappings (𝜑𝜓 )𝛼 : (𝑋, 𝛿∗ ) → (𝑋𝛼 , 𝛿𝛼 ) fuzzy soft proximity mapping. We will show that 𝛿 < 𝛿∗ , which will complete the proof. Let 𝑓𝛿∗ 𝑔 and consider any covers 𝑓 = ⊔𝑛𝑖=1 𝑓𝑖 and 𝑔 = ⊔𝑚 𝑗=1 𝑔𝑗 of 𝑓 and 𝑔, respectively. Since 𝑓 = (𝑓1 ⊔ ⋅ ⋅ ⋅ ⊔ 𝑓𝑛 )𝛿∗ 𝑔, by (𝑓𝑠𝑝4 ), there is an 𝑖 ∈ {1, . . . , 𝑛} such that 𝑓𝑖 𝛿𝑔. In the same way, since 𝑓𝑖 𝛿∗ 𝑔 = (𝑔1 ⊔⋅ ⋅ ⋅⊔𝑔𝑚 ), by (𝑓𝑠𝑝4 ), there is a 𝑗 ∈ {1, . . . , 𝑚} such

8


that 𝑓𝑖 𝛿𝑔𝑗 . From the fact that all mappings (𝜑𝜓 )𝛼 : (𝑋, 𝛿∗ ) → (𝑋𝛼 , 𝛿𝛼 ) are fuzzy soft proximity mapping, it follows that (𝜑𝜓 )𝛼 (𝑓𝑖 )𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 ) for each 𝛼 ∈ Δ. Thus, we get 𝑓𝛿𝑔. Theorem 38. A fuzzy soft mapping 𝜑𝜓 : (𝑌, 𝛿∗ ) → (𝑋, 𝛿) is a fuzzy soft proximity mapping if and only if (𝜑𝜓 )𝛼 ∘ 𝜑𝜓 : (𝑌, 𝛿∗ ) → (𝑋𝛼 , 𝛿𝛼 ) is a fuzzy soft proximity mapping for every 𝛼 ∈ Δ. Proof. The necessity is easy. We prove the sufficiency. Suppose that (𝜑𝜓 )𝛼 ∘ 𝜑𝜓 is a fuzzy soft proximity mapping for every 𝛼 ∈ Δ. Let 𝑓𝛿∗ 𝑔 and let 𝜑𝜓 (𝑓) = ⊔𝑛𝑖=1 𝑓𝑖 and 𝜑𝜓 (𝑔) = ⊔𝑚 𝑗=1 𝑔𝑗 . Then, we have 𝑓⊑

𝑛

⨆ 𝜑𝜓−1 𝑖=1

(𝑓𝑖 ) ,

𝑚

−1

𝑔 ⊑ ⨆ (𝜑𝜓 ) 𝑔𝑗 . 𝑗=1

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Since 𝑓𝛿∗ 𝑔, by (𝑓𝑠𝑝4 ), there exist 𝑖, 𝑗 such that 𝜑𝜓−1 (𝑓𝑖 )𝛿∗ 𝜑𝜓−1 (𝑔𝑗 ). Because (𝜑𝜓 )𝛼 ∘ 𝜑𝜓 ∘ 𝜑𝜓−1 (𝑓𝑖 ) ⊑ 𝜑𝜓 (𝑓𝑖 ) , (𝜑𝜓 )𝛼 ∘ 𝜑𝜓 ∘ 𝜑𝜓−1 (𝑔𝑗 ) ⊑ 𝜑𝜓 (𝑔𝑗 ) ,

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it follows from our hypothesis that (𝜑𝜓 )𝛼 (𝑓𝑖 )𝛿𝛼 (𝜑𝜓 )𝛼 (𝑔𝑗 ) for every 𝛼 ∈ Δ. This shows that 𝜑𝜓 (𝑓)𝛿𝜑𝜓 (𝑔). Definition 39. Let {(𝑋𝛼 , 𝛿𝛼 ) : 𝛼 ∈ Δ} be a family of fuzzy soft proximity spaces and let 𝑋 = ∏𝛼∈Δ 𝑋𝛼 and 𝐸 = ∏𝛼∈Δ 𝐸𝛼 be product sets. An initial fuzzy soft proximity structure 𝛿 = ∏𝛼∈Δ 𝛿𝛼 on 𝑋 with respect to all the fuzzy soft projection mappings (𝑝𝑋𝛼 )𝑞𝐸 , where 𝑝𝑋𝛼 : 𝑋 → 𝑋𝛼 and 𝑞𝐸𝛼 : 𝐸 → 𝛼 𝐸𝛼 , is called the product fuzzy soft proximity structure. (𝑋, 𝛿) is said to be a product fuzzy soft proximity space. From Theorems 37 and 38, we obtain the following corollary. Corollary 40. Consider {(𝑋𝛼 , 𝛿𝛼 ) : 𝛼 ∈ Δ} be a family of fuzzy soft proximity spaces. Let 𝑋 = ∏𝛼∈Δ 𝑋𝛼 and 𝐸 = ∏𝛼∈Δ 𝐸𝛼 be sets and for each 𝛼 ∈ Δ let (𝑝𝑋𝛼 )𝑞𝐸 be a fuzzy soft mapping. 𝛼 For any 𝑓, 𝑔 ∈ 𝐹𝑆(𝑋, 𝐸), define 𝑓𝛿𝑔 if and only if, for every finite families {𝑓𝑖 : 𝑖 = 1, . . . , 𝑛} and {𝑔𝑗 : 𝑗 = 1, . . . , 𝑚}, where 𝑓 = ⊔𝑛𝑖=1 𝑓𝑖 and 𝑔 = ⊔𝑚 𝑗=1 𝑔𝑗 , there exist an 𝑓𝑖 and a 𝑔𝑗 such that (𝑝𝑋𝛼 )𝑞𝐸 (𝑓𝑖 )𝛿𝛼 (𝑝𝑋𝛼 )𝑞𝐸 (𝑔𝑗 ) for each 𝛼 ∈ Δ. Then, 𝛼

𝛼

(i) 𝛿 = ∏𝛼∈Δ 𝛿𝛼 is the coarsest fuzzy soft proximity on 𝑋 for which all mappings (𝑝𝑋𝛼 )𝑞𝐸 (𝛼 ∈ Δ) are fuzzy soft 𝛼 proximity mapping; (ii) a fuzzy soft mapping 𝜑𝜓 : (𝑌, 𝛿∗ ) → (𝑋, 𝛿) is a fuzzy soft proximity mapping if and only if (𝑝𝑋𝛼 )𝑞𝐸 ∘ 𝜑𝜓 : 𝛼 (𝑌, 𝛿∗ ) → (𝑋𝛼 , 𝛿𝛼 ) is a fuzzy soft proximity mapping for every 𝛼 ∈ Δ.

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 𝑋 be a set and let 𝐴 be a subset of 𝑋. Then, a mapping 𝜒̃𝐴 : 𝐸 → 𝐼𝑋 is a fuzzy soft set on 𝑋 defined as the following: 𝜒̃𝐴 (𝑒) = 𝜒𝐴

for every 𝑒 ∈ 𝐸,

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where 𝜒𝐴 is a characteristic function of 𝐴. Example 42. Let 𝑋 = {𝑥1 , 𝑥2 , 𝑥3 }, 𝐴 = {𝑥1 , 𝑥2 }, and 𝐸 = {𝑒1 , 𝑒2 }. Then, 𝜒̃𝐴 = {(𝑒1 , {

𝑥1 𝑥2 𝑥3 𝑥 𝑥 𝑥 , , }) , (𝑒2 , { 1 , 2 , 3 })} 1 1 0 1 1 0

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is a fuzzy soft set on 𝑋. Lemma 43. For any subsets 𝐴, 𝐵 of 𝑋, (i) 𝜒̃𝐴 ⊓ 𝜒̃𝐵 = 𝜒̃𝐴∩𝐵 ; (ii) 𝜒̃𝐴 ⊔ 𝜒̃𝐵 = 𝜒̃𝐴∪𝐵 ; (iii) (𝜒̃𝐴 )𝑐 = 𝜒̃𝐴𝑐 . Proof. Straightforward. Theorem 44. Let (𝑋, 𝛿) be a proximity space. By letting, for 𝑓, 𝑔 ∈ 𝐹𝑆(𝑋, 𝐸), 𝑓𝛿𝑖 𝑔 if and only if there exist subsets 𝐴, 𝐵 of 𝑋 such that 𝑓 ⊑ 𝜒̃𝐴 , 𝑔 ⊑ 𝜒̃𝐵 , and 𝐴𝛿𝐵, one defines a fuzzy soft proximity on 𝑋. Proof. We will show that 𝛿𝑖 satisfies axioms (𝑓𝑠𝑝1 )–(𝑓𝑠𝑝5 ). ̃ 𝑖 𝑓. (𝑓𝑠𝑝1 ) From 0̃ ⊑ 𝜒̃0 , 𝑓 ⊑ 𝜒̃𝑋 , and 0𝛿𝑋, it follows that 0𝛿 (𝑓𝑠𝑝2 ) Let 𝑓𝛿𝑖 𝑔. Then, there are subsets 𝐴 and 𝐵 of 𝑋 such that 𝑓 ⊑ 𝜒̃𝐴, 𝑔 ⊑ 𝜒̃𝐵 , and 𝐴𝛿𝐵. By 𝐴𝛿𝐵, we have 𝐴 ∩ ̃ Thus, we get 𝑓 ⊓ 𝑔 = 0. ̃ 𝐵 = 0, so that 𝜒̃𝐴 ⊓ 𝜒̃𝐵 = 0. (𝑓𝑠𝑝3 ) It is clear because 𝐴𝛿𝐵 implies 𝐵𝛿𝐴. (𝑓𝑠𝑝4 ) It is easy to see that if 𝑓𝛿𝑖 (𝑔 ⊔ ℎ), then 𝑓𝛿𝑖 𝑔 and 𝑓𝛿𝑖 ℎ. Conversely, suppose that 𝑓𝛿𝑖 𝑔 and 𝑓𝛿𝑖 ℎ. Then, there exist subsets 𝐴 and 𝐵 of 𝑋 such that 𝑓 ⊑ 𝜒̃𝐴, 𝑔 ⊑ 𝜒̃𝐵 , and 𝐴𝛿𝐵. Likewise, there exist subsets 𝐶 and 𝐷 of 𝑋 such that 𝑓 ⊑ 𝜒̃𝐶, ℎ ⊑ 𝜒̃𝐷, and 𝐶𝛿𝐷. Since 𝑓 ⊑ 𝜒̃𝐴 ⊓ 𝜒̃𝐶 = 𝜒̃𝐴∩𝐶, 𝑔 ⊔ ℎ ⊑ 𝜒̃𝐵 ⊔ 𝜒̃𝐷 = 𝜒̃𝐵∪𝐷, and (𝐴 ∩ 𝐶)𝛿(𝐵 ∪ 𝐷), we conclude that 𝑓𝛿𝑖 (𝑔 ⊔ ℎ). (𝑓𝑠𝑝5 ) If 𝑓𝛿𝑖 𝑔, then there are subsets 𝐴 and 𝐵 of 𝑋 such that 𝑓 ⊑ 𝜒̃𝐴 , 𝑔 ⊑ 𝜒̃𝐵 , and 𝐴𝛿𝐵. Since 𝐴𝛿𝐵, by (𝑝5 ), there is a 𝐶 ⊆ 𝑋 such that 𝐴𝛿𝐶 and 𝐵𝛿(𝑋 − 𝐶). Therefore, for ̃ − 𝜒̃𝐶), a fuzzy soft set 𝜒̃𝐶, we obtain 𝑓𝛿𝑖 𝜒̃𝐶 and 𝑔𝛿𝑖 (𝑋 which completes the proof.

Theorem 45. Let (𝑋, 𝛿∗ ) be a fuzzy soft proximity space. (a) There is a proximity relation 𝛿 on 𝑋 such that 𝛿∗ = 𝛿𝑖 . (b) If 𝑓𝛿∗ 𝑔, then there exist subsets 𝐴 and 𝐵 of 𝑋 such that 𝑓 ⊑ 𝜒̃𝐴 , 𝑔 ⊑ 𝜒̃𝐵 , and 𝜒̃𝐴 𝛿∗ 𝜒̃𝐵


9

(c) The relation 𝐴𝛿𝐵 if and only if 𝜒̃𝐴 𝛿∗ 𝜒̃𝐵 is a proximity on 𝑋. Then, (𝑎) and (𝑏) are equivalent and they imply (𝑐). Proof. The implication (a) ⇒ (b) is obvious. To prove that (b) ⇒ (c), since the other axioms are readily verified, it is enough to show that 𝛿 satisfies (𝑝5 ). Let 𝐴𝛿𝐵 for any subsets 𝐴, 𝐵 of 𝑋. Because 𝜒̃𝐴𝛿∗ 𝜒̃𝐵 , there is an 𝑓 ∈ ̃ − 𝑓). By hypothesis, 𝐹𝑆(𝑋, 𝐸) such that 𝜒̃𝐴 𝛿∗ 𝑓 and 𝜒̃𝐵 𝛿∗ (𝑋 there exist subsets 𝐶, 𝐷, 𝐺, 𝐻 of 𝑋 such that 𝜒̃𝐴 ⊑ 𝜒̃𝐶, 𝑓 ⊑ ̃𝐺, 𝑋 ̃ − 𝑓 ⊑ 𝜒̃𝐻, 𝜒̃𝐶𝛿∗ 𝜒̃𝐷, and 𝜒̃𝐺𝛿∗ 𝜒̃𝐻. Because ̃𝐷, 𝜒̃𝐵 ⊑ 𝑋 𝑋 ̃𝐷 and 𝑋 ̃ − 𝑓 ⊑ 𝜒̃𝐻, we have 𝜒̃(𝑋−𝐷) ⊑ 𝜒̃𝐻. Now, from 𝑓⊑𝑋 ∗ 𝜒̃𝐶𝛿 𝜒̃𝐷 and 𝜒̃𝐴 ⊑ 𝜒̃𝐶, it follows that 𝜒̃𝐴 𝛿∗ 𝜒̃𝐷 and so 𝐴𝛿𝐷. Likewise, since 𝜒̃𝐺𝛿∗ 𝜒̃𝐻, 𝜒̃𝐵 ⊑ 𝜒̃𝐺, and 𝜒̃(𝑋−𝐷) ⊑ 𝜒̃𝐻, we have 𝜒̃𝐵 𝛿∗ 𝜒̃(𝑋−𝐷) and this implies that 𝐵𝛿(𝑋 − 𝐷). Thus, 𝛿 has the axiom (𝑝5 ). In order to prove (b) ⇒ (a), let us define the binary relation 𝛿 on the power set of 𝑋 as follows: 𝐴𝛿𝐵

iff 𝜒̃𝐴 𝛿∗ 𝜒̃𝐵 .

(31)

We have shown that 𝛿 is a proximity on 𝑋. To complete the proof, it will suffice to prove that 𝛿∗ = 𝛿𝑖 , that is, that 𝑓𝛿𝑖 𝑔 if and only if 𝑓𝛿∗ 𝑔. Assume that 𝑓𝛿∗ 𝑔. Then, by hypothesis, there exist subsets 𝐴 and 𝐵 of 𝑋 such that 𝑓 ⊑ 𝜒̃𝐴 , 𝑔 ⊑ 𝜒̃𝐵 , and 𝜒̃𝐴𝛿∗ 𝜒̃𝐵 . Since 𝜒̃𝐴 𝛿∗ 𝜒̃𝐵 , by definition, we have 𝐴𝛿𝐵. Thus, 𝑓𝛿𝑖 𝑔. On the other hand, suppose that 𝑓𝛿𝑖 𝑔. Using the definition of 𝛿𝑖 , we obtain subsets 𝐴, 𝐵 of 𝑋 such that 𝑓 ⊑ 𝜒̃𝐴, 𝑔 ⊑ 𝜒̃𝐵 , and 𝐴𝛿𝐵. Therefore, we get 𝜒̃𝐴𝛿∗ 𝜒̃𝐵 . Thus, since 𝑓 ⊑ 𝜒̃𝐴 , 𝑔 ⊑ 𝜒̃𝐵 , and 𝜒̃𝐴 𝛿∗ 𝜒̃𝐵 , we obtain 𝑓𝛿∗ 𝑔 and the proof is concluded.

6. Conclusion Each proximity space determines in a natural way a topological space with beneficial properties. Also, this theory possesses deep results, rich machinery, and tools. With the development of topology, the theory of proximity makes a great progress. Hence, the concept of proximity has been studied by many authors in both the fuzzy setting and the soft setting. In the present work, we mainly establish some properties of fuzzy soft proximity spaces in Katsaras’s sense. We have shown that each fuzzy soft proximity determines a fuzzy soft topology by using fuzzy soft closure operator. Also, we present an alternative description of the concept of fuzzy soft proximity, which is called fuzzy soft 𝛿-neighborhood. We believe that these results will help the researchers to advance and promote the further study on fuzzy soft topology to carry out a general framework for their applications in practical life.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

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