Erzincan Üniversitesi Erzincan University Fen Bilimleri Enstitüsü Dergisi Journal of Science and Technology 2020, 13(2), 788-802 2020, 13(2), 788-802 ISSN: 1307-9085, e-ISSN: 2149-4584 DOI: 10.18185/erzifbed.728964 Araştırma Makalesi Research Article

Near Soft Connected

Alkan ÖZKAN1* , Metin DUMAN1

1 Department of Mathematics, Faculty of Arts and Sciences, Iğdır University, Iğdır, Turkey

Geliş / Received: 29/04/2020, Kabul / Accepted: 26/08/2020

Abstract

Near soft sets are considered as mathematical tools for dealing with ambiguities. In this study, we describe the near soft connectedness in near soft topological spaces, and introduce its concerned properties.

Keywords: Near , soft set, near soft set, near soft topological spaces, near soft connectedness

Yakın Esnek Bağlantılı Uzaylar

Öz

Yakın esnek kümeler, belirsizlikler ile başa çıkmak için matematiksel araçlar olarak kabul edilir. Bu çalışmada, yakın esnek topolojik uzaylarda yakın esnek bağlantılığı tanımlamakta ve ilgili özelliklerini sunmaktayız.

Anahtar Kelimeler: Yakın küme, esnek küme, yakın esnek küme, yakın esnek topolojik uzaylar, yakın esnek bağlantılılık.

1. Introduction theory of near sets. We need to have some knowledge about these tools to begin. Classical mathematics requires certain ideas. However, there are imprecise concepts in Being one of them, the theory of fuzzy sets other fields such as engineering, medicine was introduced by Zadeh in 1965. In 2001, and economics. For example, "high (or low) Maji et al. combined fuzzy sets and soft sets salary" in the economy is not certain. Such and introduced the concept of fuzzy soft sets, uncertain concepts are called ambiguous. The and they presented an application of fuzzy noted theories on mathematical tools to deal soft sets in a decision making problem. with dubiousness are: theory of fuzzy sets, Recently, many authors (Aygunoğlu, et al., theory of rough sets, theory of soft sets and 2019; Beaula and Gunaseeli, 2014; Beaula

*Corresponding Author: [email protected] 788

Near Soft Connected Spaces and Priyanga, 2015; Beaula and Raja, 2015), Proximity lets us create correlation between etc. studied on fuzzy soft sets. two things in our daily life. The first entry in the concept of intimacy in mathematics was Rough set theory proposed by Pawlak in made in 1908 by Riesz with his article on the 1982 is a mathematical method used in intimacy of two sets (Riesz, 1908). The reasoning and knowledge extraction for “Near Set” theory was developed in 2002 as expert systems (Grzymala-Busse, 1988; a generalization of approximated sets by Orlowska, 1994). The rough set theory has Pawlak (Pawlak and Peters, 2002-2007). In emerged as a new mathematical approach to near sets, data is obtained using real-valued uncertainty. Theory essence of each object is functions. Studies on near sets gained location of thought to be paired with momentum after Peters' article on near sets information. This thought leads us to “General Theory about Nearness of Objects” information systems and is the first point of (Peters, 2007). Peters and Wasilewski origin of theory. Areas of application for the published the foundations of near sets rough set approach include data mining, (Peters, 2009) and Wolski published the new economics, robotics, chemistry, biology, approaches for near sets and approximated medicine and . sets (Wolski, 2010). In addition, Peters conducted researches on computerized Molodtsov explained the notion of soft set as applications (Peters, 2005; Peters and Henry, a set theory in 1999 (Molodtsov, 1999). This 2006; Peters and Ramanna, 2009; Peters, et new theory quickly attracted attention, and al., 2007; Peters and Naimpally, 2012) and many researchers focused on the application especially image analysis (Hassanien, et al., of soft set theory to the links between other 2009; Henry, 2010; Peters, 2010; Henry and branches of mathematics. In 2003, the basic Peters, TR-2010-017; Henry and Smith, TR- set-theoretic process and properties of the 2012-021). Then, many researchers soft set given by Maji et al. (2002). Ali et al. published on approach spaces and near sets. (2009), Maji et al. (2003) examined and Ozturk and Inan identified near groups and expanded the process and features in a more studied their key features (Inan and Ozturk, detailed way. In the continuation of this 2012). study, Aygunoğlu and Aygun (2011), Cagman et al. (2011), Zorlutuna et al. (2012) Many new set of definitions have been made and then Georgiou et al. (2013) made through these sets. One of these is the near important contributions to the development soft set that Tasbozan et al. defined in 2017 of the notion of soft topological . Pei using the feature of the concept of near set. and Miao (2005) stated that each information Tasbozan et al. studied basic properties the system is a soft set and defined some special concepts of near soft set and near soft soft sets and their relationship with topological spaces in 2017 (Tasbozan et al., information systems. First, Maji et al. (2002) 2017). In 2019, the continue of the showed that soft sets are a very useful tool in theoretical studies of near soft sets and near decision making problems. Aktas and soft topological spaces were introduced by Cagman (2007) and Acar et al. (2010) gave Ozkan (Ozkan, 2019). the concept of soft group and soft ring respectively and examined their related Intuitively when we say a space is connected features. to indicate that it has “one piece”, whereas a

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space is disconnected when it has several taken 푟 at a time to define the relation ~퐵푟. disjoint, “independent pieces” (Vu, 2019). The overlap function 풱푁푟 is defined by Following authors, such as Cantor (1883) for 풱푁푟: 풫(풪) × 풫(풪) → [0,1], where 풫(풪) is cantor set; Zhao (1968) for fuzzy sets, the powerset of 풪. The overlap function 풱푁푟 Pawlak (1982) for rough sets, Peters (2007) maps a pair of sets to a number in [0,1], for near sets, Hussain (2014) for soft sets, representing the degree of overlap between Mahanta et al. (2012) for fuzzy soft sets sets of objects with features defined by he defined connectedness, which is one of the probe functions 퐵푟 ⊆ 퐵. This relation defines important topics in mathematics. a partition of 풪 into non-empty, pairwise In this study, we aim to study the concepts of disjoint subsets that are equivalence classes denoted by ā , where 푎 = {푎′ ∈ near soft connectedness in near soft 퐵푟 퐵푟 ′ topological spaces. Moreover, it is thought 풪: 푎 ~퐵푟 푎 }. These classes form a new set that these structures, whose basic properties called the quotient set 풪/~퐵푟 , where are examined, will form the basis for future 풪/~퐵푟 = {푎퐵푟: 푎 ∈ 풪} (Peters, 2007). studies on near soft sets.

Definition 2.1 A soft set 퐹퐴 on the universe 2. Preliminaries 푈, 퐸 is a set of parameters, 풫(푈) is the power set of 푈, and 퐴 ⊂ 퐸, is defined by the In this section, we remember the definitions set of ordered pairs 퐹 = {(푒, 푓 (푒)): 푒 ∈ and conclusions. 퐴 퐴 퐸, 푓퐴(푒) ∈ 풫(푈)}, where 푓퐴: 퐸 → 풫(푈) such Let 풪 be a set of perceived objects and 푋 ⊆ that 푓퐴(푒) = ∅ if 푒 ∉ 퐴. 풪. An object description is defined via a Here, 푓 is called an approximate function of tuple of function values 휙(푥) regarding 퐴 the soft set 퐹 . The value of 푓 (푒) may be object 푥 ∈ 푋. The important point to be 퐴 퐴 arbitrary. Some of them may be empty, some considered is to select of function 휙 ∈ 퐵 is 푖 may have non-empty intersection (Cağman used to identify the corresponding object. and Enginoğlu, 2010). (Inan and Ozturk, 2012). Let 푁 (퐵)(푋) be a family of neighborhoods A nearness approximation space (푁퐴푆) is 푟 of a set 푋 ⊆ 풪. denoted by 푁퐴푆 = (풪, ℱ, ~퐵푟 , 푁푟, 풱푁푟) which is defined with a set of perceived Proposition 2.2 Every family of objects 풪, a set of probe functions ℱ neighborhoods may be considered a as soft representing object features, an set (Tasbozan et al., 2017). indiscernibility relation ~퐵푟 defined relative to 퐵푟 ⊆ 퐵 ⊆ ℱ, a collection of partitions Let 풪 is an initial universe set in soft set and (families of neighborhoods) 푁푟(퐵), and a ℱ is a collection of all possible parameters neighborhood overlap function 푁푟. The with respect to 풪 and 퐴, 퐵 ⊆ ℱ. relation ~퐵 is the usual indiscernibility 푟 Definition 2.3 Let 휎 = 퐹 be a soft set over relation from rough set theory restricted to a 퐵 풪 and 푁퐴푆 = (풪, ℱ, ~퐵푟, 푁푟, 풱푁푟) be a subset 퐵푟 ⊆ 퐵. The subscript 푟 denotes the nearness approximation space. The lower and cardinality of the restricted subset 퐵푟, where upper near approximation of 휎 = 퐹 with we consider (|퐵|), i.e., |퐵| functions 𝑖 ∈ ℱ 퐵 푟 respect to 푁퐴푆, respectively, are denoted by

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∗ ∗ 푁푟∗(휎) = 퐹퐵∗ and 푁푟 (휎) = 퐹퐵, which are (휙3, {푎2, 푎3, 푎4, 푎5})}. soft sets over with the set-valued mappings given by Then for 푟 = 1;

( ) ( ) ( ) 푎1 = {푎1, 푎3, 푎4}, 푎2 = {푎2, 푎5}, 퐹∗ 휙 = 푁푟∗(퐹 휙 ) =∪ {푎 ∈ 풪: 푎퐵푟 ⊆ 퐹 휙 }, 휙1 휙1 푎 = {푎 , 푎 }, 푎 = {푎 , 푎 , 푎 }, 1휙2 1 2 3휙2 3 4 5 ∗( ) ∗ ( ) 퐹 휙 = 푁푟 (퐹 휙 ) =∪ {푎 ∈ 풪: 푎퐵푟 ∩ 퐹(휙) ≠ ∅} 푎 = {푎 , 푎 }, 푎 = {푎 , 푎 }, 푎 = 1휙3 1 2 3휙3 3 4 5휙3 {푎5} where all 휙 ∈ 퐵. For two operators 푁푟∗ and ∗ 푁푟 on soft set, we say that these are the lower 푁∗(휎) = {(휙1, ∅), (휙2, ∅), (휙3, {푎3, 푎4, 푎5})} and upper near approximation operators, ∗ 푁 (휎) = {(휙1, {푎1, 푎2}), (휙2, {푎1, 푎3}), respectively. If 퐵푛푑푁푟(퐵)(휎) ≥ 0, then the soft set 휎 is said to be a near soft set (휙 , {푎 , 푎 , 푎 })}. (Tasbozan et al., 2017). 3 1 3 5 퐵푛푑 (휎) ≥ 0, and then 휎 is a near soft We consider only near soft sets 퐹 over a 푁푟(퐵) 퐵 set. universe 풪 in which all the parameter sets 퐵 are the same. But 휆 = 퐺퐵, 휆 = {(휙1, {푎1, 푎4, 푎5}),

훮푆푆(풪퐵) indicates the set of all near soft sets (휙2, {푎1, 푎3}), (휙3, {푎1})} is not a near soft over 풪. set because 푁∗(휆) = ∅.

Example 2.4 Let 풪 = { 푎1, 푎2, 푎3, 푎4, 푎5}, Definition 2.5 Let 퐹퐵 ∈ 푁푆푆(풪퐵) and 푎 ∈ 풪. 퐵 = {휙1, 휙2, 휙3} ⊂ ℱ = { 휙1, 휙2, 휙3, 휙4} We say that 푎 ∈ 퐹퐵 read as a belongs to the denote a set of perceptual objects and a set of near soft set 퐹퐵, whenever 푎 ∈ 퐹(휙) for all functions, respectively. Sample values of the 휙 ∈ 퐵. 휙푖, 𝑖 = 1,2,3,4 functions are shown in Table

1. Note that for 푎 ∈ 풪, 푎 ∉ 퐹퐵 if 푎 ∉ 퐹(휙) for some 휙 ∈ 퐵 (Ozkan, 2019). a1 a2 a3 a4 a5

Definition 2.6 Let 퐹퐵, 퐺퐵 ∈ 푁푆푆(풪퐵). Then 휙1 2 0,09 2 2 0,09 퐹퐵 is a near soft subset of 퐺퐵, denoted by 퐹퐵 ⊆ 퐺퐵if 푁∗(퐹퐵) ⊆ 푁∗(퐺퐵) for all 휙 ∈ 퐵, 휙2 0,01 0,01 1 1 1 i.e., 푁∗(퐹(휙), 퐵) ⊆ 푁∗(퐺(휙), 퐵) for all 휙 ∈ 퐵. 휙3 2 2 3 3 0

퐹퐵 is called a near soft superset of 퐺퐵; 휙4 0,09 0,09 2 2 2 denoted by 퐹퐵 ⊇ 퐺퐵; if 퐺퐵 is a near soft subset of 퐹퐵 (Tasbozan et al., 2017). Table 1

Definition 2.7 Let 퐹퐵, 퐺퐵 ∈ 푁푆푆(풪퐵). If near Let 휎 = 퐹 , 퐵 = {휙 , 휙 , 휙 } be a soft set 퐵 1 2 3 soft sets 퐹퐵 and 퐺퐵 are subsets of each other, defined by then they are called equal, denoted by 퐹퐵 = 퐺퐵 (Ozkan, 2019). 퐹퐵 = {(휙1, {푎2, 푎3}), (휙2, {푎2, 푎4, 푎5}),

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Near Soft Connected Spaces

Definition 2.8 Let 풪 be an initial universe set Definition 2.10 The relative complement of a 푐 and 퐹 be a universe set of parameters. near soft set 퐹퐵 denoted by 퐹퐵, is defined 푐 Let 퐹퐵, 퐺퐵 ∈ 푁푆푆(풪퐵) and 퐴, 퐵 ⊂ ℱ. where 퐹 (휙) = 풪 − 퐹(휙) for all 휙 ∈ 퐵 (Tasbozan et al., 2017). 1) The extended intersection of 퐹퐴 and 퐺퐵 over 풪 is the near soft set 퐻퐶, where Definition 2.11 Let 퐹퐵 ∈ 푁푆푆(풪퐵). The 퐶 = 퐴 ∪ 퐵, and ∀휙 ∈ 퐶, 풫(퐹퐵) set is defined as the near soft power set of 퐹퐵 as follows: 퐻(휙) 푖 푖 퐹(휙) 𝑖푓 휙 ∈ 퐴 − 퐵 풫(퐹퐵) = {퐹퐵: 퐹퐵 ⊆ 퐹퐵, 𝑖 ∈ 퐼 ⊆ ℕ} = {퐺(휙) 𝑖푓 휙 ∈ 퐵 − 퐴 ∑휙∈퐵|퐹(휙)| 퐹(휙) ∩ 퐺(휙) 𝑖푓휙 ∈ 퐴 ∩ 퐵 We use the formula |풫(퐹퐵)| = 2 to find the number of elements of this near soft We write 퐹퐴 ∩ 퐺퐵 = 퐻퐶. power set. Where |퐹(휙)| is the cardinality of 퐹(휙). 2) The restricted intersection of 퐹퐴 and 퐺퐵 over 풪 is the near soft set 퐻퐶, where Example 2.12 Let = { 푎1, 푎2, 푎3, 푎4, 푎5} ⊃ 퐶 = 퐴 ∩ 퐵, and 퐻(휙) = 퐹(휙) ∩ 퐺(휙) for 푋 = {푎1, 푎2, 푎5}, 퐵 = {휙1, 휙3} ⊂ ℱ = all 휙 ∈ 퐶. We write 퐹퐴 ∩ 퐺퐵 = 퐻퐶. { 휙1, 휙2, 휙3, 휙4}. According to Table 1;

3) The extended union of 퐹퐴 and 퐺퐵 Let 휎 = 퐹퐵, 퐵 = {휙1, 휙2} be a soft set over 풪 is the near soft set 퐻퐶, where 퐶 = 퐴 ∪ defined by 퐵, and ∀휙 ∈ 퐶, 퐹퐵 = {(휙1, {푎1, 푎2}), (휙2, {푎2, 푎5})}. Since 퐻(휙) 퐵푛푑푁푟(퐵)푋(휎) ≥ 0, so 휎 is a near soft set. 퐹(휙) 𝑖푓 휙 ∈ 퐴 − 퐵 Then, since |퐹(휙1)| = 2 and |퐹(휙2)| = 2, = {퐺(휙) 𝑖푓 휙 ∈ 퐵 − 퐴 (2+2) |풫(퐹퐵)| = 2 = 16. All subsets of 퐹퐵 are 퐹(휙) ∪ 퐺(휙) 𝑖푓휙 ∈ 퐴 ∩ 퐵 given below.

We write 퐹퐴 ∪ 퐺퐵 = 퐻퐶. 1 퐹퐵 = {(휙1, {푎1})},

4) The restricted union of 퐹퐴 and 퐺퐵 2 퐹퐵 = {(휙1, {푎2})}, over 풪 is the near soft set 퐻퐶, where 퐶 = 퐴 ∩ 3 퐵, and 퐻(휙) = 퐹(휙) ∪ 퐺(휙) for all 휙 ∈ 퐶. 퐹퐵 = {(휙1, {푎1, 푎2})},

4 We write 퐹퐴 ∪ 퐺퐵 = 퐻퐶 (Ozkan, 2019). 퐹퐵 = {(휙2, {푎2})},

5 Definition 2.9 Let 퐹퐵 ∈ 푁푆푆(풪퐵). Then 퐹퐵 퐹퐵 = {(휙2, {푎5})}, is called: 6 퐹퐵 = {(휙2, {푎2, 푎5})}, 1) a null near soft set, denoted by ∅푁 7 if 퐹(휙) = ∅, for all 휙 ∈ 퐵. 퐹퐵 = {(휙1, {푎1}), (휙2, {푎2})},

8 2) a whole near soft set, denoted by 퐹퐵 = {(휙1, {푎1}), (휙2, {푎5})}, 풪 if 퐹(휙) = 풪, for all 휙 ∈ 퐵 (Tasbozan et 푁 9 al., 2017). 퐹퐵 = {(휙1, {푎1}), (휙2, {푎2, 푎5})},

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10 퐹퐵 = {(휙1, {푎2}), (휙2, {푎2})}, are called discrete near soft and near soft indiscrete topology, respectively. 11 퐹퐵 = {(휙1, {푎2}), (휙2, {푎5})}, Example 2.17. If we consider the near soft 12 퐹퐵 = {(휙1, {푎2}), (휙2, {푎2, 푎5})}, sets given in the Example 2.12, then 휏 = 1 5 8 9 11 14 13 {∅푁, 퐹퐵, 퐹퐵 , 퐹퐵 , 퐹퐵 , 퐹퐵 , 퐹퐵 , 퐹퐵 } is a near 퐹퐵 = {(휙1, {푎1, 푎2}), (휙2, {푎2})}, soft topology on 퐹퐵. 퐹14 = {(휙 , {푎 , 푎 }), (휙 , {푎 })}, 퐵 1 1 2 1 5 Definition 2.18 Let (풪, 퐵, 휏) be a nsts and

15 16 퐹퐵 ∈ 푁푆푆(풪퐵). Then: 퐹퐵 = ∅푁, 퐹퐵 = 퐹퐵.

1) The set ∩ {퐺퐵 ⊇ 퐹퐵: 퐺퐵 is a near Definition 2.13 Let 퐹퐵 ∈ 푁푆푆(풪퐵). A near soft closed set of 풪퐵} is called the near soft soft topology on 퐹퐵, denoted by 휏, is a closure of 퐹퐵 in 풪퐵, denoted by 푐푙푛(퐹퐵). collection of near soft subsets of 퐹퐵, 퐵 is the non-empty set of parameters, and then 풪퐵 is 2) The set ∪ {퐶퐵 ⊆ 퐹퐵: 퐶퐵 is a near said to be a near soft topology on 퐹퐵 if the soft open set of 풪퐵} is called the near soft following properties are satisfied: interior of 퐹퐵 in 풪퐵, denoted by 𝑖푛푡푛(퐹퐵) (Ozkan, 2019). 1) ∅푁, 퐹퐵 ∈ 휏 where ∅(휙) = ∅ and 퐹(휙) = 퐹, for all 휙 ∈ 퐵. Definition 2.19 Let 퐹퐵 ∈ 푁푆푆(풪퐵). If for the 2) The intersection of any two near element 휙 ∈ 퐵, 퐹(휙) ≠ ∅ and 퐹(휙′) = ∅ for soft sets in 휏 belongs to 휏. all 휙′ ∈ 퐵 − {휙}, then 퐹퐵 is called a near soft 푁 point in 풪퐵, denoted by 휙퐹 (Ozkan, 2019). 3) The union of any number of near soft sets in 휏 belongs to 휏. Definition 2.20 Let (풪, 퐵, 휏) be a nsts and 퐺퐵 ∈ 푁푆푆(풪퐵). If there exists a near soft 푁 The triplet (풪, 퐵, 휏) is said to be a near soft open set 퐻퐵 such that 휙퐹 ∈ 퐻퐵 ⊆ 퐺퐵, then (Tasbozan et al., 2017). 퐺퐵 is called a near soft neighbourhood (written near soft nbd) of the near soft point We type nsts in place of near soft topological 푁 휙퐹 ∈ 풪퐵 (Ozkan, 2019). space. Definition 2.21 Let (풪, 퐵, 휏) be a nsts and Definition 2.14 Let (풪, 퐵, 휏) be a nsts over 퐺퐵 ∈ 푁푆푆(풪퐵). If there exists a near soft 풪퐵, then the members of 휏 are said to be near open set 퐻퐵 such that 퐹퐵 ⊆ 퐻퐵 ⊆ 퐺퐵, then soft open sets 풪 (Tasbozan et al., 2017). 퐺퐵 is called a near soft nbd of the near soft set 퐹 (Ozkan, 2019). Definition 2.15 Let (풪, 퐵, 휏) be a nsts over 퐵

풪퐵. 퐴 near soft subset of 풪퐵 is called near Definition 2.22 Let 푁푆푆(풪퐴) and 푁푆푆(푉퐵) soft closed if its complement is open and a be the families of all near soft sets over 풪 member of 휏 (Tasbozan et al., 2017). and 푉, respectively. The mapping 푓 is called a near soft mapping from 풪 to 푉, denoted by Example 2.16. If we consider the near soft 푓: 푁푆푆(풪 ) → 푁푆푆(푉 ), where 푢: 풪 → 푉 sets given in the Example 2.12, then 휏 = 퐴 퐵 and 푝: 퐴 → 퐵 are two mappings. 풫(퐹퐵) and 휏 = {∅푁, 퐹퐵} sets are a near soft topology on 퐹퐵. These near soft

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푁 1) Let 퐹퐴 be a near soft set in 1) 푓 is near soft continuous at 휙퐹 ∈ 푁 푁푆푆(풪퐴). Then for all 휔 ∈ 퐵 the image of 풪퐴 if for each 퐺퐵 ∈ 푁휏∗(푓(휙퐹 )), there exists 푁 퐹퐴 under 푓, written as 푓(퐹퐴) = a 퐻퐴 ∈ 푁휏(휙퐹 ) such that 푓(퐻퐴) ⊆ 퐺퐵. (푓(퐹), 푝(퐴)), is a near soft set in 푁푆푆(푉퐵) defined as follows: 2) 푓 is near soft continuous on 풪퐴 if 푓 is near soft continuous at each near soft point

푓(퐹)(휔) in 풪퐴 (Ozkan, 2019). ⋃ 푢(퐹(휙)), 푝−1(휔) ∩ 퐴 ≠ ∅ Theorem 2.25 Let (풪, 퐴, 휏) and (푉, 퐵, 휏∗) be = {휙휖푝−1(휔)∩퐴 two nstss. For a function 푓: 푁푆푆(풪 ) → ∅, 표푡ℎ푒푟푤𝑖푠푒. 퐴 푁푆푆(푉퐵), the following are equivalent:

2) Let 퐺퐵 be a near soft set in 1) 푓 is near soft continuous; 푁푆푆(푉퐵). Then for all 휙 ∈ 퐴 the near soft inverse image of 퐺 under 푓, written as 퐵 2) ∀퐹 ∈ 푁푆푆(풪 ), and the inverse −1 −1 −1 퐴 퐴 푓 (퐺퐵) = (푓 (퐺), 푝 (퐵)), is a near soft image of every near soft nbd of 푓(퐹퐴) is a set in 푁푆푆(풪퐴) defined as follows: near soft nbd of 퐹퐴; 푓−1(퐺)(휙) 3) ∀퐹퐴 ∈ 푁푆푆(풪퐴) and for each near 푢−1 (퐺(푝(휙))) , 푝(휙) ∈ 퐵 = { soft nbd 퐻퐵 of 푓(퐹퐴), there is a near soft nbd ∅, 표푡ℎ푒푟푤𝑖푠푒 퐺퐴 of 퐹퐴 such that 푓(퐺퐴) ⊆ 퐻퐵 (Ozkan, 2019). (Ozkan, 2019). Definition 2.26 Let (풪, 퐴, 휏) and (푉, 퐵, 휏∗) Definition 2.23 A near soft map be two nstss and 푓: 풪 → 푉. 푓 be a near soft 푓: 푁푆푆(풪퐴) → 푁푆푆(푉퐵) is said to be mapping on 풪퐴. If the image of each near injective (resp. surjective, bijective) if 푓 and soft open (resp. near soft closed) set in 풪퐴 is 푝 are injective (resp. surjective, bijective). a near soft open (resp. near soft closed) set in 푉 , then 푓 is called a near soft open (resp. Throughout the paper, the space 푂 and 푉 퐵 near soft closed) function (Ozkan, 2019). stand for near soft topological spaces with ∗ the assumed ((풪, 퐴, 휏) and (푉, 퐵, 휏 )) unless 3. Near Soft Connected Spaces otherwise stated and a near soft mapping 푓: 풪 → 푉 stands for a mapping, where We introduce the notions of near soft sets, 푓: (풪, 퐴, 휏) → (푉, 퐵, 휏∗), 푢: 풪 → 푉, and their essential features, and actions like near 푝: 퐴 → 퐵 are assumed mappings unless soft disjoint sets, near soft separated sets and otherwise stated and 퐴, 퐵 ⊂ ℱ. near soft connected spaces.

Definition 2.24 Let (풪, 퐴, 휏) and (푉, 퐵, 휏∗) Thanks to the following definition, be two nstss. Let 푢 be a mapping from 풪 to 푉 the theoretical studies of the near soft set will and 푝 be a mapping from 퐴 to 퐵. Let 푓 be a be accelerated and applications will be more mapping from 푁푆푆(풪퐴) to 푁푆푆(푉퐵) and comfortable. 푁 휙퐹 ∈ 풪퐴. Then: Definition 3.1 Two near soft sets 퐹퐵 and 퐺퐵 over 풪퐵 are near soft disjoint sets, if 퐹퐵 ∩ 퐺퐵 = ∅푁.

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The near soft disjoint sets denoted by 퐹퐵|퐺퐵. Conversely, let 퐹퐵, 퐺퐵 are both near soft Example 3.2 Let (풪, 퐵, 휏) be a nsts. Then for disjoint sets and near soft closed sets then so 푐 each 퐹퐵 ∈ 푁푆푆(풪퐵), 퐹퐵|퐹퐵. that 푐푙푛(퐹퐵) ∩ 퐺퐵 = ∅푁 and 퐹퐵 ∩ 푐푙푛(퐺퐵) = Definition 3.3 Let (풪, 퐵, 휏) be a nsts. Two ∅푁. Showing that 퐹퐵, 퐺퐵 are near soft non-null subsets 퐹퐵, 퐺퐵 over 풪퐵 are said to separated. If 퐹퐵, 퐺퐵 are both near soft 푐 be near soft separated sets of 풪퐵 if 퐹퐵 ∩ disjoint sets and near soft open sets then 퐹퐵, 푐 푐 푐푙푛(퐺퐵) = ∅푁 and 푐푙푛(퐹퐵) ∩ 퐺퐵 = ∅푁. 퐺퐵 are both near soft closed. Then 퐹퐵 ⊆ 퐺퐵, 푐 푐 푐 Example 3.4 According to the 휏 topology on 퐺퐵 ⊆ 퐹퐵 , 푐푙푛(퐹퐵) ⊆ 푐푙푛(퐺퐵) = 퐺퐵 and 푐 푐 퐹퐵 given in Example 2.17, and hence 푐푙푛(퐺퐵) ⊆ 푐푙푛(퐹퐵) = 퐹퐵. Then 푐푙푛(퐹퐵) ∩ (풪, 퐵, 휏) is a nsts. Clearly, the near soft 퐺퐵 = ∅푁 and 푐푙푛(퐺퐵) ∩ 퐹퐵 = ∅푁, so 퐹퐵 and closed sets 퐺퐵 are near soft separated sets. 1 푐 5 푐 8 푐 9 푐 11 푐 14 푐 ∅푁, 퐹퐵, (퐹퐵 ) , (퐹퐵 ) , (퐹퐵 ) , (퐹퐵 ) , (퐹퐵 ) , (퐹퐵 ) Definition. 3.9 Let (풪, 퐵, 휏) be a nsts and Then, let us take 퐻퐵 = {(휙1, {푎1})}; then 퐹퐵 ∈ 푁푆푆(풪퐵). Then 퐹퐵 is said to be near 1 푐 푐푙(퐻퐵) = (퐹퐵 ) and 퐺퐵 = soft connected set, if there does not exist a {(휙1, {푎2}), (휙2, {푎5})}; then 푐푙(퐺퐵) = pair 퐺퐵 and 퐻퐶 of non-null near soft disjoint 11 푐 (퐹퐵 ) and 퐻퐵 ∩ 푐푙푛(퐺퐵) = ∅푁 and subsets of (풪, 퐵, 휏) such that 퐹퐵 = 퐺퐵 ∪ 퐻퐶 푐푙푛(퐻퐵) ∩ 퐺퐵 = ∅푁, so 퐻퐵 and 퐺퐵 are near and 퐺퐵 ∩ 퐻퐶 = ∅푁, otherwise 퐹퐵 is said to soft separated sets and also near soft disjoint be near soft disconnected set. sets of 풪퐵. In above definition, if we take 풪퐵 instead of 퐹퐵, then the (풪, 퐵, 휏) is called near soft Remark 3.5 In a nsts any two near soft connected (disconnected) space. separated sets are near soft disjoint sets but We write non-empty near soft set instead of two near soft disjoint sets are not necessary non-null near soft set. for soft separated sets. Example 3.10 From Example 2.17 퐹퐵 =

Example 3.6 According to the 휏 topology on {(휙1, {푎2, 푎3}), (휙2, {푎2, 푎4, 푎5}), (휙3, {푎2, 푎3, 푎4, 푎5}) 퐹퐵 given in Example 2.17, is a near soft set, 퐺퐴 = 퐻퐵 = {(휙1, {푎1}), (휙2, {푎2, 푎5})}, 휉{(휙1, {푎2}), (휙2, {푎2, 푎4}), (휙3, {푎3, 푎4})} 퐺퐵 = {(휙1, {푎5}), (휙2, {푎1})} and 퐿푃 = {(휙1, {푎3}), (휙2, {푎5}), (휙3, {푎2, 푎5})} are so 퐻퐵 and 퐺퐵 are near soft disjoint sets and near soft disjoint subsets according to Table but not near soft separated of 풪 . 퐵 1. Then 퐹퐵 = 퐺퐴 ∪ 퐿푃 and 퐺퐴 ∩ 퐿푃 = ∅푁. Example 3.7 Although each near soft set is Hence 퐹퐵 is a near soft disconnected set. disjoint with its own complement, it may not be a near soft separated set. Example 3.11 Each near soft indiscrete The following theorem gives us in space is near soft connected, and that each which case the near soft disjoint set can be a near soft discrete non-trivial space is not near near soft separated set. soft connected. Theorem 3.8 Two near soft closed (near soft The next result gives an equivalent open) subsets FB, GB of a nsts (풪, B, τ) are near soft separated sets if and only if they are formulation of near soft connectedness. near soft disjoint sets. Theorem 3.12 Let (풪, B, τ) be a nsts and Proof. It’s clear that any two near soft F ∈ NSS(풪 ). F is a near soft connected separated sets are near soft disjoint sets. B B B

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set if and only if FB cannot be written as a The following technical theorem and union of two near soft disjoint sets. its corollary are very useful in working with

Proof. If 퐹퐵 = ∅푁, it is clear. So let’s be near soft connectedness in near soft 퐹퐵 ≠ ∅푁. Let 퐹퐵 be a near soft connected set subspaces. and near soft disjoint sets of 퐺퐵|퐻퐶 meet the Theorem 3.14 Let (풪, 퐵, 휏) be a nsts over 풪퐵 퐹퐵 ⊆ 퐺퐵 ∪ 퐻퐶 condition. and 푉 be a non-empty subset of 풪. If 퐹퐵 and 퐺퐵 ∩ 퐻퐶 ⊆ 퐺퐵 ∩ 푐푙푛(퐻퐶) = ∅푁 and 퐺퐵 ∩ 퐺퐵 are near soft sets in 푉퐵, then 퐹퐵 and 퐺퐵 퐻퐶 ⊆ 푐푙푛(퐺퐵) ∩ 퐻퐶 = ∅푁. are near soft separation sets of 푉퐵 if and only It is obtained from the coverage that the 퐹퐵 is if 퐹퐵 and 퐺퐵 are near soft separation sets of near soft disconnected. But this situation 풪퐵. contradicts by hypothesis. So 퐹퐵 cannot be written as a union of two near soft disjoint Proof. From Lemma 3.13, we have 푉퐵 풪B sets. 푐푙푛 (퐺퐵) = 푉퐵 ∩ 푐푙푛 (퐺퐵) = ∅푁, so 푉퐵 풪B 푐푙푛 (퐺퐵) ∩ 퐹퐵 = ∅푁 iff 퐹퐵 ∩ 푐푙푛 (퐺퐵 ∩ Conversely, suppose that 퐹 should not be 퐵 푉 ) = ∅ iff (퐹 ∩ 푉 ) ∩ 푐푙풪B(퐺 ) = ∅ iff written as a union of two near soft connected 퐵 푁 퐵 퐵 푛 퐵 푁 풪B( ) sets, but it is a near soft disconnected. Then, (푐푙푛 퐺퐵 ∩ 퐹퐵) = ∅푁. there are 퐺 and 퐻 in 풪 , such that 퐹 = 퐵 퐶 퐵 퐵 Similar, we have 퐺퐵 ∪ 퐻퐶 and 퐺퐵 ∩ 퐻퐶 = ∅푁. 퐺퐵 ∩ 푐푙푛(퐻퐶) = 푐푙푛(퐺퐵) ∩ 퐻퐶 = ∅푁 is obtained 푉퐵 풪B 푐푙푛 (퐹퐵) ∩ 퐺퐵 = (푐푙푛 (퐹퐵) ∩ 푉퐵) ∩ 퐺퐵 due to the closure of a near soft closed set is = (푐푙풪B(퐹 ) ∩ 퐺 ) itself. This contradicts by hypothesis. 푛 퐵 퐵 Thus, the theorem holds. Lemma 3.13 Suppose then FB ⊆ GB ⊆ 퐺퐵 풪B (풪, 퐵, 휏), then 푐푙푛 (퐹퐵) = 퐺퐵 ∩ 푐푙푛 (퐹퐵). Attention: According to Theorem 3.14, 퐻B 퐺퐵 (푐푙푛 (퐹퐵): near soft closure of 퐹퐵 in 퐺퐵 and is near soft disconnected iff 퐻B = FB ∪ GB 푐푙풪퐵 (퐹 ): near soft closure of 퐹 in 풪 .) 푛 퐵 퐵 퐵 where FB and GB are non-empty near soft separated sets in 퐻B iff 퐻B = FB ∪ GB where 풪B Proof. 퐺퐵 ∩ 푐푙푛 (퐹퐵) is a near soft closed set FB and GB are non-empty near soft separated in GB that contains 퐹B, so sets in 풪B. Theorem 3.14 is very useful 퐺퐵 풪B 푐푙푛 (퐹퐵) ⊆ 퐺퐵 ∩ 푐푙푛 (퐹퐵) …………(1). because it means that we don't have to On the other hand, suppose 푎 ∈ 퐺퐵 ∩ distinguish here between “near soft separated 풪B GB 푐푙푛 (퐹퐵). To show that 푎 ∈ 푐푙푛 (퐹퐵), pick a in HB” and “near soft separated in 풪B”- near soft open set 퐻퐵 in 퐺퐵 in that contains because these are equivalent. 푎. We need to show 퐹퐵 ∩ 퐻퐵 ≠ ∅푁. There is In contrast, if we say that 퐻B is near soft a near soft open set 퐿퐵 in 풪B such that 퐿퐵 ∩ disconnected when 퐻B is the union of two 풪B near soft disjoint sets, non-empty near soft GB = 퐻퐵. Since 푎 ∈ 푐푙푛 (퐹퐵), we have that open (or closed) sets FB, GB in 퐻B, then ∅푁 ≠ 퐿퐵 ∩ 퐹퐵 = 퐿퐵 ∩ (퐺퐵 ∩ 퐹퐵) = phrase “in 퐻B” cannot be omitted: the sets (퐿퐵 ∩ 퐺퐵) ∩ 퐹퐵 = 퐻퐵 ∩ 퐹퐵. Thus, GB FB, GB, might not be near soft open (or 푎 ∈ 푐푙푛 (퐹퐵) ...... (2). closed) in 풪B. As a result, 푐푙퐺퐵(퐹 ) = 퐺 ∩ 푐푙풪B(퐹 ) is 푛 퐵 퐵 푛 퐵 The following lemma makes a simple obtained from (1) and (2). but very useful observation.

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Lemma 3.15 Let (풪, 퐵, 휏) be a nsts over 푂퐵, Proof. 1)⇒2). Let (풪, 퐵, 휏) have a near soft and 푉 is a non-empty subset of 풪 such that separation 퐹퐵 and 퐺퐵. Then 퐹퐵 ∩ 퐺퐵 = ∅푁 (푉, 퐵, 휏∗) is a near soft connected space. If and

퐹퐵 and 퐺퐵 are a near soft separation of 풪퐵 푐푙푛(퐹퐵) = 푐푙푛(퐹퐵) ∩ (퐹퐵 ∪ 퐺퐵) = such that 푉퐵 ⊆ 퐹퐵 ∪ 퐺퐵, then 푉퐵 ⊆ 퐹퐵 or (푐푙푛(퐹퐵) ∩ 퐹퐵) ∪ (푐푙푛(퐹퐵) ∩ 퐺퐵) = 퐹퐵. 푉퐵 ⊆ 퐺퐵.

Hence, 퐹퐵 is a near soft closed set in 풪퐵. Proof. Since 푉퐵 ⊆ 퐹퐵 ∪ 퐺퐵, we have 푉퐵 = Similar, we can see that 퐺퐵 is also a near soft (푉퐵 ∩ 퐹퐵) ∪ (푉퐵 ∩ 퐺퐵). Theorem 3.14, 푉퐵 ∩ closed set in 풪퐵. 퐹퐵 and 푉퐵 ∩ 퐺퐵 are a near soft separation of 푉 . Since (푉, 퐵, 휏∗) is a near soft connected 퐵 2)⇒3). Let (풪, 퐵, 휏) has a near soft space, we have 푉퐵 ∩ 퐹퐵 = ∅푁 or 푉퐵 ∩ 퐺퐵 = separation 퐹퐵 and 퐺퐵 such that 퐹퐵 and 퐺퐵 are ∅푁. Thus, 푉퐵 ⊆ 퐹퐵 or 푉퐵 ⊆ 퐺퐵. 푐 푐 near soft closed in 풪퐵. Then 퐹퐵 and 퐺퐵 are Near soft connectedness in the near soft sets in 풪퐵. Then it is easy to see that 푐 푐 푐 푐 subspace can also be characterized by the 퐹퐵 ∩ 퐺퐵 = ∅푁 and 퐹퐵 ∪ 퐺퐵 = 풪퐵. following theorem. 3)⇒4). Let (풪, 퐵, 휏) has a near soft separation 퐹 and 퐺 such that 퐹 and 퐺 are Theorem 3.16 Let (풪, 퐵, 휏) be a nsts over 푂퐵 퐵 퐵 퐵 퐵 푐 푐 and (푉, 퐵, 휏∗) be a near soft connected near soft open in 풪퐵. Then 퐹퐵 and 퐺퐵 are near soft sets in 풪 . Then 퐹 and 퐺 are also subspace of (풪, 퐵, 휏). If 푉퐵 ⊆ 퐴퐵 ⊆ 푐푙푛(푉퐵), 퐵 퐵 퐵 near soft closed in 풪 . then (퐴, 퐵, 휏퐴) is a near soft connected 퐵 subspace of (풪, 퐵, 휏). In particular, 푐푙 (푉 ) 푛 퐵 4)⇒1). Let (풪, 퐵, 휏) has a proper near soft is a near soft connected subspace of (풪, 퐵, 휏). open and near soft closed 퐹퐵 in 풪퐵. Put 퐻퐵 = 퐹푐. Then 퐻 and 퐹 are non-empty near soft Proof. Let 퐺퐵 and 퐿퐵 be a near soft 퐵 퐵 퐵 closed set in 풪 , 퐹 ∩ 퐻 = ∅ and 퐹 ∪ separation sets of (퐴, 퐵, 휏퐴). By Lemma 3.15, 퐵 퐴 퐵 푁 퐴 퐻 = 풪 . Hence, 퐻 and 퐹 is a near soft we have 퐹퐵 ⊆ 퐺퐵 or 퐹퐵 ⊆ 퐿퐵. Without loss 퐵 퐵 퐵 퐵 separation of 풪 . of generality, we may assume that 퐹퐵 ⊆ 퐺퐵. 퐵 By Theorem 3.14, we have 푐푙 (퐺 ) ∩ 퐿 = 푛 퐵 퐵 Theorem 3.18 Let (풪, B, τ) be a nsts over ∅푁, which is a contradiction. 풪퐵. (풪, B, τ) is a near soft connected (respt. Theorem 3.17 Let (풪, 퐵, 휏) be a nsts. Then near soft disconnected) space if and only if the following provisions are equivalent: there does not exists (respt. exist) non-empty near soft set; which is both near soft open 1) (풪, 퐵, 휏) has a near soft separation, and near soft closed in (풪, B, τ).

Proof. Let (풪, 퐵, 휏) be a nsts over 풪퐵. Let us 2) There exist two near soft closed disjoint assume that there is both a near soft open sets 퐹 and 퐺 such that 퐹 ∪ 퐺 = 풪 , 퐵 퐵 퐵 퐵 퐵 and a near soft closed set of 퐹퐵 in (풪, 퐵, 휏). Then 퐹 ∪ 퐹푐 = 풪 and 퐹 ∩ 퐹푐 = ∅ . 3) There exist two near soft open disjoint sets 퐵 퐵 퐵 퐵 퐵 푁 Since 퐹 is a near soft closed set 퐹푐 ∈ 풪 퐹 and 퐺 such that 퐹 ∪ 퐺 = 풪 , 퐵 퐵 퐵. 퐵 퐵 퐵 퐵 퐵 This contradicts (풪, 퐵, 휏) being near soft 4) (풪, 퐵, 휏) has a proper near soft open and connected space. Conversely, let there does not exist both near near soft closed set in 풪퐵. soft open and near soft closed set in

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−1 −1 (풪, 퐵, 휏). Nevertheless, let (풪, 퐵, 휏) be near 퐹퐵 ⊆ 푓 (푓(퐹퐵)) ⊆ 푓 (퐺퐵 ∪ 퐻퐶) −1 −1 soft disconnected space. = 푓 (퐺퐵) ∪ 푓 (퐻퐶) In this case, there exist a pair 퐺퐵 and 퐻퐶 of and −1 −1 −1 non-empty near soft disjoint subsets of 푓 (퐺퐵 ∩ 퐻퐶) = 푓 (퐺퐵) ∩ 푓 (퐻퐶) = ∅푁. (풪, 퐵, 휏) such that 퐹퐵 = 퐺퐵 ∪ 퐻퐶 and 퐺퐵 ∩ Since 푓 is a near soft , −1 −1 퐻퐶 = ∅푁. 푓 (퐺퐵), 푓 (퐻퐶) ∈ 풪퐴. Therefore, this 푐 If 퐹퐵 = 퐺퐵 is taken from these two situation contradicts with the near soft equations, a situation that contradicts the connected of the 퐹퐵. hypothesis occurs. So (풪, 퐵, 휏) is near soft Remark 3.21 In Theorem 3.20, if we take

connected space. the 풪A instead of FB and if we take 푓 onto Note: By the Theorem 3.18, the near soft mapping, whereas (풪, A, τ) nsts is near soft topological space in Example 2.14 is a near connected, (V, B, 휏∗) nsts is also near soft soft disconnected space since the near soft set connected space. 퐹퐵 is near soft open and near soft closed in Remark 3.22 A inverse image of a near soft 풪퐵. connected space under a near soft continuous function does not need to be near soft Theorem 3.19 Let (풪, B, τ) be a near soft connected. connected space and σ ⊆ τ. Then, (풪, B, σ) is a near soft connected space. Example 3.23 Let (풪, A, τ) and (V, B, 휏∗) be

two nstss, GB ∈ ΝSS(풪A) and 푓: 풪 → 푉 is a Proof. 퐹퐵, 퐺퐵 ∈ 휎 cannot be found such that near soft continuous function, where 휏 = 퐹퐵 = 퐺퐵 ∪ 퐻퐶 and 퐺퐵 ∩ 퐻퐶 = ∅푁, since it is ∗ 풫(퐹퐴) and 휏 is a single element near soft a near soft connected space. Since 휎 ⊆ 휏 is −1 space. In this case, 푓 (GB) is not near soft near soft open sets that satisfy these two connected. conditions are not in the 휎 family. Hence

(풪, 퐵, 휎) is a near soft connected space. Theorem 3.24 If FB is a near soft subset in nsts (풪, B, τ) that provides FB ⊆ GB ⊆ Under a continuous function, the cln(FB), GB is near soft connected, below theorem indicates that near soft specifically cln(FB) near soft connected. connectedness in maintained. Proof. Let 퐻퐵 be both a near soft open and a near soft closed subset in 퐺 that is non- Theorem 3.20 Let (풪, A, τ) and (V, B, 휏∗) be 퐵 empty. Since 퐺퐵 ⊆ 푐푙푛(퐹퐵), 퐻퐵 contains at two nstss, FB ∈ ΝSS(풪A) and 푓: 풪 → 푉 be a least one point of 퐹 . Because every point of near soft continuous function. If FB is a near 퐵 퐺 is a point of 푐푙 (퐹 ) and 퐻 is a near soft soft connected set, then 푓(퐹퐵) ∈ 훮푆푆(푉퐵) is 퐵 푛 퐵 퐵 a near soft connected set. open set. Similarly, if 퐻퐵 is non-empty, cuts 퐹퐵. 퐻퐵 ∩ 퐹퐵 is both near soft open and near Proof. Suppose that 퐹퐵 ∈ 훮푆푆(풪퐴) is a near soft closed, and non-empty. From Theorem soft connected set and 푓(퐹퐵) is a near soft 3.18, then 퐻퐵 ∩ 퐹퐵 = 퐹퐵. For this reason 푐 disconnected set. Then, there exist two near 퐹퐵 ⊆ 퐻퐵. Thus 퐻퐵 ∩ 퐹퐵 = ∅푁, hence 퐻퐵 = soft sets open sets 퐺퐵 and 퐻퐶 such that 퐹퐵 = 퐺퐵. According to Theorem 3.18, 퐺퐵 is near 퐺퐵 ∪ 퐻퐶 and 퐺퐵 ∩ 퐻퐶 = ∅푁. Therefore, from soft connected. This completes the proof. Theorem 2.25

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Near Soft Connected Spaces

Corollary 3.25 If FB is a near soft connected that consist of all the elements equal to each subspace of a nsts (풪, B, τ), then cln(FB) is a other. near soft connected. Now let's define the quotient space Theorem 3.30 The inverse image of each with near soft sets. subset of the quotient near soft set under the near soft quotient mapping is always a Definition 3.26 Let (풪, A, τ), (V, B, 휏∗) be saturated near soft set. nstss. Then (V, B, 휏∗) is said to be a near soft Proof: Now 푉 ⊂ 풪/~ ⇒ 푎 ∈ 푓−1(푉) and quotient space of (풪, A, τ), by denoted 퐵푟 ∗ 푎~퐵 푏 ⇒ 푓(푎) ∈ 푉 (풪/~퐵푟, 퐵, 휏 ) if there exists a surjective 푟 mapping 푓: 풪 → 푉 with the following 푓(푎) = 푓(푏) ⇒ 푏 ∈ 푓−1(푉) property (1).

∗ and this proves what is desired. For each subset 푈퐵 of 푉퐵, 푈퐵 ∈ 휏 ⟺ −1 푓 (푈퐵) ∈ 휏………(1) Conversely, if 푉 ⊂ 풪 is a saturated near soft set, there is a 푉 ⊂ 풪/~ to be 풪 = 푓−1(푉). A surjective mapping 푓 with above property 퐵푟 is said to be a near soft quotient mapping. Indeed, the set 푉 = 푓(풪) will provide the desired. Because Remark 3.27 From Definition 3.26 it is clear −1 that every near soft quotient mapping is a 풪 saturated⇒푓 (풪) ∘ 푓(풪) near soft continuous map. obtained. Also, since 푓 is a near soft Remark 3.28 Property (1) is near soft surjection, there will be equivalent to property (2). 푓(U) ∘ 푓−1(U) = 푈

For each subset 푈퐵 of 푉퐵, 푈퐵 is near soft ∗ −1 for every 푈 ⊂ 풪/~퐵푟 . closed in (풪/~퐵푟 , 퐵, 휏 ) ⟺ 푓 (푈퐵) is near soft closed in (풪, A, τ). …..(2) Using these features, we can say the following proposition. The 푓-quotient mapping defined from nsts (풪, B, τ) to near soft quotient space Proposition 3.31 Near soft open sets of near ∗ (풪/~퐵푟, 퐵, 휏 ) does not have to be a near soft soft quotient topology consists of images open (or near soft closed) mapping. But this under the near soft quotient mapping of mapping portrays some near soft open sets of saturated near soft open sets. space 풪 over near soft open sets of quotient Of course, the above proposition does space 풪/~퐵푟. Let's examine this feature. not mean that the near soft quotient mapping Definition 3.29 If an 푉 ⊂ 풪 subset has the is a near soft open mapping. Because we know that this mapping will depict saturated following properties (according to the ~퐵푟), then it is called a saturated near soft set: near soft open sets on near soft open sets. It may not depict any near soft open set over a near soft open set. Similarly, we can say that 푎 ∈ 풪 and 푏~퐵푟푎 ⇒ 푏 ∈ 풪. the near soft quotient mapping does not have So, a saturated near soft set is a near soft set to be a near soft closed mapping.

799

Near Soft Connected Spaces

Theorem 3.32 Each near soft quotient space 4. Conclusion of a near soft connected space is also a near soft connected space. This study introduces the notions of near soft connectedness, near soft connected Proof: Let (풪, B, τ) be a near soft connected topological spaces, near soft disjoint sets and some of their properties. In addition, it space, the correlation ~퐵푟 is an equivalence relation defined on 풪퐵. Since the near soft should be noted that the definitions and quotient mapping defined from (풪, B, τ) to theorems presented in this paper are accepted ∗ as a general theory for the near soft set (풪/~퐵푟, 퐵, 휏 ) is a near soft continuous function (see Remark 3.27), the desired thing theory. With this paper, it is considered to is obtained from Theorem 3.20. pave the way for new studies (theoretical or applications) as a result of defining the There is no opposite to this theorem, connectedness of near soft set theory. but there is the following feature. Moreover, the contribution of this study is that it provides a near approach to proximity Proposition 3.33 If (풪, B, τ) is a near soft as an ambiguous concept that can be topological space and according to the ~퐵푟 approximated to the state of objects and field relation, each of the equivalence classes is a knowledge. We foresee that problems of near soft connected set, then (풪, B, τ) space many fields that include uncertainties and will also be near soft connected. will provide further study on near soft topology to fulfil general skeleton for the Proof: If 풪 was not near soft connected, near applications in practical life can be consulted soft sets 퐹퐵 and 퐺퐵 would exist such that with the findings in this study. 풪퐵 = 퐹퐵 ∪ 퐺퐵, 퐹퐵 ∩ 퐺퐵 = ∅푁. According to the ~퐵푟, 퐹퐵 and 퐺퐵 are two saturated near Acknowledgment soft sets, because if there were 푎퐵 ∩ 퐺퐵 ≠ 푟 The writers are thankful to the anonymous ∅ for one 푎 ∈ 퐹 since 푎 would be 푁 퐵푟 퐵 퐵푟 arbiters and editor for the detailed and their equal to the combination of sets 푎 ∩ 퐹 and 퐵푟 퐵 helpful remarks that amended this study. 푎퐵푟 ∩ 퐺퐵, which are open and disjoint in subspace 푎퐵푟, the equivalence class 푎퐵푟 References could not be connected. Since there can be no Acar, U., Koyuncu, F. and Tanay B. 2010. contradiction, 퐹 and 퐺 are two near soft 퐵 퐵 “Soft Sets and Soft Rings”, Comp. Math. saturated sets. Thus, in accordance with Appl., 59, 3458-3463. Proposition 3.31, the 푓(퐹퐵) and 푓(퐺퐵) images of these two near soft sets are near Aktas, H. and Cağman, N. 2007. “Soft Sets soft open under the quotient mapping 푓. and Soft Groups”, Inform. Sciences, 177, 2726-2735. Also, these two near soft sets do not intersect. They are not empty and their Ali, I. M., Feng, F., Liu, X. Y., Min, W. K. composition is equal to 풪/~퐵푟. This is and Shabir, M., 2009. “On Some New contrary to the near soft connected of the Operations in Soft Set Theory”, Comp. Math. Appl. 57, 1547-1553. quotient space /~퐵푟. Since this contradiction does not exist, the 풪 space is near soft connected.

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