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788 Near Soft Connected Spaces Alkan ÖZKAN1* , Metin DUMAN1 1

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788 Near Soft Connected Spaces Alkan ÖZKAN1* , Metin DUMAN1 1 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 Spaces 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 set, 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 image analysis. 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 space. 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 789 Near Soft Connected Spaces 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 ∈ ℱ 퐵 푟 respect to 푁퐴푆, respectively, are denoted by 790 Near Soft Connected Spaces ∗ ∗ 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.
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