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Soft Separation Axioms and Fixed Soft Points Using Soft Semiopen Sets

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Soft Separation Axioms and Fixed Soft Points Using Soft Semiopen Sets Hindawi Journal of Applied Mathematics Volume 2020, Article ID 1746103, 11 pages https://doi.org/10.1155/2020/1746103 Research Article Soft Separation Axioms and Fixed Soft Points Using Soft Semiopen Sets T. M. Al-shami Department of Mathematics, Sana'a University, Sana'a, Yemen Correspondence should be addressed to T. M. Al-shami; [email protected] Received 1 August 2020; Revised 28 October 2020; Accepted 30 October 2020; Published 16 November 2020 Academic Editor: Bruno Carpentieri Copyright © 2020 T. M. Al-shami. 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. The importance of soft separation axioms comes from their vital role in classifications of soft spaces, and their interesting properties are studied. This article is devoted to introducing the concepts of tt-soft semi-Tiði =0,1,2,3,4Þ and tt-soft semiregular spaces with respect to ordinary points. We formulate them by utilizing the relations of total belong and total nonbelong. The advantages behind using these relations are, first, generalization of existing comparable properties on general topology and, second, eliminating the stability shape of soft open and closed subsets of soft semiregular spaces. By some examples, we show the relationships between them as well as with soft semi-Tiði =0,1,2,3,4Þ and soft semiregular spaces. Also, we explore under what conditions they are kept between soft topology and its parametric topologies. We characterize a tt-soft semiregular space and demonstrate that it guarantees the equivalence of tt-soft semi-Tiði =0,1,2Þ. Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces, and sum of soft topological spaces. Finally, we define a concept of semifixed soft point and study its main properties. 1. Introduction In 2011, Shabir and Naz [6] defined a soft topological space over a family of soft sets. They constructed the In daily life, human beings face different kinds of uncer- fundamental notions of soft topological spaces such as the tainties in fields such as economics, environmental and social operators of soft closure and interior, soft subspace, and soft sciences, engineering, medicine etc. To tackle such uncer- separation axioms. Min [7] continued studying soft separa- tainties, different approaches were proposed like fuzzy and tion axioms and corrected some alleged results in [6]. Then, rough set theories. However, they have their own limitations. Aygünoglu and Aygün [8] scrutinized the concept of com- Molodtsov [1] introduced an effective tool that is free from pactness on soft setting. By exploiting one of the diverges these limitations to solve uncertainties, namely soft set. This between soft sets and crisp sets, Hida [9] studied a strong tool does not require specific type of parameters. Instead, it type of soft compactness. Al-shami [10, 11] revised many accommodates all types of parameters such as words, num- results given in many studies of soft separation axioms. Then, bers, sentences, and functions. After that, Maji et al. [2] estab- Al-shami et al. [12] introduced a weak type of soft compact- lished some fundamental operations between two soft sets ness, namely almost soft compactness. The authors of [13] and then tackled one of their applications in decision- presented soft maps by using two crisp maps, one of them making problems [3]. But the authors of [4] demonstrated between the sets of parameters and the second one between some shortcomings of the operations defined in [2]. They the universal sets. However, the authors of [14] introduced also explored novel operations between soft sets which soft maps by using the concept of soft points. Some applica- became a goal of many studies on algebraic soft structures. tions of different types of soft maps were the goal of some Al-shami and El-Shafei [5] defined new types of soft subset articles (see [13–15]). and equality relations and then applied them on soft linear Until 2018, the belong and nonbelong relations that equations. utilized in these studies are those given by [6]. In 2018, the 2 Journal of Applied Mathematics authors of [16] came up new relations of belong and nonbe- In this study, we use a symbol GP to refer a soft set, and ∈ long between an ordinary points and soft set, namely partial we identify it as ordered pairs GP = fðp, GðpÞÞ: p P and belong and total nonbelong relations. In fact, these relations GðpÞ ∈ 2Xg. widely open the door to study and redefine many soft topo- A class of all soft sets defined over X with P is symbolized logical notions. This leads to obtain many fruitful properties by SðXPÞ. and changes which can be seen significantly on the study of – soft separation axioms as it was shown in [16 18]. fi Das and Samanta [19] studied the concept of a soft metric De nition 2. [33] GP is said to be a subset of HP, symbolized ⊆~ ⊆ ∈ based on the soft real set and soft real numbers given in [20]. by GP HP,ifGðpÞ HðpÞ for each p P. Wardowski [14] tackled the fixed point in the setup of soft ⊆~ ⊆~ topological spaces. Abbas et al. [21] presented soft contrac- We write GP = HP provided that GP HP and HP GP. tion mappings and established a soft Banach fixed point the- orem in the framework of soft metric spaces. Recently, many fi ∈ De nition 3. [6, 16] For a soft set GP and w X, we say that researchers explored fixed point findings in soft metric type – ∈ spaces (see, for example, [22 25]). Some results related to (i) w GP, it is read as follows: w totally belongs to GP, intuitionistic fuzzy soft sets were investigated in [26, 27]. if w ∈ GðpÞ for each p ∈ P Recently, Alcantud [28] has discussed the properties of topo- ∈ logical separability on soft setting. (ii) w/GP, it is read as follows: w does not partially ∈ ∈ One of the significant ideas that helps to prove some belong to GP,ifw/GðpÞ for some p P properties and removes some problems on soft topology is (iii) w ⋐ G , it is read as follows: w partially belongs to fi fi P the concept of a soft point. It was rst de ned by Zorlutuna G ,ifw ∈ GðpÞ for some p ∈ P and Çakir [29] in order to study the interior points of a soft P ⋐ set and soft neighborhood systems. Then, [19, 30] simulta- (iv) w/GP, it is read as follows: w does not totally belong ∈ ∈ neously redefined soft points to discuss soft metric spaces. to GP,ifw/GðpÞ for each p P. In fact, the recent definition of a soft point makes similarity between many set-theoretic properties and their counterparts fi on soft setting. Two types of soft topologies, namely enriched De nition 4. [4] The relative complement of GP is another soft topology and extended soft topology, were studied in c c : ⟶ X fi c soft set GP, where a map G P 2 is de ned by G ðpÞ = [8, 30], respectively. Al-shami and Kocinac [31] discussed X \ GðpÞ for all p ∈ P. the equivalence between these two topologies and obtained interesting results. Recently, they [32] have introduced the fi concept of nearly soft Menger spaces and investigated De nition 5. [2, 16, 19, 30] A soft set ðGPÞ over X is said to be main characteristics. We organized this paper as follows: After this introduc- (i) a null soft set (resp., an absolute soft set), symbolized Φ~ ~ tion, Section 2 addresses the basic principles about the soft by (resp., X), if GðpÞ is the empty (resp., universal) ∈ sets and soft topologies. In Section (, we introduce the con- set for each p P cepts of tt-soft semi-Tiði =0,1,2,3,4Þ and tt-soft semiregu- x ∈ ∈ (ii) a soft point Sp if there are p P and x X so that G lar spaces with respect to ordinary points by using the ′ ∅ ′ ∈ relations of total belong and total nonbelong. The relation- ðpÞ = fxg and Gðp Þ = for each p P \ fpg.We x ∈ ∈ ships between them and their main properties are discussed write that Sp GP if x GðpÞ with the help of interesting examples. In Section 4, we explore (iii) a stable soft set if all p-approximate are equal A and a semifixed soft point theorem and study some main proper- it is symbolized by A~. In particular, we symbolize by ties. In particular, we conclude under what conditions semi- x if A = fxg fixed soft points are preserved between a soft topology and its P parametric topologies. Finally, conclusions of the paper in (iv) a countable (resp., finite) soft set if GðpÞ is countable Section 5. (resp., finite) for each p ∈ P. Otherwise, it is said to be uncountable (resp., infinite). 2. Preliminaries fi To well understand the results obtained in this study, we shall De nition 6. [2, 4] The intersection and union of GP and HP mind some essential concepts, definitions, and properties are defined as follows: from the literature. ∩~ (i) Their intersection, symbolized by GP HP, is a soft : ⟶ X set UP, where a mapping U P 2 is given by 2.1. Soft Sets UðpÞ = GðpÞ ∩ HðpÞ fi X ~∪ De nition 1. [1] A map G of P into the power set 2 of a non- (ii) Their union, symbolized by GP HP, is a soft set UP, empty set X is called a soft set over X, where X is the universal where a mapping U : P ⟶ 2X is given by UðpÞ = set and P is a set of parameters.
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