Comments on some results related to soft separation axioms
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Comments on some results related to soft separation axioms T. M. Al-shami1 Received: 13 August 2019 / Accepted: 15 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract Separation axioms are among the most widespread, significant and motivating concepts via classical topology. They can be utilized to approach problems related to digital topology and to establish more restricted families of topological spaces. This matter applies to them via soft topology as well. Therefore many research studies about soft separation axioms and their properties have been carried out. However, we observe existing some errors over these studies which it can be attributed to the different types of belong and non-belong relations which were defined via the soft set theory, and to the chosen objects of study: are they ordinary points or soft points? Our desire of removing confusions and constructing accurate framework motivates us to do this investigation. Through this paper, we show some alleged findings obtained in Bayramov and Aras (TWMS J Pure Appl Math 9(1):82–93, 2018), Hussain and Ahmad (Hacet J Math Stat 44(3):559–568, 2015), Matejdes (Int J Pure Appl Math 116(1):197–200, 2017), Singh and Noorie (Ann Fuzzy Math Inform 14(5):503–513, 2017) by giving convenient examples and then we formulate the right forms of these findings. In the last section, we demonstrate the relationships among soft T4 -spaces introduced in the previous studies and prove that all types of soft Ti -spaces are preserved under finitely soft product space in the cases of i = 0, 1, 2. Keywords Soft set · Soft point · Soft separation axioms Mathematics Subject Classification 54D10 · 54D15
1 Introduction Many practical problems in different directions involve data which are not always all deterministic and crisp. This leads to appear various forms of uncertainties in these problems and thus failure to utilize classical methods to approach them. This new reality prompts researchers to initiate alternative theories for dealing with uncertainties such as theory of probability and fuzzy sets. However, these theories have their own demerits or difficulties which possibly attributed to the inadequacy of the parametrization tool of these theories. To overcome these obstacles, Molodtsov [23] has proposed a new vital mathematical tool,
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T. M. Al-shami [email protected] Department of Mathematics, Sana’a University, Sana’a, Yemen
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T. M. Al-Shami
which is free from the above difficulties, to copy with uncertainties and handle it efficiency. He called this tool soft set. Molodtsov in his pioneer work [23] has pointed out its merits compared to the theory of probability and fuzzy sets, and has elaborated its rich scope for applications in many fields. Later than, more studies have been done in order to construct and discuss the theoretical and applications aspects of soft sets. In the year 2011, Shabir and Naz [25] employed soft sets in defining the concept of soft topological spaces. They pr
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