
<abstract><p>To contribute to soft topology, we originate the notion of soft bioperators $ \tilde{\gamma} $ and $ {\tilde{\gamma}}^{'} $. Then, we apply them to analyze soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open sets and study main properties. We also prove that every soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open set is soft open; however, the converse is true only when the soft topological space is soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-regular. After that, we define and study two classes of soft closures namely $ Cl_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $ and $ \tilde{\tau}_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $-$ Cl $ operators, and two classes of soft interior namely $ Int_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $ and $ \tilde{\tau}_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $-$ Int $ operators. Moreover, we introduce the notions of soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ g $.closed sets and soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ T_{\frac{1}{2}} $ spaces, and explore their fundamental properties. In general, we explain the relationships between these notions, and give some counterexamples.</p></abstract>
Counterexample, soft \((\tilde{\gamma}, {\tilde{\gamma}}^{'})\)-open sets, Social Sciences, Management Science and Operations Research, Multi-Criteria Decision Making, Space (punctuation), $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ g $, Decision Sciences, $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open set, Lower separation axioms (\(T_0\)--\(T_3\), etc.), Fuzzy Logic and Residuated Lattices, Topological Spaces, $ \tilde{\gamma} $ and $ {\tilde{\gamma}}^{'} $, QA1-939, FOS: Mathematics, $ (\tilde{\gamma}, Topology (electrical circuits), Fuzzy topology, Physics, bioperators \(\tilde{\gamma}\) and \({\tilde{\gamma}}^{'}\) on \(\tilde{\tau}\), Application of Soft Set Theory in Decision Making, Computer science, Tilde, Operating system, Computational Theory and Mathematics, soft \((\tilde{\gamma}, {\tilde{\gamma}}^{'})\)-\(T_{\frac{1}{2}}\) spaces, Combinatorics, Computer Science, Physical Sciences, Topological spaces and generalizations (closure spaces, etc.), Soft Set Theory, soft \((\tilde{\gamma}, {\tilde{\gamma}}^{'})\)-\(g\).closed sets, {\tilde{\gamma}}^{'}) $-$ t_{\frac{1}{2}} $ spaces, Mathematics
Counterexample, soft \((\tilde{\gamma}, {\tilde{\gamma}}^{'})\)-open sets, Social Sciences, Management Science and Operations Research, Multi-Criteria Decision Making, Space (punctuation), $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ g $, Decision Sciences, $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open set, Lower separation axioms (\(T_0\)--\(T_3\), etc.), Fuzzy Logic and Residuated Lattices, Topological Spaces, $ \tilde{\gamma} $ and $ {\tilde{\gamma}}^{'} $, QA1-939, FOS: Mathematics, $ (\tilde{\gamma}, Topology (electrical circuits), Fuzzy topology, Physics, bioperators \(\tilde{\gamma}\) and \({\tilde{\gamma}}^{'}\) on \(\tilde{\tau}\), Application of Soft Set Theory in Decision Making, Computer science, Tilde, Operating system, Computational Theory and Mathematics, soft \((\tilde{\gamma}, {\tilde{\gamma}}^{'})\)-\(T_{\frac{1}{2}}\) spaces, Combinatorics, Computer Science, Physical Sciences, Topological spaces and generalizations (closure spaces, etc.), Soft Set Theory, soft \((\tilde{\gamma}, {\tilde{\gamma}}^{'})\)-\(g\).closed sets, {\tilde{\gamma}}^{'}) $-$ t_{\frac{1}{2}} $ spaces, Mathematics
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