On Soft <i>ω<sub>δ</sub></i>-Open Sets and Some Decomposition Theorems
In this paper, we present a novel family of soft sets named “soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>δ</mi></msub></semantics></math><...
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MDPI AG
2024-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/6/924 |
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| author | Dina Abuzaid Samer Al-Ghour Monia Naghi |
| author_facet | Dina Abuzaid Samer Al-Ghour Monia Naghi |
| author_sort | Dina Abuzaid |
| collection | DOAJ |
| description | In this paper, we present a novel family of soft sets named “soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>δ</mi></msub></semantics></math></inline-formula>-open sets”. We find that this class constitutes a soft topology that lies strictly between the soft topologies of soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-open sets and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ω</mi><mn>0</mn></msup></semantics></math></inline-formula>-open sets. Also, we introduce certain sufficient conditions for the equivalence between this new soft topology and several existing soft topologies. Moreover, we verify several relationships that contain soft covering properties, such as soft compactness and soft Lindelofness, which are related to this new soft topology. Furthermore, in terms of the soft interior operator in certain soft topologies, we define four classes of soft sets. Via them, we obtain new decomposition theorems for soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-openness and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>-openness, and we characterize the soft topological spaces that have the soft “semi-regularization property”. In addition, via soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>δ</mi></msub></semantics></math></inline-formula>-open sets, we introduce and investigate a new class of soft functions named “soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>δ</mi></msub></semantics></math></inline-formula>-continuous functions”. Finally, we look into the connections between the newly proposed soft concepts and their counterparts in classical topological spaces. |
| first_indexed | 2024-04-24T18:03:04Z |
| format | Article |
| id | doaj.art-cf76f06fb8e74fb99d48f39ff19c1abc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T18:03:04Z |
| publishDate | 2024-03-01 |
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| spelling | doaj.art-cf76f06fb8e74fb99d48f39ff19c1abc2024-03-27T13:53:19ZengMDPI AGMathematics2227-73902024-03-0112692410.3390/math12060924On Soft <i>ω<sub>δ</sub></i>-Open Sets and Some Decomposition TheoremsDina Abuzaid0Samer Al-Ghour1Monia Naghi2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, JordanDepartment of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaIn this paper, we present a novel family of soft sets named “soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>δ</mi></msub></semantics></math></inline-formula>-open sets”. We find that this class constitutes a soft topology that lies strictly between the soft topologies of soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-open sets and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ω</mi><mn>0</mn></msup></semantics></math></inline-formula>-open sets. Also, we introduce certain sufficient conditions for the equivalence between this new soft topology and several existing soft topologies. Moreover, we verify several relationships that contain soft covering properties, such as soft compactness and soft Lindelofness, which are related to this new soft topology. Furthermore, in terms of the soft interior operator in certain soft topologies, we define four classes of soft sets. Via them, we obtain new decomposition theorems for soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>-openness and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>-openness, and we characterize the soft topological spaces that have the soft “semi-regularization property”. In addition, via soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>δ</mi></msub></semantics></math></inline-formula>-open sets, we introduce and investigate a new class of soft functions named “soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>δ</mi></msub></semantics></math></inline-formula>-continuous functions”. Finally, we look into the connections between the newly proposed soft concepts and their counterparts in classical topological spaces.https://www.mdpi.com/2227-7390/12/6/924soft <i>δ</i>-open setssoft <i>θ</i>-open setssoft <i>ω</i><sup>0</sup>-open setssuper-continuitysoft generated soft topological spaces |
| spellingShingle | Dina Abuzaid Samer Al-Ghour Monia Naghi On Soft <i>ω<sub>δ</sub></i>-Open Sets and Some Decomposition Theorems Mathematics soft <i>δ</i>-open sets soft <i>θ</i>-open sets soft <i>ω</i><sup>0</sup>-open sets super-continuity soft generated soft topological spaces |
| title | On Soft <i>ω<sub>δ</sub></i>-Open Sets and Some Decomposition Theorems |
| title_full | On Soft <i>ω<sub>δ</sub></i>-Open Sets and Some Decomposition Theorems |
| title_fullStr | On Soft <i>ω<sub>δ</sub></i>-Open Sets and Some Decomposition Theorems |
| title_full_unstemmed | On Soft <i>ω<sub>δ</sub></i>-Open Sets and Some Decomposition Theorems |
| title_short | On Soft <i>ω<sub>δ</sub></i>-Open Sets and Some Decomposition Theorems |
| title_sort | on soft i ω sub δ sub i open sets and some decomposition theorems |
| topic | soft <i>δ</i>-open sets soft <i>θ</i>-open sets soft <i>ω</i><sup>0</sup>-open sets super-continuity soft generated soft topological spaces |
| url | https://www.mdpi.com/2227-7390/12/6/924 |
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