Soft Semi <i>ω</i>-Open Sets
In this paper, we introduce the class of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets of a soft topological space <...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/24/3168 |
| _version_ | 1797502637959020544 |
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| author | Samer Al Ghour |
| author_facet | Samer Al Ghour |
| author_sort | Samer Al Ghour |
| collection | DOAJ |
| description | In this paper, we introduce the class of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets of a soft topological space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>, using soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets. We show that the class of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets contains both the soft topology <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>τ</mi><mi>ω</mi></msub></semantics></math></inline-formula> and the class of soft semi-open sets. Additionally, we define soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-closed sets as the class of soft complements of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets. We present here a study of the properties of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets, especially in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi><mi>ω</mi></msub><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In particular, we prove that the class of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets is closed under arbitrary soft union but not closed under finite soft intersections; we also study the correspondence between the soft topology of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets of a soft topological space and their generated topological spaces and vice versa. In addition to these, we introduce the soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-interior and soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-closure operators via soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open and soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-closed sets. We prove several equations regarding these two new soft operators. In particular, we prove that these operators can be calculated using other usual soft operators in both of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi><mi>ω</mi></msub><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and some equations focus on soft anti-locally countable soft topological spaces. |
| first_indexed | 2024-03-10T03:37:58Z |
| format | Article |
| id | doaj.art-4c35bc93a96b458192434d121b0dac15 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:37:58Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-4c35bc93a96b458192434d121b0dac152023-11-23T09:25:09ZengMDPI AGMathematics2227-73902021-12-01924316810.3390/math9243168Soft Semi <i>ω</i>-Open SetsSamer Al Ghour0Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, JordanIn this paper, we introduce the class of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets of a soft topological space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>, using soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets. We show that the class of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets contains both the soft topology <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>τ</mi><mi>ω</mi></msub></semantics></math></inline-formula> and the class of soft semi-open sets. Additionally, we define soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-closed sets as the class of soft complements of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets. We present here a study of the properties of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets, especially in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi><mi>ω</mi></msub><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In particular, we prove that the class of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets is closed under arbitrary soft union but not closed under finite soft intersections; we also study the correspondence between the soft topology of soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open sets of a soft topological space and their generated topological spaces and vice versa. In addition to these, we introduce the soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-interior and soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-closure operators via soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-open and soft semi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-closed sets. We prove several equations regarding these two new soft operators. In particular, we prove that these operators can be calculated using other usual soft operators in both of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi><mi>ω</mi></msub><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and some equations focus on soft anti-locally countable soft topological spaces.https://www.mdpi.com/2227-7390/9/24/3168soft ω-opensoft semi-opensoft semi interiorsoft semi interiorsoft generated soft topological spacesoft induced topological spaces |
| spellingShingle | Samer Al Ghour Soft Semi <i>ω</i>-Open Sets Mathematics soft ω-open soft semi-open soft semi interior soft semi interior soft generated soft topological space soft induced topological spaces |
| title | Soft Semi <i>ω</i>-Open Sets |
| title_full | Soft Semi <i>ω</i>-Open Sets |
| title_fullStr | Soft Semi <i>ω</i>-Open Sets |
| title_full_unstemmed | Soft Semi <i>ω</i>-Open Sets |
| title_short | Soft Semi <i>ω</i>-Open Sets |
| title_sort | soft semi i ω i open sets |
| topic | soft ω-open soft semi-open soft semi interior soft semi interior soft generated soft topological space soft induced topological spaces |
| url | https://www.mdpi.com/2227-7390/9/24/3168 |
| work_keys_str_mv | AT sameralghour softsemiiōiopensets |