Weaker Forms of Soft Regular and Soft <i>T</i><sub>2</sub> Soft Topological Spaces
Soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-local indiscreetness as a weaker form of both soft local countability and soft local in...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/17/2153 |
| _version_ | 1797521125471682560 |
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| author | Samer Al Ghour |
| author_facet | Samer Al Ghour |
| author_sort | Samer Al Ghour |
| collection | DOAJ |
| description | Soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regularity as a weaker form of both soft regularity and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-local indiscreetness is defined and investigated. Additionally, soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula> as a new soft topological property that lies strictly between soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula> and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>1</mn></msub></semantics></math></inline-formula> is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-locally indiscreetness (resp. soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regularity) and soft locally indiscreetness (resp. soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regularity). Additionally, it is proved that the induced topological spaces of a soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-locally indiscrete (resp. soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular, soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula>) soft topological space are (resp. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula>) topological spaces. Additionally, it is proved that the generated soft topological space of a family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-locally indiscrete (resp. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula>) topological spaces is soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula> soft topological spaces are obtained. Moreover, it is proved that soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula> are hereditarily under soft subspaces. |
| first_indexed | 2024-03-10T08:07:11Z |
| format | Article |
| id | doaj.art-bdc6a3cdb2794f469a5b577a79994ece |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T08:07:11Z |
| publishDate | 2021-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-bdc6a3cdb2794f469a5b577a79994ece2023-11-22T10:58:41ZengMDPI AGMathematics2227-73902021-09-01917215310.3390/math9172153Weaker Forms of Soft Regular and Soft <i>T</i><sub>2</sub> Soft Topological SpacesSamer Al Ghour0Department of Mathematics, Jordan University of Science and Technology, Irbid 22110, JordanSoft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regularity as a weaker form of both soft regularity and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-local indiscreetness is defined and investigated. Additionally, soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula> as a new soft topological property that lies strictly between soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula> and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>1</mn></msub></semantics></math></inline-formula> is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-locally indiscreetness (resp. soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regularity) and soft locally indiscreetness (resp. soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regularity). Additionally, it is proved that the induced topological spaces of a soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-locally indiscrete (resp. soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular, soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula>) soft topological space are (resp. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula>) topological spaces. Additionally, it is proved that the generated soft topological space of a family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-locally indiscrete (resp. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula>) topological spaces is soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula> soft topological spaces are obtained. Moreover, it is proved that soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-regular and soft <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mn>2</mn></msub></semantics></math></inline-formula> are hereditarily under soft subspaces.https://www.mdpi.com/2227-7390/9/17/2153soft local indiscreetnesssoft regularitysoft <i>T2</i> soft topological spacessoft productsoft subspacesoft generated soft topological space |
| spellingShingle | Samer Al Ghour Weaker Forms of Soft Regular and Soft <i>T</i><sub>2</sub> Soft Topological Spaces Mathematics soft local indiscreetness soft regularity soft <i>T2</i> soft topological spaces soft product soft subspace soft generated soft topological space |
| title | Weaker Forms of Soft Regular and Soft <i>T</i><sub>2</sub> Soft Topological Spaces |
| title_full | Weaker Forms of Soft Regular and Soft <i>T</i><sub>2</sub> Soft Topological Spaces |
| title_fullStr | Weaker Forms of Soft Regular and Soft <i>T</i><sub>2</sub> Soft Topological Spaces |
| title_full_unstemmed | Weaker Forms of Soft Regular and Soft <i>T</i><sub>2</sub> Soft Topological Spaces |
| title_short | Weaker Forms of Soft Regular and Soft <i>T</i><sub>2</sub> Soft Topological Spaces |
| title_sort | weaker forms of soft regular and soft i t i sub 2 sub soft topological spaces |
| topic | soft local indiscreetness soft regularity soft <i>T2</i> soft topological spaces soft product soft subspace soft generated soft topological space |
| url | https://www.mdpi.com/2227-7390/9/17/2153 |
| work_keys_str_mv | AT sameralghour weakerformsofsoftregularandsoftitisub2subsofttopologicalspaces |