A Novel Class of Separation Axioms, Compactness, and Continuity via <i>C</i>-Open Sets
In this paper, we originate a new class of open sets, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula&g...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/23/4729 |
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| author | Mesfer H. Alqahtani Hind Y. Saleh |
| author_facet | Mesfer H. Alqahtani Hind Y. Saleh |
| author_sort | Mesfer H. Alqahtani |
| collection | DOAJ |
| description | In this paper, we originate a new class of open sets, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets, and we review its important properties. Then, some separation axioms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets are introduced and investigated. In addition, we define the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-compact and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="monospace">C</mi><mo>′</mo></msup></semantics></math></inline-formula>-compact spaces via <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets, and the theorems based on them are discussed with counterexamples. Moreover, we entitle the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-continuous and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="monospace">C</mi><mo>′</mo></msup></semantics></math></inline-formula>-continuous functions by applying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets. In particular, several inferred properties of them and their connection with the other topological spaces are studied theoretically. Many examples are given to explain the concepts lucidly. The results established in this research paper are new in the field of topology. |
| first_indexed | 2024-03-09T01:46:55Z |
| format | Article |
| id | doaj.art-6645935ead644d29bc6b9ac1c84549d4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T01:46:55Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-6645935ead644d29bc6b9ac1c84549d42023-12-08T15:21:36ZengMDPI AGMathematics2227-73902023-11-011123472910.3390/math11234729A Novel Class of Separation Axioms, Compactness, and Continuity via <i>C</i>-Open SetsMesfer H. Alqahtani0Hind Y. Saleh1Department of Mathematics, University College of Umluj, University of Tabuk, Tabuk 48322, Saudi ArabiaDepartment of Mathematics, College of Basic Education, University of Duhok, Duhok 42001, IraqIn this paper, we originate a new class of open sets, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets, and we review its important properties. Then, some separation axioms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets are introduced and investigated. In addition, we define the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-compact and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="monospace">C</mi><mo>′</mo></msup></semantics></math></inline-formula>-compact spaces via <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets, and the theorems based on them are discussed with counterexamples. Moreover, we entitle the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-continuous and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="monospace">C</mi><mo>′</mo></msup></semantics></math></inline-formula>-continuous functions by applying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets. In particular, several inferred properties of them and their connection with the other topological spaces are studied theoretically. Many examples are given to explain the concepts lucidly. The results established in this research paper are new in the field of topology.https://www.mdpi.com/2227-7390/11/23/4729<tt>C</tt>-open set<tt>C</tt>-regular space<tt>C</tt>-normal space<tt>C</tt>-compact space<tt>C</tt>′-compact space<tt>C</tt>-continuous function |
| spellingShingle | Mesfer H. Alqahtani Hind Y. Saleh A Novel Class of Separation Axioms, Compactness, and Continuity via <i>C</i>-Open Sets Mathematics <tt>C</tt>-open set <tt>C</tt>-regular space <tt>C</tt>-normal space <tt>C</tt>-compact space <tt>C</tt>′-compact space <tt>C</tt>-continuous function |
| title | A Novel Class of Separation Axioms, Compactness, and Continuity via <i>C</i>-Open Sets |
| title_full | A Novel Class of Separation Axioms, Compactness, and Continuity via <i>C</i>-Open Sets |
| title_fullStr | A Novel Class of Separation Axioms, Compactness, and Continuity via <i>C</i>-Open Sets |
| title_full_unstemmed | A Novel Class of Separation Axioms, Compactness, and Continuity via <i>C</i>-Open Sets |
| title_short | A Novel Class of Separation Axioms, Compactness, and Continuity via <i>C</i>-Open Sets |
| title_sort | novel class of separation axioms compactness and continuity via i c i open sets |
| topic | <tt>C</tt>-open set <tt>C</tt>-regular space <tt>C</tt>-normal space <tt>C</tt>-compact space <tt>C</tt>′-compact space <tt>C</tt>-continuous function |
| url | https://www.mdpi.com/2227-7390/11/23/4729 |
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