Topologies on <em>Z</em><sup>n</sup> that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological Space
The present paper deals with two types of topologies on the set of integers, <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula>: a quasi-discrete topolo...
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MDPI AG
2019-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/11/1072 |
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| author | Sang-Eon Han Saeid Jafari Jeong Min Kang |
| author_facet | Sang-Eon Han Saeid Jafari Jeong Min Kang |
| author_sort | Sang-Eon Han |
| collection | DOAJ |
| description | The present paper deals with two types of topologies on the set of integers, <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula>: a quasi-discrete topology and a topology satisfying the <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msub> </semantics> </math> </inline-formula>-separation axiom. Furthermore, for each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </semantics> </math> </inline-formula>, we develop countably many topologies on <inline-formula> <math display="inline"> <semantics> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </semantics> </math> </inline-formula> which are not homeomorphic to the typical <i>n</i>-dimensional Khalimsky topological space. Based on these different types of new topological structures on <inline-formula> <math display="inline"> <semantics> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </semantics> </math> </inline-formula>, many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on. |
| first_indexed | 2024-12-10T21:27:56Z |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-10T21:27:56Z |
| publishDate | 2019-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-cb8345719cb2452aab348e6f294fae2b2022-12-22T01:32:56ZengMDPI AGMathematics2227-73902019-11-01711107210.3390/math7111072math7111072Topologies on <em>Z</em><sup>n</sup> that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological SpaceSang-Eon Han0Saeid Jafari1Jeong Min Kang2Department of Mathematics Education, Institute of Pure and Applied Mathematics Jeonbuk National University, Jeonju-City 54896, Jeonbuk, KoreaCollege of Vestsjaelland South Herrestraede 114200 Slagelse, DenmarkMathematics, School of Liberal, Arts Education, University of Seoul, Seoul 02504, KoreaThe present paper deals with two types of topologies on the set of integers, <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula>: a quasi-discrete topology and a topology satisfying the <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msub> </semantics> </math> </inline-formula>-separation axiom. Furthermore, for each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </semantics> </math> </inline-formula>, we develop countably many topologies on <inline-formula> <math display="inline"> <semantics> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </semantics> </math> </inline-formula> which are not homeomorphic to the typical <i>n</i>-dimensional Khalimsky topological space. Based on these different types of new topological structures on <inline-formula> <math display="inline"> <semantics> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </semantics> </math> </inline-formula>, many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on.https://www.mdpi.com/2227-7390/7/11/1072khalimsky topologyquasi-discrete (clopen or pseudo-discrete)t12-separation axiomalexandroff topologydigital topology |
| spellingShingle | Sang-Eon Han Saeid Jafari Jeong Min Kang Topologies on <em>Z</em><sup>n</sup> that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological Space Mathematics khalimsky topology quasi-discrete (clopen or pseudo-discrete) t12-separation axiom alexandroff topology digital topology |
| title | Topologies on <em>Z</em><sup>n</sup> that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological Space |
| title_full | Topologies on <em>Z</em><sup>n</sup> that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological Space |
| title_fullStr | Topologies on <em>Z</em><sup>n</sup> that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological Space |
| title_full_unstemmed | Topologies on <em>Z</em><sup>n</sup> that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological Space |
| title_short | Topologies on <em>Z</em><sup>n</sup> that Are Not Homeomorphic to the <em>n</em>-Dimensional Khalimsky Topological Space |
| title_sort | topologies on em z em sup n sup that are not homeomorphic to the em n em dimensional khalimsky topological space |
| topic | khalimsky topology quasi-discrete (clopen or pseudo-discrete) t12-separation axiom alexandroff topology digital topology |
| url | https://www.mdpi.com/2227-7390/7/11/1072 |
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