Connectivity with respect to α-discrete closure operators
We discuss certain closure operators that generalize the Alexandroff topologies. Such a closure operator is defined for every ordinal α>0\alpha \gt 0 in such a way that the closure of a set AA is given by closures of certain α\alpha -indexed sequences formed by points of AA. It is shown that conn...
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-08-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2022-0046 |
| _version_ | 1818025203793395712 |
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| author | Šlapal Josef |
| author_facet | Šlapal Josef |
| author_sort | Šlapal Josef |
| collection | DOAJ |
| description | We discuss certain closure operators that generalize the Alexandroff topologies. Such a closure operator is defined for every ordinal α>0\alpha \gt 0 in such a way that the closure of a set AA is given by closures of certain α\alpha -indexed sequences formed by points of AA. It is shown that connectivity with respect to such a closure operator can be viewed as a special type of path connectivity. This makes it possible to apply the operators in solving problems based on employing a convenient connectivity such as problems of digital image processing. One such application is presented providing a digital analogue of the Jordan curve theorem. |
| first_indexed | 2024-12-10T04:12:23Z |
| format | Article |
| id | doaj.art-7834025308184bc69b59d19cb6c44517 |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-12-10T04:12:23Z |
| publishDate | 2022-08-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-7834025308184bc69b59d19cb6c445172022-12-22T02:02:41ZengDe GruyterOpen Mathematics2391-54552022-08-0120168268810.1515/math-2022-0046Connectivity with respect to α-discrete closure operatorsŠlapal Josef0Institute of Mathematics, Brno University of Technology, 616 69 Brno, Czech RepublicWe discuss certain closure operators that generalize the Alexandroff topologies. Such a closure operator is defined for every ordinal α>0\alpha \gt 0 in such a way that the closure of a set AA is given by closures of certain α\alpha -indexed sequences formed by points of AA. It is shown that connectivity with respect to such a closure operator can be viewed as a special type of path connectivity. This makes it possible to apply the operators in solving problems based on employing a convenient connectivity such as problems of digital image processing. One such application is presented providing a digital analogue of the Jordan curve theorem.https://doi.org/10.1515/math-2022-0046closure operatorordinal (number)ordinal-indexed sequenceconnectivitydigital jordan curve54a0554d05 |
| spellingShingle | Šlapal Josef Connectivity with respect to α-discrete closure operators Open Mathematics closure operator ordinal (number) ordinal-indexed sequence connectivity digital jordan curve 54a05 54d05 |
| title | Connectivity with respect to α-discrete closure operators |
| title_full | Connectivity with respect to α-discrete closure operators |
| title_fullStr | Connectivity with respect to α-discrete closure operators |
| title_full_unstemmed | Connectivity with respect to α-discrete closure operators |
| title_short | Connectivity with respect to α-discrete closure operators |
| title_sort | connectivity with respect to α discrete closure operators |
| topic | closure operator ordinal (number) ordinal-indexed sequence connectivity digital jordan curve 54a05 54d05 |
| url | https://doi.org/10.1515/math-2022-0046 |
| work_keys_str_mv | AT slapaljosef connectivitywithrespecttoadiscreteclosureoperators |