The structure of the poset of regular topologies on a set
We study the subposet E3(X) of the lattice L1(X) of all T1-topologies on a set X, being the collections of all T3 topologies on X, with a view to deciding which elements of this partially ordered set have and which do not have immediate predecessors. We show that each regular topology which is not R...
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| Format: | Article |
| Language: | English |
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Universitat Politècnica de València
2011-04-01
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| Series: | Applied General Topology |
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| Online Access: | http://polipapers.upv.es/index.php/AGT/article/view/1695 |
| _version_ | 1811326596453236736 |
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| author | Ofelia T. Alas Richard G. Wilson |
| author_facet | Ofelia T. Alas Richard G. Wilson |
| author_sort | Ofelia T. Alas |
| collection | DOAJ |
| description | We study the subposet E3(X) of the lattice L1(X) of all T1-topologies on a set X, being the collections of all T3 topologies on X, with a view to deciding which elements of this partially ordered set have and which do not have immediate predecessors. We show that each regular topology which is not R-closed does have such a predecessor and as a corollary we obtain a result of Costantini that each non-compact Tychonoff space has an immediate predecessor in E3. We also consider the problem of when an R-closed topology is maximal R-closed. |
| first_indexed | 2024-04-13T14:52:43Z |
| format | Article |
| id | doaj.art-af9ae1eec3894277a81d035d92b89b73 |
| institution | Directory Open Access Journal |
| issn | 1576-9402 1989-4147 |
| language | English |
| last_indexed | 2024-04-13T14:52:43Z |
| publishDate | 2011-04-01 |
| publisher | Universitat Politècnica de València |
| record_format | Article |
| series | Applied General Topology |
| spelling | doaj.art-af9ae1eec3894277a81d035d92b89b732022-12-22T02:42:33ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472011-04-0112111310.4995/agt.2011.16951362The structure of the poset of regular topologies on a setOfelia T. Alas0Richard G. Wilson1Universidade de Sao PauloUniversidad Autónoma MetropolitanaWe study the subposet E3(X) of the lattice L1(X) of all T1-topologies on a set X, being the collections of all T3 topologies on X, with a view to deciding which elements of this partially ordered set have and which do not have immediate predecessors. We show that each regular topology which is not R-closed does have such a predecessor and as a corollary we obtain a result of Costantini that each non-compact Tychonoff space has an immediate predecessor in E3. We also consider the problem of when an R-closed topology is maximal R-closed.http://polipapers.upv.es/index.php/AGT/article/view/1695Lattice of T1-topologiesPoset of T3-topologiesUpper topologyLower topologyR-closed spaceR-minimal spaceSubmaximal spaceMaximal R-closed spaceDispersed space |
| spellingShingle | Ofelia T. Alas Richard G. Wilson The structure of the poset of regular topologies on a set Applied General Topology Lattice of T1-topologies Poset of T3-topologies Upper topology Lower topology R-closed space R-minimal space Submaximal space Maximal R-closed space Dispersed space |
| title | The structure of the poset of regular topologies on a set |
| title_full | The structure of the poset of regular topologies on a set |
| title_fullStr | The structure of the poset of regular topologies on a set |
| title_full_unstemmed | The structure of the poset of regular topologies on a set |
| title_short | The structure of the poset of regular topologies on a set |
| title_sort | structure of the poset of regular topologies on a set |
| topic | Lattice of T1-topologies Poset of T3-topologies Upper topology Lower topology R-closed space R-minimal space Submaximal space Maximal R-closed space Dispersed space |
| url | http://polipapers.upv.es/index.php/AGT/article/view/1695 |
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