Symmetries of two-point sets
A two-point set is a subset of the plane which meets every planar line in exactly two-points. We discuss the problem “What are the topological symmetries of a two-point set?”. Our main results assert the existence of two-point sets which are rigid and the existence of two-point sets which are invari...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Elsevier
2008
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| Subjects: |
| _version_ | 1797054465453326336 |
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| author | Chad, B Suabedissen, R |
| author_facet | Chad, B Suabedissen, R |
| author_sort | Chad, B |
| collection | OXFORD |
| description | A two-point set is a subset of the plane which meets every planar line in exactly two-points. We discuss the problem “What are the topological symmetries of a two-point set?”. Our main results assert the existence of two-point sets which are rigid and the existence of two-point sets which are invariant under the action of certain autohomeomorphism groups. We pay particular attention to the isometry group of a two-point set, and show that such groups consist only of rotations and that they may be chosen to be any subgroup of S<sup>1</sup> having size less than <em>c</em> . We also construct a subgroup of S<sup>1</sup> having size c that is contained in the isometry group of a two-point set. |
| first_indexed | 2024-03-06T18:57:40Z |
| format | Journal article |
| id | oxford-uuid:126680e1-d0c8-4888-872b-7f36a75e7f45 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T18:57:40Z |
| publishDate | 2008 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:126680e1-d0c8-4888-872b-7f36a75e7f452022-03-26T10:07:49ZSymmetries of two-point setsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:126680e1-d0c8-4888-872b-7f36a75e7f45Analytic Topology or TopologyAlgebraic topologyEnglishOxford University Research Archive - ValetElsevier2008Chad, BSuabedissen, RA two-point set is a subset of the plane which meets every planar line in exactly two-points. We discuss the problem “What are the topological symmetries of a two-point set?”. Our main results assert the existence of two-point sets which are rigid and the existence of two-point sets which are invariant under the action of certain autohomeomorphism groups. We pay particular attention to the isometry group of a two-point set, and show that such groups consist only of rotations and that they may be chosen to be any subgroup of S<sup>1</sup> having size less than <em>c</em> . We also construct a subgroup of S<sup>1</sup> having size c that is contained in the isometry group of a two-point set. |
| spellingShingle | Analytic Topology or Topology Algebraic topology Chad, B Suabedissen, R Symmetries of two-point sets |
| title | Symmetries of two-point sets |
| title_full | Symmetries of two-point sets |
| title_fullStr | Symmetries of two-point sets |
| title_full_unstemmed | Symmetries of two-point sets |
| title_short | Symmetries of two-point sets |
| title_sort | symmetries of two point sets |
| topic | Analytic Topology or Topology Algebraic topology |
| work_keys_str_mv | AT chadb symmetriesoftwopointsets AT suabedissenr symmetriesoftwopointsets |