Reconstructing compact metrizable spaces
The deck, D(X), of a topological space X is the set D(X) = {[X\{x}]: x ∈ X}, where [Y] denotes the homeomorphism class of Y. A space X is (topologically) reconstructible if whenever D(Z) = D(X), then Z is homeomorphic to X. It is known that every (metrizable) continuum is reconstructible, whereas th...
| Main Authors: | , , |
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| 格式: | Journal article |
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American Mathematical Society
2016
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| _version_ | 1826277093195907072 |
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| author | Gartside, P Pitz, M Suabedissen, R |
| author_facet | Gartside, P Pitz, M Suabedissen, R |
| author_sort | Gartside, P |
| collection | OXFORD |
| description | The deck, D(X), of a topological space X is the set D(X) = {[X\{x}]: x ∈ X}, where [Y] denotes the homeomorphism class of Y. A space X is (topologically) reconstructible if whenever D(Z) = D(X), then Z is homeomorphic to X. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible. The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point x there is a sequence <bx n="" n:="" ℕ="" ∈=""> of pairwise disjoint clopen subsets converging to x such that Bxn and Byn are homeomorphic for each n and all x and y. In a non-reconstructible compact metrizable space the set of 1-point components forms a dense Gδ. For h-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense Gδ set of 1-point components is presented, some reconstructible and others not reconstructible.</bx> |
| first_indexed | 2024-03-06T23:23:42Z |
| format | Journal article |
| id | oxford-uuid:69a78f6d-d401-41fd-9ebb-e9b4e50e8d87 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T23:23:42Z |
| publishDate | 2016 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:69a78f6d-d401-41fd-9ebb-e9b4e50e8d872022-03-26T18:52:20ZReconstructing compact metrizable spacesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:69a78f6d-d401-41fd-9ebb-e9b4e50e8d87Symplectic Elements at OxfordAmerican Mathematical Society2016Gartside, PPitz, MSuabedissen, RThe deck, D(X), of a topological space X is the set D(X) = {[X\{x}]: x ∈ X}, where [Y] denotes the homeomorphism class of Y. A space X is (topologically) reconstructible if whenever D(Z) = D(X), then Z is homeomorphic to X. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible. The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point x there is a sequence <bx n="" n:="" ℕ="" ∈=""> of pairwise disjoint clopen subsets converging to x such that Bxn and Byn are homeomorphic for each n and all x and y. In a non-reconstructible compact metrizable space the set of 1-point components forms a dense Gδ. For h-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense Gδ set of 1-point components is presented, some reconstructible and others not reconstructible.</bx> |
| spellingShingle | Gartside, P Pitz, M Suabedissen, R Reconstructing compact metrizable spaces |
| title | Reconstructing compact metrizable spaces |
| title_full | Reconstructing compact metrizable spaces |
| title_fullStr | Reconstructing compact metrizable spaces |
| title_full_unstemmed | Reconstructing compact metrizable spaces |
| title_short | Reconstructing compact metrizable spaces |
| title_sort | reconstructing compact metrizable spaces |
| work_keys_str_mv | AT gartsidep reconstructingcompactmetrizablespaces AT pitzm reconstructingcompactmetrizablespaces AT suabedissenr reconstructingcompactmetrizablespaces |