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On the Reconstruction of Topological Spaces from Their Groups of Homeomorphisms
| Content Provider | Scilit |
|---|---|
| Author | Rubin, Matatyahu |
| Copyright Year | 1989 |
| Abstract | For various classes of topological spaces we prove that if and have isomorphic homeomorphism groups, then and are homeomorphic. Let denote a subgroup of the group of homeomorphisms of a topological space . A class of 's is faithful if for every , if is a group isomorphism, then there is a homeomorphism between and such that for every . Theorem 1: The following class is faithful: is a locally finite-dimensional polyhedron in the metric or coherent topology or is a Euclidean manifold with boundary) and for every is an accumulation point of is a differentiable or a -manifold and contains the group of differentiable or piecewise linear homeomorphisms is a manifold over a normed vector space over an ordered field. This answers a question of Whittaker , who asked about the faithfulness of the class of Banach manifolds. Theorem 2: The following class is faithful: is a locally compact Hausdorff space and for every open and and is somewhere dense. Note that this class includes Euclidean manifolds as well as products of compact connected Euclidean manifolds. Theorem 3: The following class is faithful: (1) is a 0-dimensional Hausdorff space; (2) for every there is a regular open set whose boundary is ; (3) for every there are such that , and (4) for every nonempty open there is such that . Note that (2) is satisfied by 0-dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from . Theorem 4: We prove (Theorem 2.23.1) that for an appropriate class of trees is first-order interpretable in . |
| Related Links | http://www.ams.org/tran/1989-312-02/S0002-9947-1989-0988881-4/S0002-9947-1989-0988881-4.pdf |
| Ending Page | 538 |
| Page Count | 52 |
| Starting Page | 487 |
| ISSN | 00029947 |
| e-ISSN | 10886850 |
| DOI | 10.2307/2001000 |
| Journal | Transactions of the American Mathematical Society |
| Issue Number | 2 |
| Volume Number | 312 |
| Language | English |
| Publisher | Duke University Press |
| Access Restriction | Open |
| Subject Keyword | Logic Groups of Homeomorphisms Reconstruction of Topological Spaces Topological Spaces From Their Groups |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics |
