New Degree Spectra of Polish Spaces
| Autor: | Alexander G. Melnikov |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Siberian Mathematical Journal. 62:882-894 |
| ISSN: | 1573-9260 0037-4466 |
| DOI: | 10.1134/s0037446621050116 |
| Popis: | The main result is as follows: Fix an arbitrary prime $ q $ . A $ q $ -divisible torsion-free (discrete, countable) abelian group $ G $ has a $ \Delta^{0}_{2} $ -presentation if, and only if, its connected Pontryagin–van Kampen Polish dual $ \widehat{G} $ admits a computable complete metrization (in which we do not require the operations to be computable). We use this jump-inversion/duality theorem to transfer the results on the degree spectra of torsion-free abelian groups to the results about the degree spectra of Polish spaces up to homeomorphism. For instance, it follows that for every computable ordinal $ \alpha>1 $ and each $ {\mathbf{a}}>0^{(\alpha)} $ there is a connected compact Polish space having proper $ \alpha^{th} $ jump degree $ {\mathbf{a}} $ (up to homeomorphism). Also, for every computable ordinal $ \beta $ of the form $ 1+\delta+2n+1 $ , where $ \delta $ is zero or is a limit ordinal and $ n\in\omega $ , there is a connected Polish space having an $ X $ -computable copy if and only if $ X $ is $ non $ - $ low_{\beta} $ . In particular, there is a connected Polish space having exactly the $ non $ - $ low_{2} $ complete metrizations. The case when $ \beta=2 $ is an unexpected consequence of the main result of the author’s M.Sc. Thesis written under the supervision of Sergey S. Goncharov. |
| Databáze: | OpenAIRE |
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