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COMPUTABILITY OF POLISH SPACES UP TO HOMEOMORPHISM
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 4 / December 2020
- Published online by Cambridge University Press:
- 26 October 2020, pp. 1664-1686
- Print publication:
- December 2020
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- Article
- Export citation
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We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a
$\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal 
$\alpha $, an effectively closed set not homeomorphic to any 
$0^{(\alpha )}$-computable Polish space; this answers a question of Nies. We also prove analogous results for compact Polish groups and locally path-connected spaces.
