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.

中文翻译：

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同态的波兰空间的可计算性

我们研究可计算的波兰空间和波兰群直至同胚。我们证明了 Stone 对偶的自然有效类比，并且我们还开发了一种有效的可定义性技术，该技术适用于同胚。作为一个应用程序，我们证明有一个$\三角洲^0_2$波兰空间不与可计算空间同胚。我们应用我们的技术来构建任何可计算的序数$\阿尔法$, 一个不同胚于任何的有效闭集$0^{(\alpha)}$- 可计算的波兰空间；这回答了 Nies 的一个问题。我们还证明了紧凑波兰群和局部路径连接空间的类似结果。