Computability of Polish spaces up to homeomorphism

Computability of Polish spaces up to homeomorphism

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 Polish space not homeomorphic to a compu...

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Bibliographic Details
Main Authors: Harrison-Trainor, Matthew, Melnikov, Alexander, Ng, Keng Meng
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2022
Subjects:
Science::Mathematics
Computer Analysis
Applications of Computable Structure Theory
Online Access:https://hdl.handle.net/10356/159280
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Institution: Nanyang Technological University
Language: English
Description
Summary: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 Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal, an effectively closed set not homeomorphic to any -computable Polish space; this answers a question of Nies. We also prove analogous results for compact Polish groups and locally path-connected spaces.

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