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COMPUTABILITY OF POLISH SPACES UP TO HOMEOMORPHISM

Published online by Cambridge University Press:  26 October 2020

MATTHEW HARRISON-TRAINOR
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTON6140NEW ZEALANDE-mail: matthew.harrisontrainor@vuw.ac.nz
ALEXANDER MELNIKOV
Affiliation:
MASSEY UNIVERSITY AUCKLAND PRIVATE BAG 102904, NORTH SHORE AUCKLAND 0745, NEW ZEALANDE-mail: alexander.g.melnikov@gmail.com
KENG MENG NG
Affiliation:
SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES DIVISION OF MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY, SINGAPOREE-mail: selwyn.km.ng@gmail.com

Abstract

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.

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Articles
Copyright
© The Association for Symbolic Logic 2020

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