Separable space

Pure mathematics

Discrete mathematics

Topological space

Computable analysis

Operating system

Duality (order theory)

Computer science

Computability

Homeomorphism (graph theory)

Computable number

Mathematics

Mathematical analysis

Path (computing)

Polish space

Combinatorics

Programming language

Space (punctuation)

Locally compact space

Journal of Symbolic Logic

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.

COMPUTABILITY OF POLISH SPACES UP TO HOMEOMORPHISM