TY - JOUR

T1 - Computable Stone spaces

AU - Bazhenov, Nikolay

AU - Harrison-Trainor, Matthew

AU - Melnikov, Alexander

N1 - Melnikov and Harrison-Trainor were partially supported by Rutherford Discovery Fellowship (RDF-VUW1902), Royal Society Te Apārangi.The work of Bazhenov and Melnikov is supported by the Mathematical Center in Akademgorodok under agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation.

PY - 2023/10/1

Y1 - 2023/10/1

N2 - We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space X, the Banach space C(X;R) has a computable presentation if, and only if, X is homeomorphic to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl.

AB - We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space X, the Banach space C(X;R) has a computable presentation if, and only if, X is homeomorphic to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85163195822&origin=inward&txGid=a754f561df57eb3b8612bacca92b7e77

UR - https://www.mendeley.com/catalogue/a3551d3b-ff7c-39c0-9855-df5aa528bbb9/

U2 - 10.1016/j.apal.2023.103304

DO - 10.1016/j.apal.2023.103304

M3 - Article

VL - 174

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 9

M1 - 103304

ER -