TY - JOUR

T1 - Computably and punctually universal spaces

AU - Bagaviev, Ramil

AU - Batyrshin, Ilnur I.

AU - Bazhenov, Nikolay

AU - Bushtets, Dmitry

AU - Dorzhieva, Marina

AU - Koh, Heer Tern

AU - Kornev, Ruslan

AU - Melnikov, Alexander G.

AU - Ng, Keng Meng

N1 - Computably and punctually universal spaces / R. Bagaviev, I. I. Batyrshin, N. Bazhenov [et al.] // Annals of Pure and Applied Logic. – 2025. – Vol. 176, No. 1. – P. 103491. – DOI 10.1016/j.apal.2024.103491. – EDN VIPSYQ. The work of R. Bagaviev and I.I. Batyrshin is performed under the development program of Volga Region Mathematical Center (agreement No. 075-02-2024-1438). N. Bazhenov and R. Kornev were supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2022-282 with the Ministry of Science and Higher Education of the Russian Federation. M. Dorzhieva was supported by Rutherford Discovery Fellowship (RDF-VUW1902) of A.G. Melnikov. K.M. Ng was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (RG23/19). A. Melnikov was supported by Rutherford Discovery Fellowship (Wellington) RDF-VUW1902, Royal Society Te Aparangi. We are grateful to the anonymous reviewer for their helpful comments and suggestions. Part of the research contained in the paper was carried out in the student project “Computably universal spaces” at the Second Workshop of the Mathematical Center in Akademgorodok (Summer 2021).

PY - 2025/1

Y1 - 2025/1

N2 - We prove that the standard computable presentation of the space C[0,1] of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s. We prove that the original Urysohn's construction of the universal separable Polish space U is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space 2ω; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.

AB - We prove that the standard computable presentation of the space C[0,1] of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s. We prove that the original Urysohn's construction of the universal separable Polish space U is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space 2ω; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.

KW - Computable Polish space

KW - Continuous functions on the unit interval

KW - Primitive recursive analysis

KW - Punctual

KW - Universal spaces

KW - Universal spaces

KW - Punctual

KW - Computable Polish space

KW - Continuous functions on the unit interval

KW - Primitive recursive analysis

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85200913507&origin=inward&txGid=91f4204e4737a9462e1b7b10407409d2

UR - https://www.elibrary.ru/item.asp?id=80686308

UR - https://www.mendeley.com/catalogue/201223d0-9465-375e-b32f-b1fcbc65209a/

U2 - 10.1016/j.apal.2024.103491

DO - 10.1016/j.apal.2024.103491

M3 - Article

VL - 176

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 1

M1 - 103491

ER -