Computably and punctually universal spaces

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 19...

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Bibliographic Details
Main Authors: Bagaviev, Ramil, Batyrshin, Ilnur I., Bazhenov, Nikolay, Bushtets, Dmitry, Dorzhieva, Marina, Koh, Heer Tern, Kornev, Ruslan, Melnikov, Alexander G., Ng, Keng Meng
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2024
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Online Access:https://hdl.handle.net/10356/180629
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Institution: Nanyang Technological University
Language: English
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Summary: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.