Computably and punctually universal spaces

• 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
• Subjects: Mathematical Sciences; Universal spaces; Computable Polish space
• Online Access: https://hdl.handle.net/10356/180629
• Institution: Nanyang Technological University

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.