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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Main Authors: | , , , , , , , , |
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Other Authors: | |
Format: | Article |
Language: | English |
Published: |
2024
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Subjects: | |
Online Access: | https://hdl.handle.net/10356/180629 |
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Institution: | Nanyang Technological University |
Language: | English |
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. |
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