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    • Computably and Punctually Universal Spaces

    Ramil Bagaviev,Ilnur I. Batyrshin,Nikolay Bazhenov, Dmitry Bushtets, Marina Dorzhieva, Heer Tern Koh,Ruslan Kornev,Alexander G. Melnikov,Keng Meng Ng

    ANNALS OF PURE AND APPLIED LOGIC（2025）

    Kazan Fed Univ | Kazan Federal University | Sobolev Inst Math | Novosibirsk State Univ | Victoria Univ Wellington | Nanyang Technol Univ

    Cited 0|Views8
    Abstract
    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 Sierpi & nacute;ski 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 2 omega; omega ; 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. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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    Key words
    Universal spaces,Punctual,Computable Polish space,Continuous functions on the unit,interval,Primitive recursive analysis
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    【要点】：论文证明了单位区间上连续实值函数空间C[0,1]的标准可计算呈现是可计算且逐点普适的，解决了Sierpiński在20世纪40年代提出的问题，并展示了有效紧致、逐点Stone空间可逐点同构嵌入Cantor空间。
    

    【方法】：通过构建数学证明，利用现代可计算性理论来分析并验证空间C[0,1]和Urysohn构造的普适可分Polish空间U的普适性。
    

    【实验】：本文主要进行理论研究和证明，未涉及具体实验。无数据集名称和结果。