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Computably isometric spaces. (English) Zbl 1332.03008

Computable categoricity has been extensively studied in computable algebra and computable model theory. The paper under review extends these studies to the analytic context, exploring computable categoricity in computable metric spaces and Banach spaces. For example, it shows that Cantor space, Urysohn space, and Hilbert spaces are computably categorical; but the spaces \(\mathcal C[0,1]\) and \(l_1\) are not. The paper also makes interesting connections between the Banach space structure on a computable metric space, and its uniqueness as a computable object. The paper also studies subsets of \({\mathbb R}^n\).

MSC:

03C57 Computable structure theory, computable model theory
03D32 Algorithmic randomness and dimension
03D45 Theory of numerations, effectively presented structures
03C35 Categoricity and completeness of theories
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