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arXiv · 2504.07020

Computably discrete represented spaces

Eike Neumann·Arno Pauly·Cécilia Pradic·Manlio Valenti
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Abstract

In computable topology, a represented space is called computably discrete if its equality predicate is semidecidable. While any such space is classically isomorphic to an initial segment of the natural numbers, the computable-isomorphism types of computably discrete represented spaces exhibit a rich structure. We show that the widely studied class of computably enumerable equivalence relations (ceers) corresponds precisely to the computably Quasi-Polish computably discrete spaces. We employ computably discrete spaces to exhibit several separating examples in computable topology. We construct a computably discrete computably Quasi-Polish space admitting no decidable properties, a computably discrete and computably Hausdorff precomputably Quasi-Polish space admitting no computable injection into the natural numbers, a two-point space which is computably Hausdorff but not computably discrete, and a two-point space which is computably discrete but not computably Hausdorff. We further expand an example due to Weihrauch that separates computably regular spaces from computably normal spaces.

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Eike Neumann, Arno Pauly, Cécilia Pradic, Manlio Valenti. 2025-04-09. Computably discrete represented spaces. https://doi.org/10.1007/978-3-031-95908-0_25

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