Computably discrete represented spaces
Eike Neumann, Arno Pauly, Cécilia Pradic, Manlio Valenti
TL;DR
The paper develops a detailed theory of computably discrete represented spaces, showing that quotients of $\mathbb{N}$ by computably enumerable equivalence relations (ceers) exactly capture the computably discrete computably Quasi-Polish spaces. It provides a rich set of separating examples that distinguish computable analogues of classical notions such as discreteness, Hausdorffness, regularity, and normality, including finite two-point counterexamples and a construction showing no nontrivial decidable properties can exist in some computably discrete spaces. The work further analyzes precomputably Quasi-Polish spaces, including the S_A family parameterized by $A$, and investigates the computable-isomorphism types of $\mathbb{N}$, culminating in open questions about unique characterizations. By connecting computable topology with ceers and effective bases, the paper yields new structural insights and counterexamples that sharpen our understanding of discreteness, separability, and admissibility in represented spaces.
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.