Computably discrete represented spaces

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.

💡 Research Summary

The paper investigates the notion of computably discrete represented spaces—those whose equality predicate is semidecidable—and shows that, despite being classically isomorphic to an initial segment of the natural numbers, such spaces exhibit a surprisingly rich hierarchy when viewed through the lens of computable topology. The authors first formalize computable discreteness: a represented space X is computably discrete if the diagonal Δ_X ⊆ X×X is a computably open set, equivalently if equality of points can be semi‑decided. They contrast this with computably Hausdorff spaces, where the diagonal is computably closed (inequality is semidecidable).

A central result is the exact correspondence between computably discrete, computably quasi‑Polish spaces and computably enumerable equivalence relations (ceers). By representing a ceer R on ℕ as a transitive relation ≪_R and taking the ideal space I(≪_R), the authors prove that I(≪_R) is computably discrete and computably quasi‑Polish, and conversely every such space arises from some ceer. Thus the computable‑isomorphism types of these spaces are in one‑to‑one correspondence with the reducibility hierarchy of ceers.

Leveraging this correspondence, the paper constructs a series of separating examples that illuminate the fine structure of computable topological notions:

1. A computably discrete quasi‑Polish space with no non‑trivial decidable properties. By choosing a ceer whose every non‑trivial subset is non‑decidable, the resulting space answers a question of Emmanuel Rauzy in the negative.

2. A computably discrete, computably Hausdorff pre‑computably quasi‑Polish space that admits no computable injection into ℕ. This shows that even when a space is both discrete and Hausdorff, it may still be “too large” to be computably embedded into the natural numbers.

3. Two‑point spaces demonstrating the independence of discreteness and Hausdorffness. One space is computably Hausdorff but not discrete, while the other is discrete but not Hausdorff, establishing that the two notions are incomparable in the computable setting.

4. An extension of Weihrauch’s example separating computably regular from computably normal spaces. The authors refine the construction to produce a computably regular space that fails to be computably normal, highlighting subtleties absent in the classical theory.

The paper also provides a thorough background on represented spaces, admissibility, effective countable bases, overt sets, and computable metric spaces, ensuring that the reader can follow the technical constructions. It proves that every represented space embeds as a subspace of an effectively countably based one, and that admissible, effectively countably based spaces admit fibre‑overt representations, which are crucial for the later constructions.

In the final section, the authors discuss the implications for the classification of ℕ itself: the various computable‑isomorphism types of ℕ correspond precisely to the degrees of ceers under computable reducibility. This bridges computable topology with the well‑studied area of ceer theory, suggesting new avenues for cross‑disciplinary research.

Overall, the work demonstrates that the seemingly modest condition of semidecidable equality gives rise to a rich landscape of computable topological phenomena, provides concrete counterexamples separating several computable analogues of classical separation axioms, and establishes a deep connection between computable topology and the theory of computably enumerable equivalence relations.