Computably discrete represented spaces

2025-04-09

AI Summary

This paper addresses the structural analysis and classification of computably discrete representable spaces in computable topology, focusing on computational isomorphism complexity arising from semidecidability of the equality predicate. Methodologically, it establishes a bijective correspondence between such spaces and computably enumerable equivalence relations (ceers)—the first such characterization—thereby enabling a complete classification of computable quasi-Polish discrete spaces. The authors systematically construct separating counterexamples that rigorously delineate the strict inclusion hierarchy among computable separation axioms (T₀–T₃). Key contributions include: (i) a refined understanding of the essence of computable separation; (ii) identification of fundamental phenomena such as non-computable embeddings and the absence of undecidable properties; and (iii) a novel paradigm for model-theoretic and classification-theoretic investigations in computable topology. The results unify foundational concepts from computability theory and descriptive set theory within a topological framework, advancing both theoretical foundations and methodological tools.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Characterize computably discrete represented spaces in computable topology
• Link computably enumerable equivalence relations to computably Quasi-Polish spaces
• Construct separating examples to illustrate distinctions in computable topology

Innovation

• Computably discrete spaces for ceers classification
• Constructing computably discrete Quasi-Polish spaces
• Separating computable topology properties via examples