Effective bases and notions of effective second countability in computable analysis

2025-09-24

AI Summary

This paper addresses the lack of consensus on the definition of “computable topological bases” in computable analysis, systematically clarifying the logical relationships among various notions of computable second-countability and their connections to the Sierpiński representation and computable metrizability. By integrating tools from represented spaces theory, computably enumerable bases, strong computable regularity, and open choice problems, the authors prove that several ostensibly distinct base concepts become equivalent under the assumption of computable enumerability—thereby establishing a robust definition of computable second-countable spaces. Key contributions are: (1) a unifying compatibility framework and hierarchy relating diverse approaches to computable topology; (2) an effective metrization theorem precisely characterizing represented spaces embeddable into computable metric spaces; and (3) a deep connection between non-surjective open choice and effective second-countability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Social Networks and Social Media: Computational social science

Abstract

We investigate different notions of "computable topological base" for represented spaces. We show that several non-equivalent notions of bases become equivalent when we consider computably enumerable bases. This indicates the existence of a robust notion of computably second countable represented space. These spaces are precisely those introduced by Grubba and Weihrauch under the name "computable topological spaces". The present work thus clarifies the articulation between Schröder's approach to computable topology based on the Sierpinski representation and other approaches based on notions of computable bases. These other approaches turn out to be compatible with the Sierpinski representation approach, but also strictly less general. We revisit Schröder's Effective Metrization Theorem, by showing that it characterizes those represented spaces that embed into computable metric spaces: those are the computably second countable strongly computably regular represented spaces. Finally, we study different forms of open choice problems. We show that having a computable open choice is equivalent to being computably separable, but that the "non-total open choice problem", i.e., open choice restricted to open sets that have non-empty complement, interacts with effective second countability in a satisfying way.

Problem

• Investigating different notions of computable topological bases for represented spaces
• Clarifying relationships between Sierpinski representation and computable base approaches
• Studying open choice problems and their interaction with effective second countability

Innovation

• Investigates computable topological bases for represented spaces
• Shows computably enumerable bases yield robust second countability
• Revisits Effective Metrization Theorem for computable metric embeddings