Multi-representation associated to the numbering of a subbasis and formal inclusion relations
Emmanuel Rauzy
TL;DR
The paper develops a unified framework for representations based on a numbered subbasis and a strong inclusion relation, showing that relaxing Spreen's strong inclusion yields multiple representations ($\rho_{\beta}^{\min}$, $\rho_{\beta}^{\max}$, $\rho_{\beta}^{\mathring{\subseteq}}$) that can differ in computational behavior. It proves an admissibility theorem tying these representations to the generated topology, and analyzes how these representations interact with subspace embeddings, metric space structures, and equivalence notions for bases. In metric spaces, the strong inclusion approach recovers the Cauchy representation under suitable conditions, and the work also addresses semi-decidability, semi-decidable openness, and totalization, contributing robust tools for comparing bases and ensuring compatibility across subsets. The results advance computable analysis by clarifying how the choice of basis numbering and inclusion relation shapes computability and continuity of representations across general topological and metric settings, with implications for both theory and applications in computer-assisted analysis.
Abstract
We revisit Dieter Spreen's notion of a representation associated to a numbered basis equipped with a strong inclusion relation. We show that by relaxing his requirements, we obtain different classically considered representations as subcases, including representations considered by Grubba, Weihrauch and Schröder. We show that the use of an appropriate strong inclusion relation guarantees that the representation associated to a computable metric space seen as a topological space always coincides with the Cauchy representation. We also show how the use of a formal inclusion relation guarantees that when defining multi-representations on a set and on one of its subsets, the obtained multi-representations will be compatible, i.e. inclusion will be a computable map. The proposed definitions are also more robust under change of equivalent bases.