2024/11/08 by Nathaniel Bannister, Bannister, Nathaniel
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2411.05967
openalex publication_date 2024/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a transitive model of set theory and X be a space in the sense of M. Is there a reasonable way to interpret X as a space in V? A general theory due to Zapletal provides a natural candidate which behaves well on sufficiently complete spaces (for instance Čech complete spaces) but behaves poorly on more general spaces - for instance, the Zapletal interpretation does not commute with products. We extend Zapletal's framework to instead interpret locales, a generalization of topological spaces which focuses on the structure of open sets. Our extension has a number of desirable properties; for instance, localic products always interpret as spatial products. We show that a number of localic notions coincide exactly with properties of their interpretations; for instance, we show a locale is TU if and only if all its interpretations are T1, a locale is I-Hausdorff if and only if all its interpretations are T2, a locale is regular if and only if all its interpretations are T3, and a locale is compact if and only if all its interpretations are compact.