The descriptive complexity of the set of all closed zero-dimensional\n subsets of a Polish space

2018/05/17 by Gabriel Debs, Debs, Gabriel, Jean Saint Raymond +1
Mathematics · Social Sciences · #54H05 #Advanced Topology and Set Theory #FOS: Mathematics #Historical Geography and Cartography #Logic (math.LO) #Mathematical Dynamics and Fractals #Primary: 03E15

paper · pdf · doi:10.48550/arxiv.1805.06732

openalex publication_date 2018/05/17 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Given a space X we investigate the descriptive complexity class G_X of\nthe set FF₀(X) of all its closed zero-dimensional subsets, viewed as a\nsubset of the hyperspace FF(X) of all closed subsets of X. We prove that\n \max G_X; X analytic = pca and\n \sup G_X; X Borel borm \ξ \⊇ Game bora \ξ for\nany countable ordinal \ξ\≥1. In particular we prove that there exists a\none-dimensional Polish subpace of 2^ wo\× R² for which FF₀(X) is\nnot in the smallest non trivial pointclass closed under complementation and the\nSouslin operation mathcal A ,.\n

Related

• Polish G-spaces, the generalized model theory and complexity
• Descriptive complexity of subsets of the space of finitely generated groups
• Between Polish and completely Baire
• Topological complexity of ideal limit points
• Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property