arXiv subjects
Publications and source records attributed to Gabriel Debs.
We compute the exact complexity of the set of all arc-connected compact subsets of $\boldmath R^2$, which turns out to be strictly higher than the classical $\boldmath \Sigma^1_1$ and $\boldmath \Pi^1_1$ classes of analytic and coanalytic sets, but stricly lower than the class $\boldmath \Pi^1_2$ which is the exact descriptive class of the set of all arc-connected compact subsets of $\boldmath R^3$.
Given a space $X$ we investigate the descriptive complexity class $\G_X$ of the set $\FF_0(X)$ of all its closed zero-dimensional subsets, viewed as a subset of the hyperspace $\FF(X)$ of all closed subsets of $X$. We prove that $\max \{ \G_X; \ X \text{ analytic } \}=\pca $ and $\sup \{ \G_X; \ X \text{ Borel } \borm ξ\} \supseteq \Game \bora ξ$ for any countable ordinal $ξ\geq1$. In particular we prove that there exists a one-dimensional Polish subpace of $2^\wo\times \R^2$ for which $\FF_0(X)$ is not in the smallest non trivial pointclass closed under complementation and the Souslin operation $\mathcal A\,$.