The descriptive complexity of connectedness in Polish spaces
Volume 249 / 2020
Fundamenta Mathematicae 249 (2020), 261-286
MSC: Primary 03E15, 28A05, 54D05; Secondary 54H05.
DOI: 10.4064/fm754-7-2019
Published online: 20 December 2019
Abstract
We investigate the descriptive complexity of connectedness (and also pathwise connectedness and local connectedness) of Polish spaces, and prove that even in the framework of finite-dimensional euclidean spaces this complexity can be the highest possible, and much beyond the first projective classes $\boldsymbol{\Sigma}^1_1 $ and $\boldsymbol{\Pi}^1_1 $. In particular we prove that several of these notions are $\boldsymbol{\Pi}^1_2 $-complete.
