Classifying Erdős type spaces of higher descriptive complexity

Abstract

Let \({({E_n})_{n \in \omega }}\) be a sequence of zero-dimensional subsets of the reals, ℝ. The Erdős type space ɛ corresponding to this sequence is defined by ɛ = {x ∈ l 2: x n ∈ E n , n ∈ω}. The most famous examples are Erdős space, with E n equal to the rationals for each n, and complete Erdős space, with E n = {0} ∪ {1/m: m ∈ ℕ} for each n. If all sets E n are \({G_\delta }\) and the space ɛ is not zero-dimensional, then ɛ is known to be homeomorphic to complete Erdős space, and if all sets E n are \({F_{\sigma \delta }}\), then under a mild additional condition ɛ is known to be homeomorphic to Erdős space. In this paper we investigate the situation where all sets E n are Borel sets in the same multiplicative class. Many of these spaces can be linked to the Erdős type space with all sets E n equal to the element of that multiplicative Borel class which absorbs the class. Furthermore, we introduce coanalytic Erdős space and we establish a link between this space and homeomorphism groups of manifolds that leave the zero-dimensional pseudoboundary invariant. The general framework that we develop gives analogous results for nonseparable Erdős type spaces.