The descriptive complexity of series rearrangements (Q2510963)

For the discrete topological space \(\omega = \{0, 1, 2, \dots\}\) denote by \(S_\infty\) the set of all permutations \(\pi: \omega \to \omega\), endowed with the topology of pointwise convergence. Given a conditionally convergent series \(\sum_{n=0}^\infty x_n\) of vectors \(x_n \in {\mathbb R}^d\), the author studies properties of the set \(D \subset S_\infty\) of those permutations \(\pi\) for which the rearranged series \(\sum_{n=0}^\infty x_{\pi(n)}\) diverges. It is shown that both \(D\) and \(S_\infty \setminus D\) are uncountable and dense, that \(S_\infty \setminus D\) is a meager set, and that \(D\) is a \(G_{\delta \sigma}\) but not an \(F_{\sigma \delta}\) set.