On diagonal resolvable spheres (Q386846)

\textit{A. Arhangel'skii} [Quest. Answers Gen. Topology 27, No. 2, 83--105 (2009; Zbl 1191.54028)] introduced and studied the following notion: A topological space \(X\) is called a \textit{diagonal resolvable space} if for every point \(z\in X\) there exists a homeomorphism \(h_z: X\times X \to X\times X\) such that \(h_z(\Delta_X)=\{ z\}\times X\), where \(\Delta_X :=\{ (x,x): x\in X\}\) is the diagonal. Answering a question of Arhangel'skii posed in the aforementioned paper, the authors characterize diagonal resolvable spheres: The sphere \(S^n\) is diagonal resolvable if and only if \(n\in\{ 0,1,3,7\}\). The second main result of the article is the following: Every upper half even dimensional \(\mathbb{Q}\)-sphere is not diagonal resolvable.