A pseudocompact group which is not strongly pseudocompact
A topological space \(X\) is \textit{strongly pseudocompact} if for every sequence \((U_n)_{n\in\mathbb N}\) of pairwise disjoint non-empty open subsets of \(X\) there exists a sequence \((x_n)_{n\in\mathbb N}\) in \(X\) such that \(x_n\in U_n\) for every \(n\in\mathbb N\) and \(cl_X(\{x_n : n \in\mathbb N\})\setminus\left( \bigcup_{n\in\mathbb N}U_n\right)\neq\emptyset\). It is known that for a topological space \[ \text{countably compact } \Rightarrow \text{ strongly pseudocompact } \Rightarrow \text{ pseudocompact}. \] Examples are known of a pseudocompact space that is not strongly pseudocompact and of a strongly pseudocompact group that is not countably compact. Moreover, for a topological group it is equivalent to be strongly pseudocompact and ultrapseudocompact. In this paper, the authors construct a pseudocompact group that is not strongly pseudocompact.
scientific article; zbMATH DE number 6473610
scientific article