A space $X$ is called { \it selectively pseudocompact} if for each sequence $(U_{ n})_{ n\in \mathbb{ N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{ n})_{ n\in \mathbb{ N}}$ of points in $X$ such that $cl_X(\{ x_n : n < \omega\}) \setminus \big(\bigcup_{ n < \omega}U_n \big) \neq \emptyset$ and $x_{ n}\in U_{ n}$, for each $n < \omega$. Countably compact space spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of $CH$, that for every positive integer $k > 2$ there exists a topological group whose $k$-th power is countably compact but its $(k+1)$-st power is not selectively pseudocompact. This provides a positive answer to a question posed in \cite{ gt} in any model of $ZFC+CH$.
Categories: math.GN