Finite powers of selectively pseudocompact groups

Abstract

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 $c{l}_X(\{x_n:n<\omega \})\setminus \left({\bigcup }_{n<\omega }{U}_n\right)\ne \varnothing$ 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 $ZFC+CH$.

Cited by in corpus (1)

• Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups