Finite powers of selectively pseudocompact groups

TL;DR

The paper constructs topological groups with specific powers that are countably compact but not selectively pseudocompact, under the assumption of CH, addressing a previously open question.

Contribution

It demonstrates, assuming CH, the existence of groups whose finite powers exhibit countable compactness without being selectively pseudocompact, answering a known open problem.

Findings

Existence of groups with countably compact powers but not selectively pseudocompact powers

Construction under CH for all integers greater than 2

Positive resolution of a question from prior literature

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 $cl_X(\{x_n : n < \omega\}) \setminus \big(\bigcup_{n < \omega}U_n \big) \neq \emptyset$ 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 $ZFC+CH$.