Failure of the product and subgroup theorems for the covering dimension of topological groups
Volume 275 / 2026
Fundamenta Mathematicae 275 (2026), 45-61
MSC: Primary 22A05; Secondary 54F45
DOI: 10.4064/fm250720-26-4
Published online: 7 September 2026
Abstract
Strongly zero-dimensional topological groups $G_1$, $G_2$, and $G$ such that $G_1\times G_2$ has positive covering dimension and $G$ contains a closed subgroup of positive covering dimension are constructed. Moreover, all finite powers of $G_1$ are Lindelöf and $G_2$ is second-countable. An example of a Lindelöf strongly zero-dimensional space $X$ whose free, free Abelian, and free Boolean topological groups have positive covering dimension is also given.
