There are No Product and Subgroup Theorems for the Covering Dimension of Topological Groups

Ol'ga Sipacheva

paper

There are No Product and Subgroup Theorems for the Covering Dimension of Topological Groups

arXiv:2507.14889

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 strongly zero-dimensional space $X$ whose free, free Abelian, and free Boolean topological groups have positive covering dimension is also given.

This is an improved and complemented version of the preprints arXiv:2207.04961 and arXiv:2303.04593