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

TL;DR

This paper constructs examples of strongly zero-dimensional topological groups and spaces where the covering dimension behaves unexpectedly, challenging existing theorems and intuitions in topological group theory.

Contribution

It provides the first known examples showing the failure of product and subgroup theorems for covering dimension in topological groups.

Findings

Existence of strongly zero-dimensional groups with positive covering dimension in products and subgroups.

Construction of a strongly zero-dimensional space with free topological groups of positive covering dimension.

Examples demonstrating the limitations of product and subgroup theorems for covering dimension.

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 Lindel\"of 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.