No Product Theorem for the Covering Dimension of Topological Groups

TL;DR

This paper constructs examples of Lindel"of topological groups and spaces demonstrating that the product of zero-dimensional groups can have positive covering dimension, challenging assumptions about dimension preservation.

Contribution

It provides the first known examples of zero-dimensional Lindel"of topological groups whose product has positive dimension and spaces with non-zero-dimensional free topological groups.

Findings

Product of zero-dimensional groups can have positive covering dimension.

Lindel"of zero-dimensional spaces can have free topological groups that are not zero-dimensional.

Counterexamples to the Product Theorem for covering dimension.

Abstract

Two (strongly) zero-dimensional Lindel\"of topological groups whose product has positive covering dimension are constructed. An example of a Lindel\"of (strongly) zero-dimensional space whose free and free Abelian topological groups are not strongly zero-dimensional is given.