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

2025/07/20 by Sipacheva, Ol'ga
#22A05 #54F45 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2507.14889

Abstract

Strongly zero-dimensional topological groups G₁, G₂, and G such that G₁× G₂ has positive covering dimension and G contains a closed subgroup of positive covering dimension are constructed. Moreover, all finite powers of G₁ are Lindelöf and G₂ 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.

Citations

• The completeness of free Boolean topological groups
• Free Boolean Topological Groups

Related

• No Product Theorem for the Covering Dimension of Topological Groups
• No Subgroup Theorem for the Covering Dimension of Topological Groups
• Topological Groups with Strong Disconnectedness Properties
• Nonequality of Dimensions for Metric Groups
• Borsuk--Ulam theorems for elementary abelian 2-groups