Borel's Conjecture in Topological Groups

Abstract

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $κ$, let {\sf BC}$_κ$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_κ$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (2)If it is consistent that ${\sf BC}_{\aleph_1}$ holds, then it is consistent that there is an inaccessible cardinal. (3)If it is consistent that there is a 1-inaccessible cardinal with $ω$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_ω} \, +\, (\forall n<ω){\sf BC}_{\aleph_n}$ is consistent. (4)If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_ω}$. (5)If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_κ$ holds for a proper class of cardinals $κ$ of countable cofinality.