Borel's conjecture in topological groups

Abstract

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let ${\sf BC}_{\kappa}$ 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 ${\sf BC}_{\kappa}$ 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 ${\sf BC}_{\aleph_1}$, then it is consistent that there is an inaccessible cardinal.

3. If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} + (\forall n < \omega){\sf BC}_{\aleph_n}$ is consistent.

4. If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_{\omega}}$.

5. If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_{\kappa}$ for a proper class of cardinals $\kappa$ of countable cofinality.