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 ${\mathsf{BC}}_{{\aleph }_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg {\mathsf{BC}}_{{\aleph }_1}$ is equivalent to the existence of a Kurepa tree of height ${\aleph }_1$. Using the connection of ${\mathsf{BC}}_{\kappa }$ 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 ${\mathsf{BC}}_{{\aleph }_1}$. (2)If it is consistent that ${\mathsf{BC}}_{{\aleph }_1}$ holds, then it is consistent that there is an inaccessible cardinal. (3)If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg {\mathsf{BC}}_{{\aleph }_{\omega }}+(\forall n<\omega ){\mathsf{BC}}_{{\aleph }_n}$ is consistent. (4)If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\mathsf{BC}}_{{\aleph }_{\omega }}$. (5)If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\mathsf{BC}}_{\kappa }$ holds for a proper class of cardinals $\kappa$ of countable cofinality.

arXiv comment15 pages