Borel's Conjecture in Topological Groups

TL;DR

This paper generalizes Borel's Conjecture to all infinite cardinals, explores its equivalences and consistency with large cardinal hypotheses, and connects it to Kurepa's Hypothesis in topological groups.

Contribution

It introduces a new family of conjectures ${\sf BC}_{\kappa}$ for all infinite cardinals, linking them to large cardinal axioms and Kurepa's Hypothesis, and establishes their consistency results.

Findings

${\sf BC}_{\aleph_0}$ is equivalent to classical Borel's conjecture.

Assuming Borel's conjecture, ${\sf BC}_{\aleph_1}$ relates to Kurepa trees.

Various large cardinal assumptions imply the consistency of ${\sf BC}_{\kappa}$ for different cardinals.

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 ${\sf BC}_{\aleph_1}$ 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 $\neg{\sf BC}_{\aleph_{\omega}} \, +\, (\forall n<\omega){\sf BC}_{\aleph_n}$ 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 ${\sf BC}_{\kappa}$ holds for a proper class of cardinals $\kappa$ of countable cofinality.