Borel's Conjecture in Topological Groups

Details Borel's Conjecture in Topological Groups Borel's Conjecture in Topological Groups

2011-07-27

arxiv.org/abs/1107.5383v1

Mathematics

Logic

15 pages

Scientific paper

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. We obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$ holds. (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 ${\sf BC}_{\aleph_{\omega}}$ holds. (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.

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