We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCℵ0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℵ1 is equivalent to the existence of a Kurepa tree of height ℵ1. Using the connection of BCκ 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 BCℵ1.
2. If it is consistent that BCℵ1, 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 ¬BCℵω+(∀n<ω)BCℵn is consistent.
4. If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℵω.
5. If it is consistent that there is a 3-huge cardinal, then it is consistent thatBCκ for a proper class of cardinals κ of countable cofinality.
This is an author-produced, peer-reviewed version of this article. The final, definitive version of this document can be found online at Journal of Symbolic Logic, published by The Association for Symbolic Logic. Copyright restrictions may apply. DOI: 10.2178/jsl.7801110
Galvin, Fred and Scheepers, Marion. (2013). "Borel's Conjecture in Topological Groups". Journal of Symbolic Logic, 78(1), 168-184. http://dx.doi.org/10.2178/jsl.7801110