#### Document Type

Article

#### Publication Date

3-1-2013

#### Abstract

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCℵ_{0} is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℵ_{1} is equivalent to the existence of a Kurepa tree of height ℵ_{1}. 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 BCℵ_{1}.

2. If it is consistent that BCℵ_{1}, 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 ¬BCℵ_{ω}+(∀n<ω)BCℵ_{n} is consistent.

4. If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℵ_{ω}.

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.

#### Copyright Statement

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

#### Publication Information

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