We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCℕ0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℕ1 is equivalent to the existence of a Kurepa tree of height ℕ1. 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 BCℕ1.
(2) If it is consistent that BCℕ1, 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 ¬BCℕω + (∀n < ω)BCℕn is consistent.
(4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℕω
(5) If it is consistent that there is a 3-huge cardinal, then it is consistent that BCκ for a proper class of cardinals κ of countable cofinality.
