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Borel's conjecture in topological groups

  • Fred Galvin (a1) and Marion Scheepers (a2)

Abstract

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BC0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BC1 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 BC1, 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 ¬BCω + (∀n < ω)BCn is consistent.

(4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BCω

(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.

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Borel's conjecture in topological groups

  • Fred Galvin (a1) and Marion Scheepers (a2)

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