Borel's Conjecture in Topological Groups

Fred Galvin, Marion Scheepers

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

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.

Original languageAmerican English
Pages (from-to)168-184
Number of pages17
JournalJournal of Symbolic Logic
Volume78
Issue number1
DOIs
StatePublished - Mar 2013

Keywords

  • Borel Conjecture
  • Chang's Conjecture
  • Kurepa Hypothesis
  • Rothberger bounded
  • n-huge cardinal

EGS Disciplines

  • Mathematics

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