BOREL%26apos;S CONJECTURE IN TOPOLOGICAL GROUPS

作者：Galvin Fred^*; Scheepers Marion

来源：Journal of Symbolic Logic, 2013, 78(1): 168-184.

DOI：10.2178/jsl.7801110

• Rothberger bounded
• Borel Conjecture
• Kurepa Hypothesis
• Chang%26apos
• s Conjecture
• n-huge cardinal

We introduce a natural generalization of Borel%26apos;s Conjecture. For each infinite cardinal number kappa, let BC kappa denote this generalization. Then BC aleph 0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, (sic)BC aleph 1 is equivalent to the existence of a Kurepa tree of height aleph(1). Using the connection of BC kappa with a generalization of Kurepa%26apos;s Hypothesis, we obtain the following consistency results: %26lt;br%26gt;(1) If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BC aleph 1. %26lt;br%26gt;(2) If it is consistent that BC aleph 1, then it is consistent that there is an inaccessible cardinal. %26lt;br%26gt;(3) If it is consistent that there is a 1-inaccessible cardinal with omega inaccessible cardinals above it, then (sic)BC aleph omega + (for all n %26lt; omega)BC aleph n is consistent. %26lt;br%26gt;(4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BC aleph omega. %26lt;br%26gt;(5) If it is consistent that there is a 3-huge cardinal, then it is consistent that BC kappa for a proper class of cardinals kappa of countable cofinality.

• 出版日期2013-3