September 6
Corey Switzer,
Kurt Gödel Research Center
Weak and Strong Variants of Baumgartner's Axiom for Polish Spaces
(One version of) Cantor's second best theorem states that every pair of countable, dense sets of reals are isomorphic as linear orders. From the perspective of set theory it's natural to ask whether some variant of this theorem can hold consistently when 'countable' is replaced by 'uncountable'. This was shown in the affirmative by Baumgartner in 1973 who showed the consistency of 'all $\aleph_1$-dense sets of reals are order isomorphic' where a set is $\kappa$-dense for a cardinal $\kappa$ if its intersection with any open interval has size $\kappa$. The above became known as Baumgartner's axiom, denoted BA, and is an important axiom in both combinatorial set theory and set theoretic topology. BA has natural higher dimensional analogues - i.e., statements with the same relation to $\mathbb R^n$ that BA has to $\mathbb R$. It is a long standing open conjecture of Steprāns and Watson that BA implies its higher dimensional analogues.
In the talk I will describe some attempts to break the ice on this open problem mostly by looking at a family of weaker and stronger variants of BA and investigating their combinatorial, analytic and topological consequences. We will show that while some weak variants of BA have all the same consequences as BA, even weaker ones do not. Meanwhile a strengthening of BA for Baire and Polish space gives much more information.