Mini-course (25.04.2024 - 16.05.2024, 3 lectures) - 3^rd lecture:

Given a cardinal \(\kappa\), a set of reals \(A \subseteq \mathbb R\) is \(\kappa\)-dense if its intersection with any open interval has size \(\kappa\). Baumgartner's axiom (BA) — proved consistent by Baumgartner in 1973 — states that all \(\aleph_1\)-dense sets of reals are order isomorphic with the induced linear order from \(\mathbb R\). This is the most straightforward generalization to the uncountable of Cantor's proof that all countable dense linear orders without endpoints are order isomorphic. BA has variations to other topological spaces — given a topological space \(X\), a subset \(A \subseteq X\) is \(\kappa\)-dense if its intersection with each non-empty open subset has size \(\kappa\). The axiom BA(\(X\)) states that given any two \(\aleph_1\)-dense subsets of \(X\), say \(A\) and \(B\), there is an autohomeomorphism of \(X\) mapping \(A\) onto \(B\). In this parlance BA is equivalent to BA(\(\mathbb R\)). Surprisingly BA is not equivalent to BA(\(\mathbb R^n\)) for any finite \(1< n < \omega\). In fact BA does not follow from Martin's Axiom (Abraham-Rubin-Shelah) though BA(\(\mathbb R^n\)) does (in fact from \(\mathfrak p > \aleph_1\)) for each \(n > 1\) (Steprāns-Watson).