Weak Baumgartner axioms and universal spaces

Abstract

If $X$ is a topological space and $\kappa$ is a cardinal then $\mathsf{BA}_\kappa (X)$ is the statement that for each pair $A, B \subseteq X$ of $\kappa$-dense subsets there is an autohomeomorphism $h:X \to X$ mapping $A$ to $B$. In particular $\mathsf{BA}_{\aleph_1} (\mathbb R)$ is equivalent the celebrated Baumgartner axiom on isomorphism types of $\aleph_1$-dense linear orders. In this paper we consider two natural weakenings of $\mathsf{BA}_\kappa (X)$ which we call $\mathsf{BA}^-_\kappa (X)$ and $\mathsf{U}_\kappa (X)$ for arbitrary perfect Polish spaces $X$. We show that the first of these, though properly weaker, entails many of the more striking consequences of $\mathsf{BA}_\kappa (X)$ while the second does not. Nevertheless the second is still independent of $\mathsf{ZFC}$ and we show in particular that it fails in the Cohen and random models. This motivates several new classes of pairs of spaces which are ``very far from being homeomorphic" which we call ``avoiding", ``strongly avoiding", and ``totally avoiding". The paper concludes by studying these classes, particularly in the context of forcing theory, in an attempt to gauge how different weak Baumgartner axioms may be separated.