Weak Baumgartner axioms and universal spaces
Corey Bacal Switzer
TL;DR
The paper studies uncountable topological analogues of Baumgartner’s axiom by introducing $\mathsf{BA}_\kappa(X)$ and weakenings $\mathsf{BA}^-_\kappa(X)$ and $\mathsf{U}_\kappa(X)$ for perfect Polish spaces. It shows $\mathsf{BA}^-_\kappa(X)$ preserves several key consequences of $\mathsf{BA}_\kappa(X)$ (e.g., $2^{\aleph_0}=2^\kappa$ and, for $\kappa=\aleph_1$, $\mathfrak b>\aleph_1$ and $2^{\aleph_1}=2^{\aleph_0}$), while $\mathsf{U}_\kappa(X)$ remains independent of $\mathsf{ZFC}$ and can fail in Cohen and random models. The authors introduce and analyze avoiding notions—$X$ avoids $Y$, strongly avoids, and totally avoids—and show these properties can be preserved under ccc forcing iterations, enabling a robust no-step-up phenomenon where BA-type behavior at small sizes does not force universal structures at larger sizes. The work also develops forcing tools (Medini forcing and Shelah-type iterations) to separate these axioms, yielding new consistency results and raising open questions about the precise relationships between the weakenings and the full Baumgartner axiom.
Abstract
If $X$ is a topological space and $κ$ is a cardinal then $\mathsf{BA}_κ(X)$ is the statement that for each pair $A, B \subseteq X$ of $κ$-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 $\mathsf{BA}_κ(X)$ which we call $\mathsf{BA}^-_κ(X)$ and $\mathsf{U}_κ(X)$ for arbitrary perfect Polish spaces $X$. We show that the first of these, though properly weaker, entails many of the more striking consequences of $\mathsf{BA}_κ(X)$ 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.