Weak Baumgartner axioms and universal spaces

TL;DR

This paper explores weakened forms of Baumgartner axioms related to homeomorphisms between dense subsets of Polish spaces, revealing their independence from ZFC and implications for classifying space pairs.

Contribution

It introduces two natural weakenings of Baumgartner axioms, analyzes their consequences, independence, and relationships, and develops new classifications of space pairs based on these axioms.

Findings

 Weakening  entails many consequences of the original axiom.

 One weakening is independent of ZFC and fails in Cohen and random models.

 Introduces new classes of space pairs: avoiding, strongly avoiding, totally avoiding.

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.