2025/02/14 by Corey Bacal Switzer, Switzer, Corey Bacal · 3 citations
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2502.10029
openalex publication_date 2025/02/14 · openalex created_date 2025/02/18 · openalex updated_date 2026/07/28
If X is a topological space and κ is a cardinal then BAκ(X) is the statement that for each pair A, B ⊆ X of κ-dense subsets there is an autohomeomorphism h:X → X mapping A to B. In particular BAℵ1 (\mathbb R) is equivalent the celebrated Baumgartner axiom on isomorphism types of ℵ1-dense linear orders. In this paper we consider two natural weakenings of BAκ(X) which we call BA-κ(X) and Uκ(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 BAκ(X) while the second does not. Nevertheless the second is still independent of 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.