The power of trees

TL;DR

The paper advances the study of non-preservation under products for interval-topology trees by providing two consistent constructions. First, under a diamond-type principle, it builds an $ ext{R-embeddable}$ almost-Souslin $oldsymbol{ ext{aleph}_1}$-tree $T$ with $X_T$ perfectly normal yet $(X_T)^2$ fails to be cmc, demonstrating nonproductivity of cmc in this class. Second, it introduces a proxy-principle framework at an inaccessible κ to produce a κ-tree whose all $n$-derived trees are Souslin but all $(n+1)$-derived trees are special, illustrating delicate control over derived-tree properties. The methods hinge on forcing and proxy principles (including $ ext{$P_<$}$ variants) combined with canonical tree/$C$-sequence techniques, yielding new consistency results about the interaction between interval-topology, normality, and derived-tree behavior. These results illuminate the landscape of tree-based constructions with prescribed topological and combinatorial features and connect classical set-theoretic tools to topological consequences.

Abstract

We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$: 1. An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact. 2. For an inaccessible $κ$ and a positive integer $n$, a $κ$-tree such that all of its $n$-derived trees are Souslin and all of its $(n+1)$-derived trees are special.

Theorems & Definitions (71)

• Theorem A
• Theorem B
• Definition 2.1
• Definition 2.2
• Remark 2.4
• Definition 2.5: Two types of square
• Remark 2.6
• proof
• Definition 2.8
• Corollary 2.9