The power of trees

TL;DR

This paper constructs specific trees with unique topological and combinatorial properties, demonstrating significant differences between their finite powers and derived trees.

Contribution

It introduces two novel constructions of trees with contrasting properties in their finite powers and derived trees, advancing understanding in set-theoretic topology.

Findings

An $oldsymbol{ ext{aleph}_1}$-tree with a perfectly normal interval topology but non-countably metacompact square.

A $oldsymbol{ ext{kappa}}$-tree with all $n$-derived trees Souslin and all $(n+1)$-derived trees special.

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 $\kappa$ and a positive integer $n$, a $\kappa$-tree such that all of its $n$-derived trees are Souslin and all of its $(n+1)$-derived trees are special.