The power of trees

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

A polished version resulting from delivering a series of seminar talks