Joint work with Ari Meir Brodsky and Shira Yadai.
Abstract. We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$:
- An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact.
- 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.
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Submitted to Annals of Pure and Applied Logic, October 2025.
Accepted, April 2026.