A Proof of the Tree Alternative Conjecture Under the Topological Minor Relation

We prove the Tree Alternative Conjecture for the topological minor relation: letting $[T]$ denote the equivalence class of $T$ under the topological minor relation we show that: $\mathrm{\mid }[T]\mathrm{\mid }=1$ or $\mathrm{\mid }[T]\mathrm{\mid }\ge {\mathrm{\aleph }}_0$ and $\mathrm{\forall }r\in V(T)$, $\mathrm{\mid }[(T,r)]\mathrm{\mid }=1$ or $\mathrm{\mid }[(T,r)]\mathrm{\mid }\ge {\mathrm{\aleph }}_0$. In particular, by means of curtailing trees, we show that for any tree $T$ with at least one not eventually bare ray: $\mathrm{\mid }[T]\mathrm{\mid }\ge 2^{{\mathrm{\aleph }}_0}$.