The Scott topology of a rooted non-metric tree

Abstract

In this paper we prove that the Scott topology $\mathfrak S$ on a rooted non-metric tree $\mathcal T$ is strictly coarser than the weak tree topology. Moreover, for each $t\in \mathcal T$, we consider a natural order $\preceq_t$ on $\mathcal T$ under which $t$ is the root of $\mathcal T$. Then the weak tree topology is generated by the union of the Scott topologies $\mathfrak S_t$ associated to $\preceq_t$.