Corollary 2

Let C be a chamber of $\mathfrak{t}$ and $\overline{C}$ its closure. The canonical injections $\overline{C}\rightarrow \mathfrak{t}\rightarrow \mathfrak{g}$ induce by passage to the quotient homeomorphisms

$\overline{C}\rightarrow \mathfrak{t}/W\rightarrow \mathfrak{g}/$Ad(G).

The canonical maps $\overline{C}\rightarrow \mathfrak{t}$ and $\mathfrak{t}\rightarrow \mathfrak{t}/W$ are proper (General Topology, Chap. III, §4, no. 1, Prop. $2c$)$)$. The map $\overline{C}\rightarrow \mathfrak{t}/W$ is continuous, proper and bijective (Prop. $2c$)$)$; thus, it is a homeomorphism, hence the corollary in view of the Cor. of Prop. 6 of §2, no. 5.