Corollary 2

Let E be a subset of G stable under inner automorphisms. Then E is open (resp. closed, resp. dense) in G if and only if $E\cap T$ is open (resp. closed, resp. dense) in T.

This follows from Cor. 1 and the fact that the canonical maps $\mathrm{T}\to \mathrm{T}/\mathrm{W}_{G}(\mathrm{T})$ and $\mathrm{G}\to \mathrm{G}/\mathrm{Int}(\mathrm{G})$ are open (General Topology, Chap. III, p. 10, Lemma 2).

Denote the Lie algebra of $\mathrm{G}$ by $\mathfrak{g}$, and that of $\mathrm{T}$ by $\mathfrak{t}$. The operation of $\mathrm{W}_{G}(\mathrm{T})$ on $\mathrm{T}$ induces a representation, called the canonical representation, of the group $\mathrm{W}_{G}(\mathrm{T})$ on the $\mathbf{R}$-vector space $\mathfrak{t}$.