Covering space, Weyl group, flag manifold.

Question

Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We thus have a map $$ W \longrightarrow G/T \longrightarrow G/{N_{G}}(T). $$ How can I prove that this map is a covering? For example we can say that $G=U(n)$. So $U(n)/T$ is the flag manifold...