Covering space, Weyl group, flag manifold.

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...

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