solid state physics - Why is Brillouin zone a torus?

I am told that Brillouin Zone is a torus, which plays an important role in defining chern number and other topological invariants. The argument is that Bloch wave function $\psi_k$ is periodical in reciprocal lattice vector: $$\psi_{k+G}=\psi_k$$.

However, chern number (and also other invariants) is defined using the periodic part of Bloch wave function $u_k$ ($\psi_k=e^{ikr}u_k$), which is not periodic in $k$: $$u_{k+G}=e^{-iGr}u_k$$. In many literature, to prove chern number is an integer, $u_{k+G}=e^{i\theta}u_k$ (notice that $\theta$ is $r$ independent!) is used, which is not consistent with the equation above.

So, in what sense Brillouin zone is a torus and why chern number is an integer with this understanding?