Set $\Bbb T^n = \Bbb R^n \setminus \Bbb Z^n$. Define $d : \Bbb T^n \times \Bbb T^n \to \Bbb R$ by
$$d(x+\Bbb Z^n,y+\Bbb Z^n)=\inf\{\|v-w\|:v\in x+\Bbb Z^n, \text{ and } w \in y+ \Bbb Z^n\}$$
Show that $ \Bbb T^n$ is complete and compact
My thought : I can prove $\Bbb T^n$ is complete by using completeness of $\Bbb R^n$. When it comes to compactness, I tried to give a homeomorphism $\psi : S^1\times \cdots \times S^1 \to \Bbb T^n$ by $\psi((e^{2 \pi ix_1}, e^{2 \pi ix_2},\ldots, e^{2 \pi ix_n}))= x+\Bbb Z$, where $x=(x_1,x_2,\ldots,x_n).$ Clealy, $\psi$ is bijective and If we can show it is continuous, $\psi$ is homeomorphism because $S^1\times \cdots \times S^1$ is compact and $\Bbb T^n$ is hausdorff. Thus, $\Bbb T^n$ is compact
However, I was stuck in showing $\psi$ is continuous.While posting on it, I realized it suffices to show $\Bbb T^n$ is bounded and closed. Anyway, Could you give me a few hints..??