Show that the quotient space is Compact and Hausdorff

[Theorem.]

Homework Statement

Let $w_1,...,w_n$ be a set of n-linearly independent vectors in $\mathbb{R}^n$. Define an equivalence relation $\sim$ by
$p\sim q \iff p-q=m_1w_1+...+m_nw_n$ for some $m_i \in \mathbb{Z}$
Show that $\mathbb{R}^n / \sim$ is Hausdorff and compact and actually homeomorphic to $(S^1)^n$.

The Attempt at a Solution

I first showed that the quotient space was Hausdorff by showing that the set $\{(x_1,x_2): x_1\sim x_2\}$ was closed in $\mathbb{R}^n \times \mathbb{R}^n$ and that the quotient map was an open map. Using a result I proved earlier in the assignment as well as this, I was able to conclude the quotient space was hausdorff.

It is proving Compactness where I get stuck. So far I have made no progress. Any advice would be much appreciated.

Homework Statement

 

[lippyka]

Let w_1,...,w_n be a set of n-linearly independent vectors in \mathbb{R}^n. Define an equivalence relation \sim by p\sim q \iff p-q=m_1w_1+...+m_nw_n for some m_i \in \mathbb{Z}Show that \mathbb{R}^n / \sim is Hausdorff and compact and actually homeomorphic to (S^1)^n.The Attempt at a SolutionI first showed that the quotient space was Hausdorff by showing that the set \{(x_1,x_2): x_1\sim x_2\} was closed in \mathbb{R}^n \times \mathbb{R}^n and that the quotient map was an open map. Using a result I proved earlier in the assignment as well as this, I was able to conclude the quotient space was hausdorff.For compactness I am using the fact that a finite product of compact spaces is compact. This means I must show that each of the quotient spaces \mathbb{R}/ \sim_i is compact. For each of these, I will show that they are closed and bounded in \mathbb{R}. To do this I will use the fact that the equivalence classes can be written as m_iw_i + k where m_i is an integer and k is a vector in \mathbb{R}^n. This leads to the conclusion that the quotient space is of the form \{z+k : z \in \mathbb{Z}\} which is clearly both closed and bounded in \mathbb{R}. Finally I need to show that \mathbb{R}^n / \sim is homeomorphic to (S^1)^n. To do this I consider the map \phi: \mathbb{R}^n / \sim \rightarrow (S^1)^n defined by \phi([x]) = [e^{2\pi i x_1},...,e^{2\pi i x_n}] where [x] is the equivalence class