# Show that the quotient space is Compact and Hausdorff

1. Jan 29, 2013

### Theorem.

1. The problem statement, all variables and given/known data
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 $(S^1)^n$.

3. 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.
1. The problem statement, all variables and given/known data