Show that the quotient space is Compact and Hausdorff

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Homework Statement


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



The Attempt at a Solution


I first showed that the quotient space was Hausdorff by showing that the set [itex]\{(x_1,x_2): x_1\sim x_2\}[/itex] was closed in [itex] \mathbb{R}^n \times \mathbb{R}^n[/itex] 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

 

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