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Show that the quotient space is Compact and Hausdorff

  1. Jan 29, 2013 #1
    1. The problem statement, all variables and given/known data
    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].



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