1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted