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Some questions about topological groups

  1. Jul 14, 2013 #1
    So, I have a topological group ##G##. This means that the functions

    [tex]m:G\times G\rightarrow G:(x,y)\rightarrow xy[/tex]

    and

    [tex]i:G\rightarrow G:x\rightarrow x^{-1}[/tex]

    are continuous.

    I have a couple of questions that seem mysterious to me.

    Let's start with this: I've seen a statement somewhere that says that if ##H## is a subgroup of ##G## such that ##H## is open, then ##H## is actually also closed. How do I prove such a thing? I know that ##\overline{H}## is a subgroup, do I prove that ##H=\overline{H}##?
     
  2. jcsd
  3. Jul 14, 2013 #2

    jgens

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    Gold Member

    Use the openness of H combined with the fact that you can write G as a disjoint union of the cosets of H.
     
  4. Jul 15, 2013 #3
    Ah yes, I got it now. That's really a nice technique! Thanks a lot.
     
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