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Toplogical group

  1. Jul 2, 2011 #1
    1. The problem statement, all variables and given/known data
    Let H be a subspace of G. show that if H is also a subgroup of G then the closure of H is a topological group
    G is a topological group

    2. Relevant equations



    3. The attempt at a solution
    let closure of H is a subspace of G then the map of the operation of G restricted to cl(H) is continuous
    I have the A2 axiom satisfied for cl(H) but I cant prove the the A1 + A3 +A4
     
  2. jcsd
  3. Jul 2, 2011 #2
    A1-4 doesn't tell us anything, please write them down. It is hard to help you if we don't know which definition of a topological group you use.
     
  4. Jul 2, 2011 #3
    A1: for all x y in G xRy is in G
    A2 : for all x,y ,z in G xR(yRz) = (xRy)Rz
    A3 : for all x in G there exists a 0 in G such that xR0=0Rx=x
    A4: xRx^-1 = 0 = x^-1 R x
     
  5. Jul 2, 2011 #4

    micromass

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    Hi NeroKid! :smile:

    You know that

    [tex]f:G\times G\rightarrow G:(x,y)\rightarrow x*y[/tex]

    is continuous. And since H is a subgroup, you know that

    [tex]f(H\times H)\subseteq H[/tex]

    Now, what can you say about

    [tex]f(cl(H)\times cl(H))[/tex]
     
  6. Jul 2, 2011 #5
    tks i get it know ;))
     
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