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Math Help - A question in topological group

  1. #1
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    Cool A question in topological group

    Hey guys, I was just thinking of a question in topological group. Suppose A and B are two closed sets in a topological group, then A*B is closed. Can any of you help me think of a counter example of this please? Or, if you think it is true, then you may prove it.

    Thanks a lot.
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  2. #2
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    Quote Originally Posted by luoginator View Post
    Hey guys, I was just thinking of a question in topological group. Suppose A and B are two closed sets in a topological group, then A*B is closed. Can any of you help me think of a counter example of this please? Or, if you think it is true, then you may prove it.
    This need not be true even in an Abelian group. For example, a normed vector space is a topological group under addition (just ignore the scalar multiplication), and it is possible for the sum of two closed subspaces to be non-closed. A specific example of that is given here.
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    Thanks, but what about the multiplication. A*B means a*b for a belongs to A and b belongs to B. Thanks
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  4. #4
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    I have a counterexample:
    Take the multiplicative group of the reals
    G:=\mathbb{R}-\{0\}

    <br />
A:=\mathbb{Z}-\{0\} <br />
    <br />
B:=\{\frac{1}{n} : n \in \mathbb{N} , n\geq 1\}<br />
    are closed in G, but
     A\cdot B=\mathbb{Q}-\{0\}
    isn't.
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  5. #5
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    Nice Counter Example. thx
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  6. #6
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    Quote Originally Posted by luoginator View Post
    Thanks, but what about the multiplication. A*B means a*b for a belongs to A and b belongs to B. Thanks
    A vector space is an additive group, so the group operation is addition.

    Iondor's example is a lot simpler and neater than the one I gave. But the vector space example shows that you can even take A and B to be closed subgroups of a topological group, and their "product" (which is their sum in this case) can still fail to be closed.
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