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I am looking at two examples of the $n$-torus. Specifically, the cases where $n = 1$ and $n = 2$, that is, $S^1$ and $S^1 \times S^1$. I am trying to see if there is a continuous multiplication with identity element on these two spaces. The unit circle has identity element 1 in the complex plane and consists of all complex numbers $z$ such that $|z| = 1$. I am wondering if a continuous multiplication can be defined on the torus in the same way as the circle but using component-wise multiplication. Does this work?

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A product of topological groups is a topological group in a natural way, so this should work. –  Dylan Moreland Dec 27 '11 at 22:18
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More specifically, if $G$ and $H$ are topological groups, then the set $G\times H$ has a natural topology and a natural group structure, and it is fairly easy to prove that the group operation is continuous under the topology. –  Thomas Andrews Dec 27 '11 at 22:26
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Another way to think of $S^1\times S^1$ as a topological group is to think of it as $\mathbb R^2/\mathbb Z^2 \cong (\mathbb R/\mathbb Z)^2$ –  Thomas Andrews Dec 27 '11 at 22:27

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