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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