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?
Multiplication on Torus
3
$\begingroup$
general-topology
-
4A product of [topological groups](http://en.wikipedia.org/wiki/Topological_group) is a topological group in a natural way, so this should work. – 2011-12-27
-
1More 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. – 2011-12-27
-
1Another way to think of $S^1\times S^1$ as a topological group is to think of it as $\mathbb R^2/\mathbb Z^2 \cong (\mathbb R/\mathbb Z)^2$ – 2011-12-27