Let $f:S^1\rightarrow G$ be a continuous homomorphism from the torus onto a profinite group G. Is it true that $f$ must be trivial?
Note that if G is finite then this is true since the kernel is an open subgroup of $S^1$ of finite index (and therefore it is $S^1$). This is also true for a product of finite groups (because the kernel will be the intersection of the kernels on each finite group).