This question started bothering me after working on an exercise. I know that there cannot be a contiuous bijection $S^1 \to (a,b)$ because if there was it would be a homeomorphism but $S^1$ and $(a,b)$ are not homeomorphic.
But the theorem that implies this is that a continuous bijection from a compact into a Hausdorff space is a homeomorphism. Hence it cannot be applied to the opposite direction.
I still suspect that the answer will turn out to be negative, i.e. there is no continuous bijection from $(a,b) \to S^1$ but I don't see how to prove it because the inverse is not required to be continuous. So somehow there still remains a faint possibility for such a map.
Please could someone help me resolve my confusion and tell me whether there is or is not a continuous bijection $(a,b) \to S^1$?