Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true that the range of $\gamma$ is either homeomorphic to a line segment or contains a subset homeomorphic to $S^1$?
• @Richard: How does that argument apply to $\gamma(t)=\sin\pi t$? Then $S$ is not finite and doesn't have a first element (but the image of $\gamma$ is a line segment). – George Lowther Sep 27 '11 at 22:59
@Richard: Real analytic is the same as complex analytic since locally the power series expansions converge on disks. Yes, $\gamma$ is analytic (or has analytic extension) to a neighborhood of $[0,1]$.