Fundamental group of Topologists sine curve

Question

How can one prove that the fundamental group of the topologists sine curve is trivial? I haven't been able to make any progress on this. A hint in the right direction preferred over a complete solution.

Answer 1

Supposing you have the parametrization of the curve given here, try to show that any curve with base point $(0,0)$ has to be constant.

Then show that any curve with basepoint not at $(0,0)$ cannot pass through $(0,0)$, and use the fact that the topologist sine curve without $(0,0)$ is homeomorphic to the real line.