TITLE: The circle is not contractible
QUESTION [10 upvotes]: I know that the circle is not contractible because I know that $\pi_1(S^1)\cong \mathbb Z$. But something is going wrong in my head. Choose a basepoint $*$ on the circle and chose an orientation (clockwise say) and for each point on the circle we take the path from that point to the base point $*$ going clockwise and making only one tour, meaning the first time we meet $*$ we stop. This seems to give us a continuous way to contract the circle onto $*$, making the circle contractible,  what is wrong with this reasoning ?

REPLY [3 votes]: As you try this on points farther and farther along the way, you start approaching the point $*$ from the other side. To be continuous also there, you'd have to drag $*$ itself around the circle once, and then the points slightly past it once and a bit. But that's in contradiction to your rule that you only move them until you hit $*$ the first time. So your contraction will rip the circle right at the point $*$, violating continuity.