Why can't $S^1$ deformatively retract to a single point?

Question

I know it can't because otherwise the fundamental group of $S^1$ will be the trivial group. But why can't we just cut $S^1$ on any point and unfold the circle to the interval $[0,1)$, then retract $[0,1)$ to $0$? In other words, why is the following $F$ not a homotopy? $$F:S^1\times I\rightarrow S^1, F(\text{e}^{\text{i}\theta},t)=\text{e}^{i(1-t)\theta},\theta\in [0,2\pi)$$