I'm trying to prove the following statement:
Let $\gamma_1$ and $\gamma_2$ be two closed curves from $[a, b]$ to $\mathbb{C}$, and let $h: [a,b]\times[0,1] \to \mathbb{C}$ be a homotopy between them. For any fixed $y_0 \in [0, 1]$, $h(x, y_0)$ is a closed curve $[a, b] \to \mathbb{C}$.
I'm not being able to make any progress though. Actually, I'm starting to suspect that it is not true, once I'm also failing to see why "opening" the closed curve and then closing it again fails to be a homotopy - it must fail, otherwise a loop with two windings around the origin could be homotopic to a loop with only one winding, which is not true!
PS: if it is actually false, I would be interested in the following weaker result, which may be simpler to prove:
Let $\gamma_1$ and $\gamma_2$ be two homotopic closed curves from $[a, b]$ to $\mathbb{C}$. Then, there exists a homotopy $h: [a, b] \times [0, 1] \to \mathbb{C}$ that satisfies: for any fixed $y_0 \in [0,1]$, $h(x, y_0)$ is a closed curve $[a, b] \to \mathbb{C}$.
That's all I need actually, to prove the FTA. But I also suspect that the first, stronger one, can be true.