Im trying to understand the path lifting theorem (in the context of locally trivial fiber bundles presented in the book "Manifolds, Tensor Analysis and Applications", by Jerrold E. Marsden et. al.), stated as follows:
Let $\pi : E \to B$ a $C^{0}$ locally trivial fiber bundle and let $c: [0,1] \to B $ be a continuous path starting at $c(0)=b_0$. Then for each $c_0 \in \pi^{-1}(b_0)$ there is a unique continuous path $\tilde{c}:[0,1] \to E $ such that $\tilde{c}(0)=c_0$ and $\pi \circ \tilde{c} = c$.
I think a conuterexample of this statement is obtained considering the trivial bundle $\mathbb{R}^{2} \times \mathbb{R}$ with the path $c:[0,1] \to \mathbb{R}^{2}$, given by $c(t)=(\cos{2 \pi t}, \sin{2 \pi t})$. Taking $b_0 = (1,0)$ and $c_0 = (1,0,a) \in \mathbb{R}$, any lifted path $\tilde{c}(t)=(\cos{2 \pi t}, \sin{2 \pi t},f(t))$, with $f: [0,1] \to \mathbb{R}$ continuous and $f(0)=a$ would fulfill $\pi \circ \tilde{c}$ and $\tilde{c}(0)=c_0$.
In this example the lifting is not unique, and I would appreciate if someone can show me what am I missing or if there is some property or condition missing in the path lifting theorem I quoted above.
Thank you for your attention.