Curve lifting property

Question

I'm reading forster's riemann surfaces book and I read this theorem: if we have a covering map between two topological spaces X,Y then it has curve lifting property, now I want to make a counter example for the inverse,but I couldn't find. Why canonical injection on open unit disk doesn't work ?could anyone help me with this problem?

Answer 1

Let $p : S^1 \times \mathbb R \to S^1$ denote the projection. Then each curve $u : [0,1] \to S^1$ has a lift, for example $\tilde u(t) = (u(t),0)$. But $p$ is not a covering map because the fibers $p^{-1}(z) = \{z\} \times \mathbb R$ are not discrete.

Answer 2

The relevant result is that if $p: M\to N$ is a local homeomorphism between manifolds and $p$ has the curve-lifting (usually called path-lifting) property, then $p$ is a covering map. You can find a proof for instance in do Carmo's book "Geometry of curves and surfaces." In the context of Riemann surfaces $M, N$, you probably will be using a holomorphic map $p: M\to N$ that has nowhere vanishing derivative.