Connected space and path

Question

Anyone can help me verify this proof? Is it correct to assume there is a path from $b_0$ to $b$ in this proof? If it's correct, why we can assume the existence of a path from $b_0$ to $b$?

---

Answer 1

Yes, the proof assumes a path from $b_0$ to $b$. No, in general you cannot assume that a connected space is path-connected. No, we do not need path-connectedness.

In fact, the proof shows that $|p^{-}(b)|$ is locally constant (because it is constant on the small open neighbourhoods we have from the very definition of covering). Then all we need to remember is that locally constant on a connected space means constant.