Let B be path-connected and p:E→B covering map (with E as covering space). Prove that ∀a,b∈B exist 1-1 injection correspondence between p−1(a) and p−1(b)
I thought somehow taking the path between a to b and take on that the inverse means p−1(γ(t)) but that’s not well defined. How can I prove that?
I really like this proof because it uses a pretty neat trick:
Declare an equivalence relation on B by saying a∼b if there exists a bijection between p−1(a) and p−1(b). Try to prove that the equivalence class [x] of x∈B is both open and closed. Since B is path-connected (and hence connected), this implies that [x]=B, which completes the proof. To prove that [x] is open, use the definition of a covering map. To prove it’s closed, use the fact that the equivalence relation creates a partition of the set.
The set p−1(x) is often called the fiber above x, and this result is often stated by saying that all the fibers have the same cardinality.