Prove that exist bijection between inverse image of covering space [duplicate]

Let B be path-connected and p:EB covering map (with E as covering space). Prove that a,bB exist 1-1 injection correspondence between p1(a) and p1(b)

I thought somehow taking the path between a to b and take on that the inverse means p1(γ(t)) but that’s not well defined. How can I prove that?

Answer

I really like this proof because it uses a pretty neat trick:

Declare an equivalence relation on B by saying ab if there exists a bijection between p1(a) and p1(b). Try to prove that the equivalence class [x] of xB 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 p1(x) is often called the fiber above x, and this result is often stated by saying that all the fibers have the same cardinality.

Attribution
Source : Link , Question Author : Community , Answer Author : manthanomen

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