Uniqueness of Lifts Lemma

Let be a covering map, and a map.
Let be two lifts of .
Then is open and closed in .

Proof

open

Fix . Let evenly covered nbd in .
Say
Have , .
Set (open in )
As is a homeomorphism, get for all so so is open.

closed

Let ie .
Let
as before

Have ,
open in .
As and are homeomorphisms, get so is open