Good afternoon,
Please, I would like you to help me with a solution to this problem, if you want:
Let $ \widetilde{X} $ and $ \widetilde{Y} $ be two simply connected covering spaces for linearly connected and locally connected spaces $X$ and $Y$, respectively. Show that if $ X\simeq Y $ (homotopically equivalent), then $ \widetilde{X}\simeq\widetilde{Y}$.
I tried to start from the definition of the concept of covered space and from the definition of the homotopically equivalence between two spaces. Along with these, I read the definition of the concept of lift maps, with the results of existence and uniqueness, but, unfortunately, I fail to build the connection between all of these.
