Covering spaces and homotopical equivalence

Question

I have this simple question: if $X$ and $Y$ are two topological spaces homotopically equivalents, have they the "same" covering spaces? (and if yes, in which sense?) This question derive from an exercise, which asks, given four topological spaces, say which admit a covering space of infinite degree. If i found that two of these are homotopically equivalent, and one of these admit a covering space of infinite degree, i would conclude that the second also admits. Any help will be greatly appreciated.

Answer 1

According to the classification of covering spaces, basepoint-preserving isomorphism classes of covering spaces are uniquely determined by the fundamental group, which is invariant by homotopy.

For your exercice, notice that a space has an infinitely-sheeted covering iff its fundamental group is infinite, so that property is invariant by homotopy.