I am supposed to show that if
$ξ : X' → X$ is a covering of a locally path-connected space $X$ all of whose path-components are
1-connected, then
$X'$ is homeomorphic to a disjoint union of copies of $X$ but I don't have much of idea how to do this..
I was trying to sketch a proper commutative diagram.
Furthermore, if, in addition, $X'$ is 0-connected, then $ξ$ is a homeomorphism.