TITLE: Covering for connected and locally path connected spaces
QUESTION [5 upvotes]: Under the condition that the spaces (or maybe just the total) are connected and locally path connected, is then the a covering the same as a homeomorphism?

REPLY [7 votes]: Dear Down, To prove the surjectivity, you will have to use connectivity of $E'$ (otherwise the result is not true). You want to show that $f(E)=E'$. Here are some hints: (i) think about what properties you need to prove for $f(E)$ to get this equality. (This is where connectedness will be used). (ii) Prove them using the covering space properties.

– Matt E Sep 15 '10 at 18:49 

Incidentally, this statement (that $E\to B$ is surjective) is also true if we are working with a fibration. This is actually a hint: use path lifting.

– Akhil Mathew Sep 16 '10 at 2:41