I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local diffeomorphism between connected manifolds which is not a covering map ?
I found here(When is a local homeomorphism a covering map?) a nice proposition which state that a local homeo from a compact space to a connected Hausdorff space is a covering map.
I am interested in all type of counterexamples, from non-Hausdorff spaces to surfaces, to get a better picture of the differences between covering maps and local homeomorphisms.
Thank you !
Best Answer
There's an error in your statement: you need the domain to be compact, and the range to be connected.
Easy counterexample: restrict the exponential map $\mathbb{R} \rightarrow S^1$ to some interval like (0, 1.5) so that the fibers have different cardinalities.