Local homeomorphisms which are not covering map?

Question

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 !

Answer 1

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.

Answer 2

To your question:

can it be a diffeomorphism between connected manifolds ?

Of course it can't be; a diffeomorphism is automatically a homeomorphism and hence a covering map. I suspect what you meant to ask if it can be locally a homeomorphism between connected manifolds.

For this, the quotient map from the line with double origin to the ordinary line, identifying the two origins, will do.

This is also a non-Hausdorff example that you were looking for.