Re: Local Homeomorphisms

On 17.04.2008 09:26, William Elliot wrote:
On Wed, 16 Apr 2008, Jannick Asmus wrote:
On 16.04.2008 10:18, William Elliot wrote:

Let f:X -> Y be a local homeomorphism.

If X is connected, then is f a homeomorphism?
No. Counterexample: C -> C\{0}, z -> exp(z);

f:C -> C\0, z - e^z

I meant z -> e^z.

is a single counter example?

No. To provide a far simpler one: Z -> Z, x -> 0, Z equipped with the discrete topology.

I didn't know that you want to count counter examples. ;)

C\0} -> C\{0}, z -> z^k
(k>1) or any topological covering.

g:C\0 -> C\0, z -> z^k, k > 1

is another counter example?

What don't you understand here? Did you check that it is a local homeomorphism?

Don't understand "or any topological covering".

I am so sorry for my bad English, I just literally translated the expression from my mother tongue. I meant a topological covering _map_.

If U,V are open and U homeomorphic f(U), V homeomorphic f(V),
is not U \/ V homeomorphic to f(U) \/ f(V) = f(U \/ V) ?
No. Patch examples above.

I'm to use f or g as an counter example?
"Patch" means to find U and V and show the patch U \/ V fails?

Sorry about the confusion. Yes - if you mean to take *one* of the counter examples above and patch two open subsets U and V such that the map on U u V is *not* injective.

Not even if U and V are overlapping?

Do not know what you are saying here. I suspect that you were not having the counter examples above in mind when you said this.

Anyway, best wishes,
J.
.

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