Re: Local Homeomorphisms

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

is a single counter example?

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?

Don't understand "or any topological covering".

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?

Not even if U and V are overlapping?
.

Loading