Re: Local Homeomorphisms

On 18.04.2008 12:13, William Elliot wrote:
On Thu, 17 Apr 2008, Jannick Asmus wrote:

On 17.04.2008 11:45, William Elliot wrote:

The result I'm wishing for is
a bijective local homeomorphism is a homeomorphism.
This is clear: A local homeomorphism is an open map or - in other words
- the inversion map f^-1 is continuous.

Well now, that depends upon the definition.

I apply this one if nothing else is said: http://en.wikipedia.org/wiki/Local_homeomorphism

Continuous f:X -> Y is a local homeomorphism when for all x,
some open U nhood x with U homeomorphic f(U) and f(U) open.

This is what I call a local homeomorphism, as well.

Then clearly local homeomorphisms are open and when surjective
are quotient maps.

Why is it that f(U) needs to be open?

This is just a question of definition, hence convention. I do not argue about definitions.

What goes wrong if f(U) isn't open?

Locally compact is another property of the domain space.
Apparently not needed.

Oh, oh, oh, oh.

Always depends on what you mean with "local homeomorphism". Some people use the wording "(local) homeomorphism on the image of f equipped with the trace topology". This makes it clearer in some cases. I think this is what you are talking about.

Let f:X -> Y be a continuous bijection, X locally compact, Y Hausdorff.
Is f a local homeomorphism? Well clearly for all x, some
open U nhood x with U homeomorphic f(U), but is f(U) open?

Certainly not: identity map (R,discrete topology) -> (R,norm topology).

Some compact K with x in int K
f:int K -> Y closed continuous bijection.
int K homeomorphic f(int K)

Why is f(int K) open, or is it?

HTH.

HTH ?

Convention - like "nhood". I do not want to argue about something like this. ;)

--
Best wishes,
J.
.

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