Re: Local Homeomorphisms

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.

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.

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

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

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 ?

Best wishes,
J.

.

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