Re: Local Homeomorphisms

On Sun, 20 Apr 2008, [ISO-8859-1] Mariano Suárez-Alvarez wrote:
On Apr 20, 5:48 am, William Elliot <ma...@xxxxxxxxxxxxxxxxxx> wrote:
On Fri, 18 Apr 2008, Jannick Asmus wrote:
On 18.04.2008 12:13, William Elliot wrote:

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.


The key difference between local homeomorphisms and covering
maps is that the first is a definition local on the domain,
while the second is local on the codomain: given a map
f : X --> Y,

* f is a local homeo if for all x in X, there is
an open U with x in U and f : U --> f(U) an homeo; while

This does not include the requirement that f(U) be open.
This is what's puzzling me. Wikipedia says f(U) is open.
Where is the definition you've given, stated or used?

The distinction between the two is that your definition
+ open map is equivalent to Wikipedia's definition.

Open map seems like a strong extra requirement.
Is it really essential? That I dispute.

A locally bijective continuous open map is equivalent
to local homeomorphism by Wikipedia's definition.

* f is a covering map if for all y in Y, there is an
open V with y in V such that f : f^(V) --> V is a trivial
covering (ie, f^(V) is a disjoint union of open sets
each one of which is mapped homeomorphically onto V
by f)

f:[0,1) -> S^1 is a local homeomorphism that's not a covering map.

--
Riddle of the day. Is a local homeo a local homo Romeo?
.

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