Re: Local Homeomorphisms

On Sun, 20 Apr 2008, Jannick Asmus wrote:

On 20.04.2008 10:48, William Elliot wrote:
On Fri, 18 Apr 2008, Jannick Asmus wrote:
On 18.04.2008 12:13, William Elliot wrote:

What's the trace topology?

... the induced topology on a subset, cf., e.g.,
http://en.wikipedia.org/wiki/Subspace_topology.

Never heard that expression for subspaces.

Google might become your friend one day. ;)

It was once. Now Yahoo is my friend.

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).

Another counter example is f:[0,1) -> S^1.

Right.

It appears than in the notion of covering map, that the local
homeomorphism has f(U) being open, not by the definition of
local homeomorphism, but by the definition of covering map.

In other words
covering map --> local homeomorphism with open local images.

This is just convention - as I said already.

Is there any usage for the definition without f(U) being 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 ?
Convention - like "nhood". I do not want to argue about something like
this. ;)

What's the definition of HTH?

Google should become your friend.

They were until they sold their soul to the stock market.

Ever since then I've been removing bookmarks to Google's services. They
are examples of 'new and improved' = 'isn't as nice and doesn't work as
good' and 'updates are worse than blind dates'.

Now Yahoo is my friend. Yahoo should become your friend.
.

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