# Neighborhood Retract of Boundary

1. Aug 22, 2012

### jgens

Here is the problem: If M is a manifold with boundary, then find a retraction r:U→∂M where U is a neighborhood of ∂M.

I realize the Collar Neighborhood Theorem essentially provides the desired map, but I am actually using this result to prove the aforementioned theorem. My thought on how to prove the theorem is to show local existence, show that you can locally extend a retraction, and then use Zorn's Lemma to construct a neighborhood retraction of the boundary. The only difficulty I run into here is showing local extendability. Can anyone help me with this step?

2. Aug 29, 2012

### Tinyboss

I don't remember needing Zorn's Lemma when I had to do this exercise, but I think we were allowed to assume the boundary was compact.

Can you be more specific about what you mean by locally extending a retraction?

3. Aug 29, 2012

### xaos

i think the open sets on the boundary are isomorphic to half spaces, if that helps any.