Neighborhood Retract of Boundary

[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?

 

[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?

 

[xaos]

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