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Old 2013-05-21, 13:15   #1
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Dec 2003
Hopefully Near M48

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Default Manifolds Question

Let M be a compact n-manifold with boundary. Show that there is an open set U containing the boundary that is diffeomorphic to (the boundary) x [0,1)


By definition of the boundary of a manifold, every point on the boundary has an open neighborhood that is diffeomorphic to R^{n-1} x [0,infinity), which of course I can shrink to R^{n-1} x [0,1). Also, by compactness, I can cover the boundary with finitely many such neighborhoods.

The hint here is that I'm supposed to use the theorem on existence of partitions of unity. Supposedly, that allows one to stitch together local information into global. Unfortunately, the book doesn't have any examples of that.

I'd actually prefer to work this problem out on my own, so... Any other examples of how to use partitions of unity to stitch together local information into global?

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