I am reading through the new edition of Lee's book and I am stuck by the proof of Theorem 6.15. When passing to a non-compact manifold, the author begin by defining several sub-levelsets and claim that they're regular domains by 5.47. However, 5.47 only asserts the result for manifold without boundary, and the manifold in 6.15 could have non-empty boundary.
Is 5.47 true for manifold with boundary? The manifold without boundary case can be shown using the equivalence between embedded submanifolds and subsets satisfying the slice condition, such equivalence is only claimed (in the book) when the ambient manifold has no boundary (5.51)
Is there a way to fix this?