I came across the following statement in Connor and Floyd's book Differentiable Periodic Maps, page 10, proposition 3.1.
(3.1) Suppose $P$ and $Q$ are closed disjoint subsets of the compact $n$-manifold $B^n$ (with or without boundary). There exists a topological manifold $B_1^n \subset B^n$ with $P \subset B_1^n$, $B_1^n \cap Q=\emptyset$ and $B_1^n$ closed in $B^n$. Moreover $B_1^n$ can be given a differentiable structure by straightening the angle.
Here straightening the angle is also referred to as smoothing the corner, which can be used to give smooth structures to certain topological manifolds with "corners." This is quite an old book. I was wondering if this statement has a more modern proof. I also don't have any intuition about the authors' original proof. In particular, I don't understand the use of compactness and the differentiable approximations / extensions.
Any help would be appreciated. Thank you