In Professor John Lee's Introduction to Smooth Manifolds, Second Edition, Theorem 9.26 has, as its hypothesis, a smooth manifold with nonempty boundary $M$. In his errata for the book, there is an entry (dated 8/19/14) for the proof of this theorem:
There's a gap in this proof, because it is not necessarily the case that $M(a)$ is a regular domain in $\mathrm{Int}M$. To correct the problem, we have to choose our collar neighborhood more carefully. Replace the first sentence of the proof by the following paragraph:
"Theorem 9.25 shows that $\partial M$ has a collar neighborhood $C_0$ in $M$, which is the image of a smooth embedding $E_0\colon[0,1)\times\partial M\to M$ satisfying $E_0(0,x)=x$ for all $x\in\partial M$. Let $f\colon M\to\mathbb{R}^+$ be a smooth positive exhaustion function. Note that $W=\{(t,x)\colon f(E_0(t,x))>f(x)-1\}$ is an open subset of $[0,1)\times\partial M$ containing $\{0\}\times\partial M$. Using a partition of unity as in the proof of Theorem 9.20, we may construct a smooth positive function $\delta\colon\partial M\to\mathbb{R}$ such that $(t,x)\in W$ whenever $0\leq t<\delta(x)$. Define $E\colon[0,1)\times\partial M\to M$ by $E(t,x)=E_0(t\delta(x),x)$. Then $E$ is a diffeomorphism onto a collar neighborhood $C$ of $\partial M$, and by construction $f(E(t,x))>f(x)-1$ for all $(t,x)\in[0,1)\times\partial M$. We will show that for each $a\in(0,1)$, the set $E([0,a]\times\partial M)$ is closed in $M$. Suppose $p$ is a boundary point of $E([0,a]\times\partial M)$ in $M$; then there is a sequence $\{(t_i,x_i)\}$ in $[0,a]\times\partial M$ such that $E(t_i,x_i)\to p\in M$. Then $f(E(t_i,x_i))$ remains bounded, and thus $f(x_i)<f(E(t_i,x_i))+1$ also remains bounded. Since $\partial M$ is closed in $M$, $f|_{\partial M}$ is also an exhaustion function, and therefore the sequence $\{x_i\}$ lies in some compact subset of $\partial M$. Passing to a subsequence, we may assume $(t_i,x_i)\to(t_0,x_0)$, and therefore $p=E(t_0,x_0)\in E([0,a]\times\partial M)$."
Then at the end of the first paragraph of the proof, add the following sentences:
"To see that $M(a)$ is a regular domain, note first that it is closed in $M$ because it is the complement of the open set $C(a)$. Let $p\in M(a)$ be arbitrary. If $p\notin E([0,a]\times\partial M)$, then $p$ has a neighborhood in $\mathrm{Int}M$ contained in $M(a)$ by the argument above. If $p\in E([0,a]\times\partial M)$, then $p=E(a,x)$ for some $x\in\partial M$, and $C$ is a neighborhood of $p$ in which $M(a)\cap C$ is the diffeomorphic image of $[a,1)\times\partial M$."
In between the above two paragraphs we have this sentence:
For any $a\in(0,1)$, let $C(a)=\{(s,x)\in[0,1)\times\partial M\colon 0\leq s<a\}$ and $M(a)=M\setminus E(C(a))$.
where I have taken the liberty of undoing the author's effort to "simplify the notation" by identifying $C$ with $[0,1)\times\partial M$.
I understand and can verify everything stated in the errata. I am having trouble, though, proving that $M(a)$ is a regular domain in $\mathrm{Int}M$. By the errata paragraph beginning "To see that $M(a)$ is a regular domain", it appears that it was his intention to use Theorem 5.51 to show that $M(a)$ is an embedded codimension-$0$ submanifold with boundary of $\mathrm{Int}M$. Indeed, in the first case, where $p\in M(a)$ is not in $E([0,a]\times\partial M)$, and we have the $\mathrm{Int}M$-neighborhood $V$ of $p$, which is contained in $M(a)$, we can find a smooth chart $(U,\phi)$ at $p$ for $\mathrm{Int}M$ such that $U\subseteq V$, and then $M(a)\cap U=U$ so $$\phi(M(a)\cap U)=\phi(U)$$ so that $M(a)\cap U$ is an $n$-slice of U$.
The problem arises in the second case, where $p=E(a,x)$. All my attempts at finding a smooth chart $(W,\lambda)$ at $p$ for $\mathrm{Int}M$ such that $\lambda(M(a)\cap W)=\lambda(W)$ or $\lambda(W)\cap\mathbb{H}^n$ have failed. From the last sentence of the errata, I was supposed to leverage the fact that $E([a,1)\times\partial M)=M(a)\cap C$. Here is an example of one of my failed attempts:
Let $E_C$ denote the diffeomorphism $E_C\colon[0,1)\times\partial M\to C$, ($E$ with its codomain restricted to $C$) where $C$ has been given the unique smooth structure to make $E_C$ a diffeomorphism. Let $(U,\phi)$ be a smooth chart at $x$ for $\partial M$. $(0,1)\times U$ is open in $[0,1)\times\partial M$, so by Proposition 2.15(d), the restriction of $E_C$, $\tilde{E}_C\colon(0,1)\times U\to E_C((0,1)\times U)$, is also a diffeomorphism. Let $\Lambda\colon\mathbb{R}\times\mathbb{R}^{n-1}\to\mathbb{R}^{n-1}\times\mathbb{R}$ be the diffeomorphism which exchanges its arguments. Let $W=E_C((0,1)\times U))\text{ open}\subseteq C\text{ open}\subseteq M$. Since $(\{0\}\times\partial M)\cap((0,1)\times U)=\varnothing$, $W\text{ open}\subseteq\mathrm{Int}M$. Also, let $$\lambda=\Lambda\circ(\mathrm{id}_{(0,1)}\times\phi)\circ\tilde{E}_C^{-1}\colon W\to\phi(U)\times(0,1)\text{ open}\subseteq\mathbb{R}^n.$$ Then $\lambda$ is a diffeomorphism, so $(W,\lambda)$ is a smooth chart for $\mathrm{Int}M$. Crucially, $p\in W$ since $p=E(a,x)$ and $(a,x)\in(0,1)\times U$. Then $$M(a)\cap W=(M\setminus E([0,a)\times\partial M))\cap E((0,1)\times U) =E([a,1)\times U),$$ $$\lambda(M(a)\cap W)=\lambda(E([a,1)\times U)=\phi(U)\times[a,1),\text{ and}$$ $$\lambda(W)=\lambda(E((0,1)\times U))=\phi(U)\times(0,1),$$ so we don't quite have $\lambda(M(a)\cap W)=\lambda(W)$ or $\lambda(W)\cap\mathbb{H}^n$. If I try to modify this by shrinking the left end of $(0,1)$ towards $a$, I can't go past $a$ since then $p=E(a,x)$ will not be in $W$, and I can't use $[a,1)$ because that is not open in $[0,1)$. I've tried to use a diffeomorphism from $[a,1)$ to $[0,1)$, but have not been able to get that to work either. I'd really appreciate some help in proving that $M(a)$ is a codimension-$0$ embedded submanifold with boundary of $\mathrm{Int}M$.
Oh, I just figured it out! Let $\tau_a\colon\mathbb{R}\to\mathbb{R}$ be defined by $\tau_a(r)=r-a$. $\tau_a$ is a diffeomorphism. Let $W$ be defined as in the post and let $$\lambda=\Lambda\circ(\tau_a\times\phi)\circ\tilde{E}_C \colon W\to\phi(U)\times(-a,1-a)\text{ open}\subseteq\mathbb{R}^n.$$ Then $$\lambda(M(a)\cap W)=\lambda(E([a,1)\times U)=\phi(U)\times[0,1-a),\text{ and}$$ $$\lambda(W)\cap\mathbb{H}^n=\lambda(E((0,1)\times U))\cap\mathbb{H}^n=(\phi(U)\times(-a,1-a))\cap\mathbb{H}^n =\phi(U)\times[0,1-a).$$ So $\lambda(M(a)\cap W)=\lambda(W)\cap\mathbb{H}^n$ as required.