I'm dealing with the following exercise from Spivak's "Calculus on Manifolds":
Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ dimensional manifold. Show that $N=A\cup \text{Fr}A$ is an $n$-dimensional manifold with boundary.
Questions:
I'm stuck. It's easy to parametrize $A$ using the identity as a coordinate patch. However, I don't know what to do with the points from $\text{Fr} A$. I tried to extend a $n-1$ dimensional coordinate patch.
If I understood Spivak's definitions, the exercise is supposing that $\text{Fr}A$ is a manifold without boundary. Is this result still true if we suppose that this set is a manifold-with-boundary?
This not a complete answer, but I hope it still is helpful. As you said, you have a chart on $A$, so it only remains to construct charts around boundary points. By assumption, for each point $x\in FrA$, there is an open neighborhood $U$ of $x$ in $\mathbb R^n$ and a diffeomorphism $F$ from $U$ onto an open subset $V\subset\mathbb R^n$ such that $U\cap FrA=F^{-1}(V\cap(\{0\}\times\mathbb R^{n-1}))$. Now $F(U\cap A)\subset V$ is an open subset whose boundary is $V\cap(\{0\}\times\mathbb R^{n-1})$. From here it should be possible to show by elementary that (if neccesary shrinking $U$ and or flipping the first coordinate) $F(U\cap A)$ has to be the intersection of $V$ with a half space $\mathbb R^+\times\mathbb R^{n-1}$.