The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold

Suppose $n\ge 1$. Restrict each boundary chart to the face $x^n=0$. By [L1] these restrictions cover exactly $\mathrm{\partial }M$. Their images are open subsets of ${\mathbb{R}}^{n-1}$, and [L2] makes their transition maps smooth restrictions of extensions of the ambient transitions. They therefore define a smooth boundaryless $(n-1)$-manifold structure on $\mathrm{\partial }M$.

In a boundary chart the inclusion $\mathrm{\partial }M\hookrightarrow M$ is the coordinate map $(x^1,\dots ,x^{n-1})\mapsto (x^1,\dots ,x^{n-1},0)$, so it is a smooth injective immersion and a homeomorphism onto its subspace image. Thus it is a smooth embedding, and [L3] gives the asserted embedded submanifold. The complement is locally $\{x^n>0\}$, hence open, so $\mathrm{\partial }M$ is closed. When $n=0$, the boundary is empty by the stated convention.