Empty boundary is equivalent to being boundaryless

If $\mathrm{\partial }M=\mathrm{\varnothing }$ and $n=0$, every boundary-chart image lies in ${\mathbb{R}}^0$ and is already Euclidean-open. If $n>0$, [L1] and [L2] show that every point has a boundary chart whose image lies in $\{x^n>0\}$ after restricting its domain. Such images are Euclidean-open. The transition maps are restrictions of the original smooth half-space transitions; on these Euclidean-open images their local Euclidean extensions show that they and their inverses are ordinary smooth maps. Hence the restricted charts form the atlas in [L3].

Conversely, suppose a covering by boundary charts has images avoiding the model face. For $n>0$, [L1] makes every covered point interior, hence $\mathrm{\partial }M=\mathrm{\varnothing }$; for $n=0$, [L1] gives the same conclusion directly. Thus both implications hold.