Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$} with $n\ge 2$, and let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary} of dimension $n$ whose \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} $\partial M$ is nonempty. Give $\partial M$ the \reftext{def:subspace-topology-2026a}{subspace topology} inherited from $M$. Then the following hold.

1. $\partial M$ is a \reftext{def:closed-subset-topological-space-2026a}{closed subset} of $M$.

2. With the subspace topology, $\partial M$ is \reftext{def:hausdorff-topological-space-2026a}{Hausdorff} and \reftext{def:second-countable-topological-space-2026a}{second countable}.

3. Let $(U,\varphi)$ be a chart in the chosen atlas of $M$ with $U\cap\partial M\ne\varnothing$. Then $\varphi(U\cap\partial M)=\varphi(U)\cap\partial H^n$, where $\partial H^n$ is the boundary hyperplane of the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space}. Let $\pi:\mathbb{R}^n\to\mathbb{R}^{n-1}$, between \reftext{def:euclidean-space-rn-2026a}{Euclidean spaces}, denote the projection $\pi(x_1,\dots,x_n)=(x_1,\dots,x_{n-1})$, and for $c\in\mathbb{R}^{n-1}$ let $T_c:\mathbb{R}^{n-1}\to\mathbb{R}^{n-1}$ denote the translation $T_c(y)=y+c$. Then for every $p\in U\cap\partial M$ there exist an open subset $U'$ of $M$ with $p\in U'\subseteq U$ and a vector $c\in\mathbb{R}^{n-1}$ such that the pair $(U'\cap\partial M,\psi)$, with chart map $\psi=T_c\circ\pi\circ\varphi$ restricted to $U'\cap\partial M$, is a \reftext{def:chart-upper-half-space-2026a}{chart} of dimension $n-1$ on $\partial M$ whose image is contained in the set of points $y=(y_1,\dots,y_{n-1})$ of the closed upper half-space $H^{n-1}$ with $y_{n-1}>0$. Charts of this form are called \textbf{induced boundary charts} of $\partial M$.

4. Any two induced boundary charts are \reftext{def:smooth-compatible-charts-upper-half-space-2026a}{smoothly compatible}, and the collection of all induced boundary charts is a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth atlas} of dimension $n-1$ on $\partial M$, making $\partial M$ a smooth manifold with boundary of dimension $n-1$ in which every point is an interior point in the sense of \ref{def:boundary-smooth-manifold-with-boundary-2026a}; that is, the boundary of $\partial M$ is empty.

5. If $M$ is compact in the sense of \ref{def:smooth-manifold-with-boundary-2026a}, then $\partial M$ is compact, by claim 1 together with \ref{thm:closed-subset-compact-is-compact-2026a}.