I found some argument in the proof of the following lemma in Lee's smooth manifold 2ed is confusing.
$\textbf{Lemma 16.2}$ Suppose $U$ is an open subset of $\mathbb{R}^n$ or $\mathbb{H}^n$, and $K$ is a compact subset of $U$. Then there is an open $\textit{domain of integration}$ (defined as bounded subset where its boundary has measure zero) $D$ such that $K \subseteq D \subseteq \bar{D} \subseteq U$.
$\textbf{Proof.}$ For each $p \in K$, there is an open ball or half-ball containing $p$ whose closure is contained in $U$. By compactness, finitely many such sets $B_1,\dots, B_m$ cover $K$. Since the boundary of an open ball is a codimension-1 submanifold, and $\color{red}{\text{the boundary of an open half-ball is contained in a union of two such submanifolds}}$, the boundary of each has measure zero by Corollary 6.12. The set $D= B_1 \cup \cdots \cup B_m$ is the required domain of integration.
I find it difficult to follow the red color part in the proof above. How can the boundary of open half-ball can be covered by union of two submanifolds. I know that the submanifolds referred above are the boundary of open ball, which is a sphere. Can anyone explain this to me ? Thank you.
The boundary of a half-ball is contained in the union of an $(n-1)$-sphere and an $(n-1)$-dimensional linear subspace, each of which is a smooth codimension-$1$ submanifold.