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\begin{document}
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\section*{Manifold with Boundary}
\begin{defn}
A  \emph{\textbf{topological manifold}} of dimension $n$
\emph{\textbf{with boundary}} \\is a space $M$ with the property
that  $ \forall p \in M , \exists $ a coordinate chart $(U,\phi) $
of $p$ \\ s.t.$\phi $ is a homeo onto an open set of
$(\mathbb{R}^n$ or) $ \mathbb{H}^n $, \\ where $\mathbb{H}^n = \{x
= (x_1 , \cdots , x_n)\in \rb \ | \ x_1 \leq 0 \}$

\end{defn}
$M$의 boundary를 $\partial M $으로 쓴다.\\ 즉, $\partial M =\{p\in
M \ |\ \phi(p)\in  \{0\}\times\rb^{n-1}=:\partial \mathbb{H}^n
\}$\\


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$\hspace{10em}$ 그림. Manifold with Boundary\\


\textbf{Remark} (invariance of domain) \\
In $\rb^n$, 1-1 continuous image of open set is open. \\
\\
위 Remark에 따르면, coordinate transition map은 interior point를
interior point로 boundary point는 boundary point로 보내므로,
$\partial M$은
coordinate chart의 선택에 관계없이 잘 정의된다.\\

A \emph{\textbf{smooth structure}} on a manifold with boundary is defined exactly same as before.\\
(\textbf{note.} a map defined on an open subset of $\mathbb{H}^n $
is smooth if it can be extended to a smooth map on an open set of
$\rb^n$)\\

$\bullet$ $M$ : $n$-$dimensional$ $C^\infty$ manifold with boundary \\
$\Rightarrow \partial M $ : $ (n-1)$-$dimensional $ $C^\infty$
manifold without boundary

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