
\documentclass[12pt ]{article}
\setlength{\textwidth}{14 true cm} \setlength{\textheight}{20 true
cm}
\usepackage{graphicx}
\usepackage{hangul}
\usepackage{amscd,amsmath}
\usepackage{amsfonts}
\usepackage{amssymb,theorem}
\usepackage{longtable}

\newcommand{\dis}{\displaystyle}
\newcommand{\disi}{\displaystyle{\sum_{i=1}^n}}
\newcommand{\disj}{\displaystyle{\sum_{j=1}^m}}
\newcommand{\pa}{\partial}
\newcommand{\Aff}{\mbox{\it Aff}}
\newcommand{\aff}{\mbox{\it aff}}
\newcommand{\cb}{\mathbb{C}}
\newcommand{\cc}{\mathcal{C}}
\newcommand{\ccn}{\mathcal{C}^{\infty}}
\newcommand{\sbb}{\mathbb{S}}
\newcommand{\rb}{\mathbb{R}}
\newcommand{\rc}{\mathcal{R}}

\newcommand{\hhhh}{\hspace{4em}}
\newcommand{\hhh}{\hspace{3em}}
\newcommand{\hh}{\hspace{2em}}
\newcommand{\h}{\hspace{1em}}

\newcommand{\alp}{\alpha}
\newcommand{\bet}{\beta}
\newcommand{\del}{\delta}
\newcommand{\gam}{\gamma}
\newcommand{\vep}{\varepsilon}
\newcommand{\eps}{\epsilon}
\newcommand{\lam}{\lambda}
\newcommand{\kap}{\kappa}
\newcommand{\sig}{\sigma}
\newcommand{\ome}{\omega}
\newcommand{\Gam}{\Gamma}
\newcommand{\Ome}{\Omega}
\newcommand{\Sig}{\Sigma}
\newcommand{\Del}{\Delta}
\newcommand{\Lam}{\Lambda}


\newtheorem{thm}{����}
\newtheorem{cor}[thm]{��������}
\newtheorem{lem}[thm]{��������}
\newtheorem{prop}[thm]{����}
\newtheorem{cl}{Claim}

{\theorembodyfont{\rm}
\newtheorem{ex}{��}
\newtheorem{que}{����}
\newtheorem{notation}{Notation}[section]
\newtheorem{defn}{����}
\newtheorem{rem}{��}
\newtheorem{note}{Note}
}
\renewcommand{\thenote}{}
\renewcommand{\therem}{}

\newenvironment{proof}{{\bf ����}}{\hfill\framebox[2mm]{}}
\newenvironment{proof1}{{\bf ��������}}{\hfill\framebox[2mm]{}}

\begin{document}
\parindent=0cm
\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}
$M$�� boundary�� $\partial M $���� ����.\\ ��, $\partial M =\{p\in
M \ |\ \phi(p)\in  \{0\}\times\rb^{n-1}=:\partial \mathbb{H}^n
\}$\\




\begin{figure}[htb]
    \centerline{\includegraphics*[scale=0.6,clip=true]{grp1.eps}}

    \end{figure}


$\hspace{10em}$ �׸�. Manifold with Boundary\\



\textbf{Remark} (invariance of domain) \\
In $\rb^n$, 1-1 continuous image of open set is open. \\
\\
�� Remark�� ������, coordinate transition map�� interior point��
interior point�� boundary point�� boundary point�� �����Ƿ�,
$\partial M$��
coordinate chart�� ���ÿ� ������� �� ���ǵȴ�.\\

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

\end{document}
