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\begin{document}

\section*{Definition and Examples}

\begin{defn}
A topological space $M$ is an {\bf n-dimensional manifold} or
({\bf n-manifold}) if
\begin{enumerate}
\item $M$ is Hausdorff,
\item $M$ is locally Euclidean, i.e.,
$\forall x \in M,$  $\exists$ a neighborhood $U$ of $x$ such that
$U$ is homeomorphic to an open set in $ \Real^n $.
\end{enumerate}
\end{defn}
¿©±â¼­ $U$°¡ $ \Real^n $°ú homeomorphicÇÑ neighborhood ¶ó°í
¿ä±¸ÇÏ¿©µµ À§ Á¤ÀÇ¿Í equivalent ÇÏ´Ù.

For an open set $U$ of a manifold, let $\phi : U \rightarrow
\phi(U) \subset \Real ^n $ be a homeomorphism. We call $(U,\phi )$
a {\bf coordinate chart}.

\begin{defn}
Let $M$ be an n-manifold. $M$ is a {\bf differentiable manifold}
if there is a system of open coordinate charts $\{ ( U_\alpha ,
\phi_\alpha ) \} $ covering $M\mathcal{\mathcal{}}$ such that

$$g_{\bet \alp} = \phi_\bet \circ \phi_\alp^{-1} : \phi_\alp ( U_\alp \cap U_\bet ) \to \phi_\beta ( U_\alp \cap U_\bet ),
 \forall \alp, \bet $$
is differentiable. $g_{\bet \alp}$ is called a {\bf coordinate
transition map}.
\end{defn}

ÀϹÝÀûÀ¸·Î coordinate transition map¿¡ ¾î¶² ±¸Á¶¸¦ ÁÖ´À³Ä¿¡ µû¶ó¼­
MÀÇ ±¸Á¶°¡ °áÁ¤µÈ´Ù. ¿¹¸¦ µé¾î coordinate transition mapµéÀÌ
$\mathcal C\mathbb{}^k$ differentiable À̸é MÀº $\mathcal C^k$
differentiable manifold¶ó ºÎ¸£°í, real analyticÀÌ µÇ°Ô coordinate
systemÀ» Àâ¾Æ ÁÙ ¼ö ÀÖÀ¸¸é MÀº real analytic manifold°¡ µÇ°í
transitionÀÌ Euclidean rigid motion ÀÌ µÇ°Ô ÀâÀ» ¼ö ÀÖÀ¸¸é MÀº
Euclidean manifold°¡ µÈ´Ù. ¸¸ÀÏ coordinate chartµéÀ» $ \Real^n $
´ë½Å $ \mathbb C^n $ ·Î Àâ°í transitionÀ» holomorphic map À¸·Î
ÀâÀ» ¼ö ÀÖÀ¸¸é complex manifold°¡ µÈ´Ù. ƯÈ÷ 1Â÷¿ø complex
manifold¸¦ Riemann surface¶ó°í ºÎ¸¥´Ù.

\begin{ex}
Manifolds

\begin{enumerate}
\item $\Real^n$ itself.
\item A space with discrete topology, in which every set is open, is 0-manifold.
\item An open set in $\Real^n$.
\item $S^n \subset \Real^{n+1}$.
\item A smooth surface in $\Real^3$: $\{ (x,y,z) | f(x,y,z)= 0 \}$ when $\nabla f \neq 0 $.
\item An open set of n-manifold.
\item non-Hausdorff manifold: Consider the real line $\Real$ with standard topology. Add one more point $0'$ to $\Real$ set-theoretically,
 and give a topology as follows: (1)The open sets of the original real line are open.
 (2)For any open set $U$ containing 0, the set $(U \cup \{0' \}) - \{0\} $ is open.
 Then this space $\Real \cup \{ 0'\}$ is locally homeomorphic to $\Real$ but not Hausdorff
 since any two neighborhoods $U$ of $0$ and $V$ of $0'$ intersect.
 \item $\infty$(Figure 8) is not a manifold.

\end{enumerate}
\end{ex}

HW1. Show that $S^n$ is a $\mathcal C^{\infty}$ manifold.
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