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\begin{document}
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  \section*{Further Examples.}

  {\bf 3. $\rb P^n$=Real Projective space.}

$\rb P^n$=$\rb^{n+1}-\{0\}/\sim$  , where $x\sim \lam x$, $\lam\in
\rb\backslash\{0\}$.

$[x]=$equivalence class of $x=(x_1,\cdot\cdot\cdot ,x_{n+1})$,
denoted by $[x_1:\cdot\cdot\cdot:x_{n+1}]$.

���� ���� ���ǵ� $\rb P^n$�� ���� coordinate chart�� ������ ����
�ش�.

$U_i=\{[x_1:\cdot\cdot\cdot:x_{n+1}]\in\rb P^n\,|\,x_i\neq 0\}$
and $\phi_i:U_i\rightarrow\rb^n$ by


$\phi([x_1:\cdot\cdot\cdot:x_{n+1}])=(\frac{x_1}{x_i},\frac{x_2}{x_i},\cdot\cdot,\hat{\frac{x_i}{x_i}}
,\cdot\cdot,\frac{x_{n+1}}{x_i})$ (���⼭ $\wedge$�� ���ٴ�
���̴�.)



�� �� $U_i$���� $\rb P^n$�� cover�ϰ� ���� �� $\phi_i$�� $\cc^k$-
related�Ǿ� ������ ������.

$i<j$��� �����ϸ�,$\phi_i(U_i\cap U_j)$ �� ������ ���� i��°
������ ��������

$\phi_j(U_i\cap U_j)$������ �������� �����Ƿ� ������ �� �� �ִ�.


$\phi_i(U_i\cap
U_j)=\{(a_1,\cdot\cdot\cdot,a_n)\in\rb^n\,|\,a_{j-1}\neq 0\}:=V,$

$\phi_j(U_i\cap
U_j)=\{(a_1,\cdot\cdot\cdot,a_n)\in\rb^n\,|\,\,\,a_{i}\neq 0\}:=W
$ and


$(\phi_i\circ\phi_j^{-1})(a_1,\cdot\cdot,a_n)=\phi_i([a_1:\cdot\cdot1:\cdot\cdot:a_n])$
$=(\frac{a_1}{a_i},\cdot\cdot,\hat{\frac{a_i}{a_i}},\cdot\cdot,\frac{1}{a_i},
\cdot\cdot,\frac{a_{n}}{a_i})$ : $W\rightarrow V$.

����$(\phi_i\circ\phi_j^{-1})$�� rational map�̹Ƿ� $\cc^{\infty}$
�� �ƴ϶� $\cc^{\omega}$ ������ �ȴ�. $\phi_j\circ\phi_i^{-1}$��
���ؼ��� ���������̰� ���� $\cc^{\infty}$-related �Ǿ� �ִ�.\\

\begin{figure}[htb]
    \centerline{\includegraphics*[scale=0.6,clip=true]{grp10.eps}}

    \end{figure}

$\hspace{15em}$ �׸� 10\\

��� $U_i$�� $\rb P^n$���� open�̰�, $\phi_i$�� homeomorphism�ӵ�
������ �Ѵ�. $U_i$�� $\rb P^n$���� open�̶�� ����� quotient
topology �� �̿��ؼ� open���� �� �� �ְ� $\phi_i$��
homeomorphism�̶�� �͵� ���� ���� �� �ִ�. $\rb P^n$�� topology��
quotient topology�� ���� �ʰ� ������ ���� weak topology�� ����Ͽ�
�����Ͽ��� ���� ����� ��´�.\\

{\bf Note.} In this example, the topology of $\rb P^n$ is given by
the quotient topology and also can be viewed as a weak(or
coherent) topology as follows :\\

$\hspace{3em}$Let    $X=\displaystyle{\bigcup_{\alp}X_{\alp}}$,
and $X_{\alp}$
be a topological space, $\forall \alp$.\\

Assume

(1) ��� $\alp,\beta$�� ����  $X_{\alp}\cap X_{\beta}$��
topology�� $X_{\alp}$�� subspace�� ���� $X_{\beta}$�� subspace��
���� topology �� ����.

(2)$X_{\alp}\cap X_{\beta}$ is open in $X_{\alp}$ and in
$X_{\beta}$, $\forall \alp,\beta$.\\

Let $\mathcal{T}=\{A\subset X\,\,|\,\,A\cap X_{\alp}\,\,is
\,\,open\,\,in\,\,X_{\alp}\,,\,\forall\,\alp\}$, then
$\mathcal{T}$ is a unique topology of $X$, called weak(or
coherent) topology, such that

(a) original topology of $X_{\alp}$=subspace topology of
$X_{\alp}$ in $X$ with respect to $\mathcal{T}$.

(b) $X_{\alp}$ is open in $X$.\\

{\bf ����2}(optional) �� note�� ������ �����϶�.\\

���� �ٽ� $\rb P^n$���� ���ư��� set���μ��� $\rb
P^n=\displaystyle{\bigcup_i}U_i$ ���ٰ� topology �� �ش�.
$\phi_i$�� homeomorphism�� �ǵ��� �� $U_i$���ٰ�  $\rb^n$�� ����
topology�� ����. �׷��� �� �� ���� (1),(2)�� �������� �� �� �ְ�
���� $\rb P^n$�� �� $U_i$���� topology�� coherent�ϰ� �����ϴ�
weak topology�� �� �� �ְ� �Ǿ� �� $U_i$�� $\rb P^n$���� open ���� �� �� �ִ�.\\

{\bf 4. $\mathbb{C}P^n$=Complex projective space.}

��3 ���� $\rb$ ��� $\mathbb{C}$�� ���� �ȴ�. ���� ��������
$\mathbb{C}P^n$ ���� n���� complex analytic manifold�� �ǰ� ����
�̴� 2n���� real smooth manifold�� �ȴ�. quaternion $\mathbb{H}$��
���ؼ��� ���������̴�.($\mathbb{H}P^n$�� 4n ������ �ȴ�.)\\

{\bf 5. An open subset of a smooth manifold is a smooth manifold.}

manifold $M$�� atlas $\mathcal{F}=\{(U_{\alp},\phi_{\alp})\}$��
���� $M$����  open�� $N$�� �ִٰ� ����. �� ��
$\displaystyle{\mathcal{F}_N:=\{(U_{\alp}\cap
N,\phi_{\alp}|_{U_{\alp}\cap N})\}}$ �� N�� smooth structure��
�ش�. ���� ��� $Gl(n,\rb)=\{A\in M(n,\rb)\,|\,detA\neq 0\}$ ��
smooth structure�� ���� manifold�� ���µ� �� �� $Gl(n,\rb)$��
$\rb^{n^2}=M(n,\rb)$�� open subset���μ� ���� ���̴�.\\

{\bf 6. A covering of a manifold is a manifold.}

$M$�� manifold��� �ϰ� $\tilde{M}$�� $M$�� covering�̶�� ����.
�׷���, �� �� $x\in M$�� ���� evenly cover�Ǵ� coordinate chart
$U$�� �����Ѵ�. �� �� �� $\widetilde{x}\in p^{-1}(x)$�� ���� $U$��
$p$�� ���� homeomorphic �� �ٹ� $\widetilde{U}\subset p^{-1}(U)$��
��� $\phi\circ p|_{\widetilde{U}}$�� smooth chart�� �ָ� �ȴ�.
���� ��� covering $p:\mathbb{S}^n\rightarrow\rb P^n$ �� ������ ��
�ִ�. �� �� ������� $\sbb^n$���� smooth structure�� ��
section����
����� smooth structure�� ���� ����.(��� �� ���� ����.)\\

{\bf 7. $M,N$ are smooth manifolds$\Rightarrow$ $M\times N$ is a
smooth structure.}

$M$�� atlas $\{(U_{\alp},\phi_{\alp})\}$�� $N$�� atlas
$\{(V_{\beta},\psi_{\beta})\}$�� ���� $\{(U_{\alp}\times V_{\beta}
,\phi_{\alp}\times \psi_{\beta})\}$�� $M\times N$�� atlas�� �ȴ�.
(�Ϲ������� $f:X\rightarrow Y$, $g:X'\rightarrow Y'$ �� ����
$f\times g:X\times Y\rightarrow X'\times Y'$ �� $(f\times
g)(x,y)=(f(x),g(y))$�� �����Ѵ�.) �̸� ���̱� ���� ���� $
(U_{\alp}\times V_{\beta} ,\phi_{\alp}\times \psi_{\beta})$��
�Ʒ��׸�ó�� ����.
\\

\begin{figure}[htb]
    \centerline{\includegraphics*[scale=0.6,clip=true]{grp11.eps}}

    \end{figure}

$\hspace{15em}$ �׸� 11\\



���� ���������� ������� ����� �Ǵٸ� chart
$(U_{\alp'},\phi_{\alp'}),(V_{\beta'},\psi_{\beta'})$�� �ִٰ�
�����ϸ�,
$(\phi_{\alp}\times\psi_{\beta})\circ(\phi_{\alp'}\times\psi_{\beta'})^{-1}$
��
$\phi_{\alp}\circ\phi_{\alp'}^{-1}\times\psi_{\beta}\circ\psi_{\beta'}^{-1}$
�� ���� ������ $\cc^{\infty}$ �̹Ƿ� �̰� ���� $\cc^{\infty}$
�̴�. product manifold $\sbb^n\times\sbb^m$,
$\sbb^1\times\cdot\cdot\cdot\times\sbb^1=\mathbb{T}^n$(torus) ����
��� smooth manifold�� �ȴ�.
















  \end{document}
