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\begin{document}
\parindent=0cm
\section*{Topology of Manifold and Partition of Unity}
\begin{thm}{\rm
Let $M$ be a connected Hausdorff $\ccn$-manifold, then the
followings are equivalent.\\
(1) $M$ is paracompact. \\
(2) $\forall \mathcal{U} $, an open covering of $M$, $\exists $ a
partition
of unity subordinate to $\mathcal{U}$,\\
\hspace*{3ex} i.e., $\exists f_\alpha~:~ M \rightarrow [0,1], \ccn, \forall \alpha \in A$, such that\smallskip\\
 \hspace*{7ex} a) $\{suppf_\alpha~|~ \alpha \in A\}$ forms a locally
finite closed refinement of $\mathcal{U}$\\
 \hspace*{7ex}  b) $\displaystyle{\sum_\alpha} f_\alpha \equiv
 1$\\
(3) $M$ admits a Riemannian metric.\\
(4) $M$ is metrizable.\\
(5) $M$ has a compact exhaustion, \\\hspace*{3ex} i.e., $\exists$
a sequence of open sets $U_1, U_2,\cdots$  such
that\\\hspace*{7ex} $\overline{U_i}$ is compact ,
$\overline{U_i}\subset U_{i+1}$, $\forall i$ and $
M=\displaystyle{\bigcup_{i=1}^\infty U_i}$. \\
(6) $M$ is $\sigma$-compact, i.e., $M$ is a countable union of
compact sets.\\(7) $M$ is a countable union of coordinate
charts.\\(8) $M$ is $2^{\textrm{nd}} $ countable.

 }
\end{thm}
������ �����ϱ⿡ �ռ� ������ ������ ��� ������� �Ұ��Ѵ�.
\begin{defn} A topological space $X$ is
\emph{\textbf{paracompact}} if every open covering of $X$ has a
locally finite refinement.
\end{defn}

�Ϲ�������, $\mathcal{V}$�� $\mathcal{U}$�� refinement��� ����
$\mathcal{V}=\{V_\beta\}, \mathcal{U}=\{U_\alpha\}$��� ���� ��,
������ $\beta$�� ���Ͽ� $V_\beta\subset U_\alpha$�� �Ǵ�
$\alpha$�� �����Ѵٴ� ���̰�, ���⼭ open covering�� refinement��
refinement�̸鼭 ���ÿ� open covering�� �Ǵ� ���� ���Ѵ�. ����
open covering�� locally finite�ϴٴ� ���� ������ ���� ���Ͽ�
������ open neighborhood�� �����Ͽ� covering�� ���Ѱ��� open
set��� �����ٴ� ���̴�.

\begin{defn} A \emph{\textbf{Riemannian metric}} $g$ is a
$\ccn$ tensor field of type (0,2) which is symmetric and
positive definite, \\
\hspace*{3ex} i.e., $g(X,Y)=g(Y,X)$, \hh $\forall X,Y \in
\mathcal{X}$,\\
\hspace*{7ex} $g(X,X)\geq 0 $,\\
\hspace*{7ex} $g(X,X)=0$ iff $X=0$.
\end{defn}

\begin{defn} A topological space $X$ is \emph{\textbf{metrizable}}
if there exists a metric $d$ on the set $X$ where metric topology
is the same as the topology of $X$.
\end{defn}
\newpage
\begin{proof1}

{\bf ((1)$\Rightarrow$(2))} \\
{\bf Step 1.}(Shrinking lemma)\\
Given $\mathcal{U}=\{U_\alpha~|~\alpha\in A\}$, an open cover,
construct $\mathcal{V}=\{V_\alpha~|~\alpha\in A\}$, locally
finite(possibly $V_\alpha =\phi$) such that $V_\alpha\subset
\overline{V_\alpha}\subset U_\alpha$. \\
\begin{proof}
$\forall p\in M$, $\exists O_p$ s.t. $p\in O_p
\subset\overline{O_p}\subset U_\alpha $ for some $\alpha$.\\
$M$�� paracompact�ϹǷ�, $\mathcal{O}=\{O_p\}$�� locally finite
open refinement�� �����ϰ� �̸� $\mathcal{N}=\{N_\beta\}_{\beta\in
B}$��� ����. ��, ������ $N_\beta$�� ���Ͽ� $N_\beta \subset
O_p$�� �����ϴ� $p$�� �����Ѵ�. ����, $N_\beta\subset O_p
\subset\overline{O_p}\subset U_\alpha $�� �����ϴ� $\alpha$��
�����ϰ� $\alpha=\varphi(\beta)$��� �ϸ�, �Լ� $\varphi~:~ B\to
A$ �� ���ǵȴ�. (�� ������ �����ϴ� $\alpha$�� ���� �� �����ϸ� ��
��
�ϳ��� �����Ѵ�.)\\
����, ������ $\alpha\in A$�� ���Ͽ�
$V_\alpha=\bigcup\{N_\beta~|~\varphi(\beta)=\alpha\}$��
�����ϸ�(possibly $V_\alpha=\phi$),
$\mathcal{V}=\{V_\alpha~|~\alpha\in A\}$�� $\mathcal{U}$�� locally
finite open refinement�� �ȴ�. ����, ������ Claim�� ���Ͽ�
$\overline{V_\alpha}=\overline{\bigcup
N_\bet}=\bigcup{\overline{N_\bet}} \subset
\bigcup\overline{O_p}\subset U_\alpha$�� �����ϹǷ�
$V_\alpha\subset \overline{V_\alpha}\subset U_\alpha$�� �����Ѵ�.
\end{proof}
\\

Claim.  1) $\{F_i~|~ i\in I \} $, a locally finite collection of
closed sets $\Rightarrow $ $ \bigcup F_i$ is closed.\\
\hspace*{7ex} 2) $\mathcal{N}=\{N_\beta\}$ locally finite
$\Rightarrow$
$\overline{\mathcal{N}}=\{\overline{N_\beta}\}$, locally finite.\\
����. \h easy exercise\\

{\bf Step 2.}(special case)\\
Suppose $\mathcal{U}=\{U_\alpha~|~\alpha\in A\}$, where
$\overline{U_\alpha}$ is compact $\forall \alpha$. Then there
exists a partition of unity subordinate to $\mathcal{U}$.\\
\begin{proof}
step 1�� �ι� �����Ͽ� $\mathcal{V}=\{V_\alpha~|~\alpha\in A\}$��
$\mathcal{W}=\{W_\alpha~|~\alpha\in A\}$�� ��´�. ��, ������
$\alp$�� ���Ͽ�, $W_\alp\subset\overline{W_\alp}\subset
V_\alp\subset\overline{V_\alp}\subset
U_\alp\subset\overline{U_\alp}$.\\[2mm]
\underline{Claim} (bump
function) $C^{compact}\subset U^{open} \Rightarrow \exists
 \ccn$-function $f$ s.t. $f(C)=1$ and $f(U^c)=0$.\\
\begin{proof} $\forall p \in C$, $\exists f_p$ : $\ccn$ s.t.
$f_p>0$ on a neighborhood $U_p\subset U$ and 0 on $U^c$. Choose a
finite subcover $\{U_{p_1},\cdots, U_{p_n}\} $ of $C$. $f_{p_i}$��
$U_{p_i}$���� positive�̹Ƿ�, compact set $C$������
$f_{p_1}+\cdots+f_{p_n}>\delta$�� ��� $\delta$�� ���� �� �ִ�.
�Լ� $\varphi$��
\begin{displaymath}
\varphi : \ccn \h \textrm{ and }\h \varphi=\{
\begin{array}{l} 1 \h \textrm{ if } x\geq\delta \\ 0\h\textrm{ if } x\leq 0 \end{array}
\end{displaymath}
��� �ϸ�, $f=\varphi\circ(f_{p_1}+\cdots+f_{p_n})$�� ���ϰ���
�ϴ� �Լ��̴�.
\end{proof}\\[2mm]
\vspace{2mm} �� Claim�� ���Ͽ� $g_\alp(\overline{W_\alp})=1$ and
$g_\alp({V_\alp}^c)=0$�� $g_\alp$�� �����ϰ�, $supp~ g_\alp\subset
\overline{V_\alp}\subset U_\alp$�̴�. �� ��, $
\{\overline{V_\alpha}~|~\alpha\in A\}$�� locally finite
refinement�̹Ƿ�, \\ $\{supp~ g_\alpha ~|~ \alpha \in A\}$��
locally finite refinement�� �ǰ�, $f_\alp=\frac{g_\alp}{\sum
g_\alp}$�� �θ� a), b) ������ �����ϴ� partition of unity�� �ȴ�.
\end{proof}\\

{\bf Step 3.}\\
$C^{closed}\subset U^{open} \Rightarrow \exists f~:M\to [0,1]$,
$\ccn$ s.t. $f(C)=1$ and $f(U^c)=0$.\\
\begin{proof}
$M$�� locally compact�̹Ƿ� ������
$p\in C$�� ���Ͽ� $\overline{V_p}$�� compact�̰�
$\overline{V_p}\subset U$�� open neighborhood $V_p$��
���� �� �ִ�. \\
����, $M-C$�� open cover $\{V_\bet~|~\bet\in B\}$��
$\overline{V_\bet}$�� compact�̰� $V_\bet\subset M-C$�� �ǵ���
�ϳ� ������, $\{V_p, V_\bet ~|~ p\in C  , \bet\in B\}$�� $M$��
open cover�̰� Step 2�� ������ ������Ų��. ���� partition of
unity $\{f_p, f_\bet\}_{p\in C,
\bet\in B}$ subordinate to $\{V_p, V_\bet\}$�� �����Ѵ�. \\
����, $f=\displaystyle{\sum_{p\in C}}f_p$�� �θ�, $f(C)=1$ and
$f(U^c)=0$�� �����Ѵ�.
\end{proof}\\

{\bf Step 4.}(general case) \\
Step 2�� �������� Claim ��� Step 3�� ����ϸ� ���� ������� �����ȴ�. \\

{\bf ((2)$\Rightarrow$(3))} \\
$M$�� cover�ϴ� coordinate chart��
$\mathcal{U}=\{U_\alp\}_{\alp\in A}$��� �ΰ�, ������ $U_\alp$��
coordinate vector field $\frac{\partial}{\partial x_i}$��
orthonormal�ϵ��� local Riemannian metric $g_\alp$�� ����. ��,
\begin{displaymath}
g_\alp(\frac{\partial}{\partial x_i},\frac{\partial}{\partial
x_j})=\del_{ij}.
\end{displaymath}
$\{f_\alp\}_{\alp\in A}$�� partition of unity subordinate to
$\mathcal{U}$��� �ϰ�, $p\in M$���� $g$��
\begin{displaymath}
g(X,Y)=\sum_{\alp \in A} f_\alp(p) g_\alp (X,Y), \h X,Y\in T_pM
\end{displaymath}
�� �����ϸ�, $g$�� $M$�� Riemannian metric�� �ȴ�.\\
�̸� Ȯ���ϱ� ���Ͽ�, ���� $g$�� smooth, symmetric, bilinear��
���� �ڸ��ϴ�. $g_\alp (X,X)\geq 0$�� $f_\alp\geq 0$����
$g(X,X)\geq 0$���� �� �� �ִ�. ����, $g(X,X)=0$�̸�
$f_\alp(p)>0$�� $\alp$�� �����ϹǷ� $g_\alp(X,X)=0$�� �Ǿ�
$X=0$�� �ȴ�. ����, $g$�� Riemannian metric�̴�. \\

(����) (3)�� ������ �����ϴ� manifold�� Riemannian manifold���
�θ���.\\

{\bf ((3)$\Rightarrow$(4))} \\
Piecewise $\cc^1$ curve $\sig$�� ���̸� ������ ���� ��������.\\
\begin{displaymath}
l(\sig)=\int^b_a \big|\frac{d\sig}{dt}\big|dt, \hh where \
\big|\frac{d\sig}{dt}\big|=\sqrt{g\big(\frac{d\sig}{dt},\frac{d\sig}{dt}\big)}
\end{displaymath}
����, $M$�� metric $d$�� ������ ���� �����Ѵ�.
\begin{displaymath}
d(x,y):= \inf \{l(\sig)~|~all\ possible\ piecewise\ \cc^1\ curve\
\sig\
 from \ x\ to \ y\}
\end{displaymath}
�̶�, $d$�� $d(x,y)\geq 0$, $d(x,y)=d(y,x)$, $d(x,y)+d(y,z)\geq
d(x,z)$�� �����ϴ� ���� �ڸ��ϰ�, (4)�� ���̱� ���ؼ��� $d(x,y)=0
\Rightarrow x=y$�� $d$�� ���Ͽ� induce�� topology�� ����
topology��
������ ���̸� �ȴ�.\\

(a) locally, on a compact coordinate ball, \\
$\exists c,~ C>0$ s.t. $c\|X\|\leq|X|\leq C\|X\|$, $\forall
X=a_1\frac{\partial}{\partial
 x_1}+\cdots+a_n \frac{\partial}{\partial x_n}$,\\
  where
 $|X|=\sqrt{g(X,X)}$ and $\|X\|=\sqrt{a_1^2+\cdots+a_n^2}$.\\
\begin{proof}
$\mathcal{X}$�� orthonormal basis�� $\{e_i\}$���
�ϰ�, $\frac{\partial}{\partial x_j}=\sum a_{ij}e_i$�� ����. ����,
$G=(g_{ij})$ where $g_{ij}=g(\frac{\partial}{\partial
x_i},\frac{\partial}{\partial x_j})$
��� �ϸ�, $G=A^tA$�� �����Ѵ�.\\
����, \\[1mm]
$\hspace*{2em} |X|^2 = g(X,X)=\displaystyle{\sum_{i,j}}
g_{ij}a_{i}a_j
=X^tGX=X^tA^tAX=(AX)^t(AX) \\[1mm]
 \hspace*{9ex}
=\|AX\|^2\leq\|A\|^2\|X\|^2\leq
C\|X\|^2 $ for some $C$\\[1mm]
�̴�. \\ ����, $|~|$�� $\|~\|$�� ������ �ٲٸ� �ٸ� ���� �ε����
��´�.
\end{proof}\\

(b) ������ compact ball���� (a)�� ���Ͽ� $cl_E(\sig)\leq l(\sig)
\leq C l_E(\sig)$ �� �����Ѵ�. ($l_E$�� Euclidean metric�� ���� ����)\\
����, ������ $x,y\in M$�� ���Ͽ� $c\|x-y\|\leq d(x,y)\leq
C\|x-y\|$�� �����ϰ�, �� �ε�Ŀ� ����, $d(x,y)=0 \Rightarrow
x=y$�� $d$-topology=manifold topology���� �� �� �ִ�.\\

{\bf ((4)$\Rightarrow$(5))} \\
$\forall p\in M$, let $r(p)=\frac{1}{2}
\sup\{r~|~\overline{B_r(p)} \
is \ compact. \}$.\\
$M$�� locally compact�̹Ƿ�, $r(p)>0$�̰�, � $p$�� ���Ͽ�
$r(p)=\infty$�̸� �翬�� (5)�� �����ϹǷ�, $0<r(p)<\infty$���
����.\\[2mm]
{\bf Claim 1.} $r~:~M\to\rb$ is continuous.\\
\begin{proof}
$\overline{B_r(x_1)}\subset\overline{B_{r+d}(x_2)}, \
d=d(x_1,x_2)$ (by triangular inequality), $\forall r$ \\[1mm]
$\Rightarrow \overline{B_{r-d}(x_1)}\subset\overline{B_{r}(x_2)}$, $\forall r$\\
 $\Rightarrow r(x_1)\geq r(x_2)-\frac{1}{2}d$\\
$x_1$�� $x_2$�� ������ �ٲٸ�, $r(x_2)\geq r(x_1)-\frac{1}{2}d$.
����, $|r(x_1)-r(x_2)|\leq \frac{1}{2}d(x_1,x_2)$�� ���, $r$��
�������� �ȴ�.
\end{proof}\\

{\bf Claim 2.} $A\subset M$, $A$ compact $\Rightarrow
\widetilde{A}=\displaystyle{\bigcup_{a\in A}}
\overline{B_{r(a)}(a)}$ is compact.\\
\begin{proof}
 $\{z_i\}^\infty_{i=1}$�� $\widetilde{A}$�� ������ sequence���
����.\\
������ $z_i$�� ���Ͽ� $z_i\in\overline{B_{r(y_i)}(y_i)}$�� $y_i\in
A$�� �����ϰ�, $\{y_i\}^\infty_{i=1}$�� $A$�� sequence�̰�, $A$��
compact set�̹Ƿ�, $A$���� �����ϴ� subsequence�� ���´�. ��
subsequence�� $\{y_j\}$, $y_j \to y$��� �ϰ�, �̿� �����Ǵ�
$\{z_i\}$�� subsequence�� $\{z_j\}^\infty_{j=1}$��� ����.(���� $z_j$�� ���ż��� �������� �ʾҴ�.) \\
$\overline{B}:=\overline{B_{\frac{3}{2}r(y)}(y)}$��� �����ϸ�,
$r$�� �����̹Ƿ�, ����� ū $j$�� ���Ͽ�
$\overline{B_{r(y_j)}(y_j)}\subset\overline{B}$�� �����Ѵ�.\\
����, $z_j\in \overline{B_{r(y_j)}(y_j)}\subset\overline{B}$��
$z_j$�� �����ϸ�, �̴� $\overline{B}$�� ���ϴ� $\{z_j\}$��
subsequence�̰�, $\overline{B}$�� compact�̹Ƿ�,
$\overline{B}$���� �����ϴ� subsequence�� ���´�.
�� subsequence�� $\{z_k\}$, $z_k \to z$��� ����.\\
���� $0<\eps<\frac{1}{2}$�� ���Ͽ�
$\overline{B}_\eps:=\overline{B_{(1+\eps)r(y)}(y)}$��� �����ϸ�,
���������� $\{z_k\}$�� $\overline{B}_\eps$���� �����ϴ�
subsequence�� ����, ���� $z\in\overline{B}_\eps$�̴�.\\
 �ᱹ
$z\in\displaystyle{\bigcap_{0<\eps<\frac{1}{2}}\overline{B}_\eps=\overline{B_{r(y)}(y)}\subset\widetilde{A}}$
�̰�, ���� $\widetilde{A}$�� compact set�̴�. \\
($M$�� metrizable�̹Ƿ� sequentially compact�� compact��
��ġ�̴�.)
\end{proof}\\

����, $A_1=\{x_0\}$, $A_{n+1}=\widetilde{A_n}$�̶�� �ϰ�,
$M=\displaystyle{\bigcup_{n=1}^\infty} A_n$���� ��������.\\
$A=\displaystyle{\bigcup_{n=1}^\infty} A_n$ is open in $M$ :\\
Let $U_{n+1} := \displaystyle{\bigcup_{a\in A_n}} B_{r(a)}(a)$.
Then $U_n$ is open and $A_n\subset U_{n+1}\subset A_{n+1}$. \\
$\therefore A=\bigcup U_n$ : open.\\

$A=\displaystyle{\bigcup_{n=1}^\infty} A_n$ is closed in $M$ :\\
$x\in \overline{A}\Rightarrow\exists y\in A$ s.t.
$d(x,y)<\frac{2}{3}r(x)$ \\
$r(y)\geq r(x)-\frac{1}{2}d(x,y)>r(x)-\frac{1}{3}r(x)=\frac{2}{3}
r(x)>d(x,y)$\\
So, $y\in A_n\Rightarrow x\in B_{r(y)}(y)\Rightarrow
x\in\widetilde{A_n}=
A_{n+1}\subset A$\\
$\therefore A$ : closed.\\

����, $A=M$�̰�, �ᱹ $A_n$�� $M$�� compact exhaustion�� �ȴ�. \\

{\bf ((5)$\Rightarrow$(1))} \\
$M=\bigcup\overline{G_i}$, $\overline{G_i}^{compact}\subset
{G_{i+1}}^{open}$�� ����.\\
Let $\mathcal{U}=\{U_\alp\}$ be an open cover of $M$.\\
For each $i$, $\overline{G_{i+1}}-G_i$ : compact \\ $\Rightarrow
\exists$ a finite subcover
$U_{\alp_1}^{(i)},\cdots,U_{\alp_k}^{(i)}$
\\ $\Rightarrow
V_{\alp_j}^{(i)}=U_{\alp_j}^{(i)}\cap(G_{i+2}-\overline{G_{i-1}})$
: cover of $\overline{G_{i+1}}-G_i$ \\
Then $\mathcal{V}$, the collection of all such $V_{\alp_j}^{(i)}$,
is a locally finite open refinement of $\mathcal{U}$.

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

{\bf ((5)$\Rightarrow$(6))} Clear. \\


{\bf ((6)$\Rightarrow$(5))} \\
$M=\displaystyle{\bigcup^\infty_{i=1}}K_i$, where $K_i$ is
compact.\\
$M$ : locally compact Hausdorff space \\
$\Rightarrow K_1$ has a
neighborhood $U_1$ with compact $\overline{U_1}$.\\
$\Rightarrow \overline{U_i}\bigcup K_2$ has a
neighborhood $U_2$ with compact $\overline{U_2}$.\\
\hspace*{3ex}\vdots\\
 $\therefore M=\displaystyle{\bigcup^\infty_{i=1}}U_i$ with
 $\overline{U_i}\subset U_{i+1}$.\\

\bigskip
{\bf ((6)$\Rightarrow$(7))} \\
$M=\displaystyle{\bigcup^\infty_{i=1}}K_i$, where $K_i$ is
compact.\\
������ $K_n$�� ���Ѱ��� coordinate ball�� ���� �� �ְ�, ���� $M$
��ü�� countable coordinate ball�� ���� �� �ִ�.\\

{\bf ((7)$\Rightarrow$(8))} \\
������ coordinate ball�� $2^{\textrm{nd}} $ countable�̰�, $M$��
coordinate
ball�� countable union�̹Ƿ� $2^{\textrm{nd}} $ countable�̴�.\\

{\bf ((8)$\Rightarrow$(6))} \\
$\forall p\in M$, take a coordinate neighborhood $U_p$ with compact $\overline{U_p}$.\\
$\exists V_p$ : basic open set s.t. $p\in V_p\subset U_p$
$\Rightarrow$ $\overline{V_p}\subset\overline{U_p}$, and hence
$\overline{V_p} $ is compact\\
Then, $M=\bigcup\overline{V_p}$ and $\{\overline{V_p}\}$ is
countably many.
\end{proof1}
\begin{cor}
{\rm Let $M$ be a (\emph{\textbf{not necessarily connected}})
Hausdorff $\ccn$-manifold, then the
followings are equivalent.\\
(1) $M$ is paracompact. \\
(2) $\forall \mathcal{U} $ an open covering of $M$, $\exists $ a
partition
of unity subordinate to $\mathcal{U}$.\\
(3) $M$ admits a Riemannian metric.\\
(4) $M$ is metrizable.\\
(5) Each component of $M$ has a compact exhaustion.\\
(6) Each component of $M$ is $\sigma$-compact.\\
(7) Each component of $M$ is a countable union of coordinate
charts.\\
(8) Each component of $M$ is $2^{\textrm{nd}} $ countable.

 }
\end{cor}
\begin{proof}

(1)$\Leftrightarrow$Each component of M is paracompact.(Clear by definition, $M$�� locally connected�̹Ƿ� �� component�� open�̴�.) \\
(2)$\Leftrightarrow$Each component has property (2) (trivial) \\
(3)$\Leftrightarrow$Each component has property (3) (trivial) \\
(4)$\Leftrightarrow$Each component has property (4) :
\begin{sub}{   }
($\Rightarrow$) clear \\
($\Leftarrow$) $M$�� �� component�� $(M_i,d_i)$��� �ϰ�, ���ο�
metric $\widetilde{d_i}=\frac{d_i}{1+d_i}$�� �����ϸ�,
$\widetilde{d_i}$�� $d_i$�� ���� topology�� �ְ�,
$\widetilde{d_i}(x,y)<1$�̴�. \\
����, $M$�� metric $d$��
\begin{displaymath}
d(x,y) =\{
\begin{array}{ll}
\widetilde{d_i}(x,y), & \textrm{if } x,y \textrm{\hspace*{1ex}are
in
the same component $M_i$} \\
1\hspace{6ex} ,& \textrm{otherwise}
\end{array}
\end{displaymath}
�� �����ϸ�, $d$�� manifold topology�� ���� topology�� �ش�.
\end{sub}
\bigskip
�� component������ (1)~(8)�� ��ġ���� �˰� �����Ƿ� ������.
\end{proof}
\end{document}
