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\begin{document}
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\section*{CW-complex as an adjuction space}

\begin{thm}
(1) Let $X$ be a CW-complex of dimension $p$ and $\{e_{\alp}\}$ be
$p$-cells in $X$ with a characteristic function $\varphi_{\alp} :
D_{\alp}^{p} \to X$.\\
$\hspace*{1.5em}\Rightarrow X \cong \coprod D_{\alp} \cup_{f}
X^{p-1}, \hspace{1.0em} f= \coprod \varphi_{\alp}|_{\bd D_{\alp}}$

(2) Conversely, $Y$ is a CW-complex of dim $p$-1 and $f : \coprod
\bd D_{\alp}^{p} \to Y$\\
$\hspace*{1.5em}\Rightarrow X = \coprod D_{\alp} \cup_{f} Y$ is a
CW-complex with
$X^{p-1} = Y$.\\

\end{thm}

\begin{pf}

(1) Let $E = \underset{\alp}{\coprod} D_{\alp} \coprod X^{p-1}$.\\
Define $h : E \to X$ by $ h=(\coprod \varphi_{\alp}) \coprod
(incl.)$.\\
It suffices to show that $h$ is a quotient map.\\

Let $C \subset X$ with $h^{-1}(C)$ closed in $E$. Then\\
$\Rightarrow \left\{
\begin{array}{ll}
C \cap X^{p-1} = h^{-1}(C) \cap X^{p-1} \textrm{is closed in
}X^{p-1}\\
 h^{-1}(C) \cap D_{\alp} \, \textrm{is closed in } \, D_{\alp}, \forall \alp \, \textrm{ and so
 compact.}
\end{array}\right.$


$\Rightarrow \left\{
\begin{array}{ll}
C \cap \ebb = (C \cap X^{p-1})\cap \ebb \, \textrm{ is closed in }
\, \ebb
\textrm{ whenever } \, dim e_{\bet} \leq p-1\\
C \cap \eab = h(h^{-1}(C) \cap D_{\alp}) : \textrm{ compact and so
closed in } \, \eab.
\end{array}\right.$


$\Rightarrow C$ is closed in $X$.\\

(2) $\left\{
\begin{array}{ll}
Y : \textrm{Hausdorff}\\
(\coprod D_{\alp}, \coprod \bd D_{\alp}) : \textrm{collared
pair}\\
\coprod D_{\alp} \coprod Y \overset{q}{\rightarrow} X\\
\end{array}\right.$

$\Rightarrow X$: Hausdorff.

���� CW-complex�� ������ �����ϴ��� check����. �켱 (1)��
�翬�ϰ�,\\

(2) $\varphi_{\alp} = q|_{D_{\alp}}$�� �����ϸ�, characteristic
function�� �ȴ�. �ֳ��ϸ�, �տ��� $q|_{D_{\alp}-\bd D_{\alp}}$��
homeomorphism ����
�˱� �����̴�.\\

(3) $\eab = \varphi_{\alp}(D_{\alp}) = q(D_{\alp})$ and note that
$q(\bd D_{\alp})$ is compact and hence meets only finitely many cells.\\

(4) Let $X = \{e_{\bet}\}$ and $Y= \{e_{\gam}\}.$\\
$A \subset X$ with $A \cap \ebb$ closed in $\ebb, \forall \bet$\\
$\Rightarrow \left\{
\begin{array}{ll}
 q^{-1}(A) \cap Y = A \cap Y \, \textrm{ is closed in } \, Y,\\
 \hspace*{1.5em} \textrm{ since } (A \cap Y)
\cap \err = A \cap \err \, \textrm{ is closed in } \, \err.\\
 q^{-1}(A) \cap D_{\alp} = q^{-1}(A \cap \eab) \cap D_{\alp} :
\textrm{ closed in } \, D_{\alp}\\
\end{array}\right.$

$\Rightarrow q^{-1}(A)$ is closed in $\coprod D_{\alp} \coprod
Y$.\\
$\Rightarrow A$ is closed.

\end{pf}

\begin{thm}
(1) Let $X$ be a CW-complex.\\
\hspace*{1.0em} Then $X$ is a coherent union of
$X^{0} \subset X^{1} \subset X^{2} \subset \cdots$.\\
(2) Conversely, if $X$ is a coherent union of CW-complexes $X_{0}
\subset X_{1} \subset X_{2} \subset \cdots$ with $X_{p}$, the
$p$-skeleton of $X_{p+1}$, then $X$ is a CW-complex with $X^{p} =
X_{p}$.
\end{thm}

\begin{pf}
Easy from the definitions.{\bf (���� 10)}
\end{pf}



\end{document}
