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\section*{II. CW-complex}
\begin{defn}
A \key{CW-complex} $X$ is a Hausdorff space along with a family
$\{e_\alp\} $ of "open cells" such that the following conditions
are satisfied.\\
Let $X^p=\bigcup\{e_\alp | \ dim \ e_\alp \leq p\}$. ($p$-skeleton)\\
(1) $X=\coprod e_\alp$(disjoint union). \\
(2) $\forall n$-cell $e_\alp$, $\exists$ a characteristic map
$\varphi_\alp : (D^n,\bd D^n)\to (X,X^{n-1})$ such that
$\varphi_\alp|_{\overset{\circ}{D^n}}$ is a homeomorphism onto
$e_\alp$.\\
(3) (Closure finiteness) Each $\bar{e}_\alp$ is contained in
the union of finitely many open cells.\\
(4) (Weak topology) $A\subset X$ is closed if and only if
$A\cap\bar{e}_\alp$ is closed in $\bar{e}_\alp$ for all $\alp$.
\end{defn}
Note.\\
(1) Let $\dot{e}_\alp = \eab-e_\alp$. Then $\varphi_\alp :
(D^n,\bd
D^n)\to (\eab,\dot{e}_\alp)$ is onto. \\
\begin{pf}
$\varphi_\alp(D^n)$�� compact�̰� ���� closed�̴�. ������ ����
(2)�κ��� $e_\alp\subset \vph_\alp(D^n)$�̹Ƿ�,
$$ \eab\subset \vph_\alp(D^n)$$
�̴�. ���� $\vph_\alp$�� continuous�̹Ƿ�
$$
\vph_\alp(D^n)=\vph_\alp(\overline{\overset{\circ}{D^n}})\subset
\overline{\vph_\alp(\overset{\circ}{D^n})}=\eab
$$
�̴�. ���� $\vph_\alp(D^n)=\eab$�̰� $\vph_\alp(\bd
D^n)=\dot{e}_\alp$�̴�.
\end{pf}

(2) For a finite CW-complex, condition (3) and (4) are automatic.


\begin{defn}
Let $X$ be a CW-complex. A \key{subcomplex} of $X$ is a subset $Y$
along with a subfamily $\{e_\bet\}$ of the cells in $X$ such that
$Y=\bigcup e_\bet$ with $\ebb\subset Y $ for all $\bet$.
\end{defn}
Note. A subcomplex $Y$ is closed and a CW-complex in its own
right.\\
\begin{pf}
$Y$�� ���� (1)-(3)�� �����ϴ� ���� �ڸ��ϴ�.\\
\underline{Claim} If $B\subset Y$ with $B\cap \ebb$ is closed in
$\ebb$ for all $\bet$, then $B$ is closed in $X$.\\
pf) ���� (4)�� ���Ͽ� $B\cup \eab$�� closed in $X$���� ���̸�
�ȴ�. ���� (3)�� $Y$�� ������ ���Ͽ� $Y\cap \eab \subset
e_1\cap\cdots\cap e_k$�� $e_1,\cdots, e_k\subset Y$�� �����Ѵ�.
�׷��� $B\cap \eab\subset Y\cap \eab$�̹Ƿ� $B\cap \eab=((B\cap
\bar{e}_1)\cup\cdots\cup(B\cap \bar{e}_k))\cap \eab$�̰� ����
$B$�� $X$���� closed�̴�. \\

���� Claim���� ���� (4)�� �������� �� �� �ְ� ���� $B=Y$�� �ϸ�
$Y$�� closed���� ��´�.
\end{pf}

Example. $X^p=\bigcup\{e_\alp | \ dim \ e_\alp \leq p\}$=
$p$-skeleton of $X$ is a subcomplex (and hence closed).\\

\begin{defn}
$dim \ X$= $\sup\{dim \ e_\alp \ | \ e_\alp\subset X\}$
\end{defn}

\begin{thm}
Let $X$ be a CW-complex with $\coprod e_\alp= X$. \\
(1) $f:X\to Y$ is continuous if and only if $f|_{\eab}$ is
continuous for all $\alp$.\\
(2) $F:X\times I \to Y$ is continuous if and only if
$F|_{\eab\times I }$ is continuous for all $\alp$.
\end{thm}
\begin{pf}
������ (����)�� ������ ����$^*$�κ��� �ڸ��ϴ�.
\end{pf}\\
(����) \\
A space $X=\bigcup X_\alp$ has a coherent topology with respect to
$X_\alp$ (or $X$ is a coherent union of $X_\alp$) if\\
$A\subset X$ is closed $\Leftrightarrow A\cap X_\alp$ is closed in
$X_\alp$ for all $\alp$\\
 or equivalently, a natural projection
$p:\coprod X_\alp \to X$ is
a quotient map\\
or equivalently, $f:X\to Y $ is continuous $\Leftrightarrow
f|_{X_\alp}:X_\alp \to Y $ is continuous for all $\alp$.

\begin{pf} {\bf ���� 9.} \end{pf}
\bigskip

$^*$����(Theorem 20, of Munkres, p.113)\\
$p:X\to Y$, a quotient map. $C$ : locally compact, Hausdorff\\
$\Rightarrow p\times id : X\times C \to Y\times C $ is a quotient
map.



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