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\begin{document}
\parindent=0cm
\section*{I. Adjunction Space (Attaching space)}
\textbf{I.1 Construction}

\begin{defn} $X, Y$: disjoint topological spaces , $A \overset{closed}{\subset}
X$ and $ f: A\rightarrow Y$, a map.\\
Define an equivalence relation $\sim$ on $X\coprod Y$ generated by
$a\sim f(a),\hspace{1em} \forall a\in A$.\\
The quotient space $X\underset{f}{\cup} Y = X\coprod Y/\sim$ is
called the adjunction space determined by $f$ and $f$ is called an
attaching map.\end{defn}

\begin{thm}(Extension principle)\\
Let $g: X\rightarrow Z$ and $h: Y\rightarrow Z$ s.t.
$g(a)=hf(a),\hspace{1em} \forall a\in A\Rightarrow$

\[
\xymatrix @M=1ex @C=3em @R=1em @*[c]{%
 X\coprod Y \ar[r]^{g\coprod h} \ar[dd]^{p}&Z&\\
            & & \textrm{ $\exists ! k$ s.t. the diagram commute}\\
 X\underset{f}{\cup} Y \ar@{.>}[uur] &  &}
\]
\end{thm}

\begin{thm} Let $X\coprod Y \overset{p}{\rightarrow}
X\underset{f}{\cup} Y$ be the quotient map.\\
(1) $Y$ is embedded as a closed subset of $X\underset{f}{\cup}Y:\\
\hspace*{1.5em}p|_Y : Y\rightarrow p(Y)$ is a homeomorphism.\\
(2) $X-A$ is embedded as an open subset of
$X\underset{f}{\cup}Y:\\
\hspace*{1.5em}p|_{X-A} : X-A \rightarrow p(X\setminus A) $ is a
homeomorphism.\end{thm}
\begin{pf}(1) $p|_Y$ is continuous and 1-1. \\
Show $p|_Y$ is a closed map:\\
$C\subset Y$ a closed subset and show $p(C)$ is closed in
$X\underset{f}{\cup}Y$,\\ i.e., $p^{-1}p(C)= f^{-1}(C)\coprod C$
is
closed. And the assertion clearly holds.\\

(2) $p|_{X-A}$ is continuous and 1-1. Show it is an open map:\\
$U\subset X-A$ open $\Rightarrow U$ open in $X \Rightarrow p(U)$
is open since $p^{-1}p(U)=U$ is open is $X\coprod Y$.\end{pf}
\newpage
\begin{thm} (Separation Axiom)\\
$X ,Y : T_1 \Rightarrow X\underset{f}{\cup}Y : T_1\\
X, Y :$ normal $\Rightarrow X\underset{f}{\cup} Y$: normal\\

Ref. See Munkres p.210\end{thm}

\begin{defn}(Collared pair) $(X,A)$ is called a collared pair if\\
(1)$ A\subset X$ is closed.\\
(2)$ X$ is Hausdorff.\\
(3) Points in $X-A$ can be separated from $A: \forall x\in X-A,
\hspace{0.5em} \exists U, V:$ disjoint open sets s.t. $x\in U $
and $ A\subset V.$\\
(4) $A$ has a collaring $B$ in $X$ : $\exists$ open $B\supset A$
s.t. $A$ is a strong deformation retract of $B$.\end{defn}

\begin{prop} $(X, A)$: a collard pair, $Y$: Hausdorff $\Rightarrow
(X\underset{f}{\cup} Y , Y)$ : a  collard pair.\\
In fact, $B$: a collaring of $A \Rightarrow Y\cup p(B)$: a
collaring of $Y$.\end{prop}
\begin{pf} (1) : clear from ���� 2(1)\\

(2) $X\underset{f}{\cup} Y$ is Hausdorff: \\
Case 1. $z_1 , z_2 \in X\underset{f}{\cup}Y -Y \cong X-A
\Rightarrow $clear.\\

Case 2. $z_1 \in Y , z_2\notin Y \overset{\textrm{����
2(3)}}{\Rightarrow} \exists U \ni z_2 , V\supset A\\ \Rightarrow
p(U):$ open neighborhood of $z_2$ and $p(V)\cup Y$: open
neighborhood of $z_1$ gives a separation. (Note $p^{-1}(p(V)\cup
Y) = V\coprod Y$: open in $X\coprod Y.$)\\

Case 3. $z_1 , z_2 \in Y :$ Let $z_1 \in V_1 , z_2 \in V_2 $
be a separation and \\
$r :B\rightarrow A$ a strong deformation retract. Let $U_i =
r^{-1}f^{-1}(V_i )$: open in $X.\\ \Rightarrow p(U_1 )\cup p(V_1
)$ and $p(U_2) \cup p(V_2)$ give a separation for $z_1$ and $z_2$\\
(Note $p^{-1}(p(U)\cup p(V)) = U\coprod V$.)\\

(3) $ z\notin Y$. Then use ���� 2(3) to get disjoint open sets
$U\ni z$ and $ V\supset A \Rightarrow p(U)$ and $p(V)\cup Y$ give
a separation
for $z$ and $Y$.(cf. Case2.)\\

(4) Let $D: id \simeq i\cdot r$(rel $A$) be a strong deformation retract :\\
$D:B\times I \rightarrow B$ s.t. $\left\{\begin {array}{cc} D(a,t)=a ,& \forall a\in A\hspace{1em} t\in I\\
D(b,0) = b , & \forall b\in B\\
D(b,1) = r(b) \in A ,& \forall b\in B
\end{array}\right\}$\\


Define $\bar{D} : p(B)\cup Y \times I \rightarrow p(B)\cup Y$ by
$\bar{D}(z,t) = \left\{\begin {array}{ccc} z ,& z\in Y&\\
p(D(b,t)) ,& z =  p(b) ,& b\in B-A
\end{array}\right\}
$\\

\[
\xymatrix @M=1ex @C=3.5em @R=1em @*[c]{%
(B\coprod Y)\times I\ar[rr]^{D\coprod p_1 }\ar[dd]^{p\times id} && B\coprod Y\ar[dd]^{p}\\
 & \curvearrowright & \\
(p(B)\cup Y)\times I\ar[rr]^{\bar{D}} & & p(B)\cup Y}
\]
$\Rightarrow \bar{D}$ is continuous by the following fact.\\

 Fact. $p:X\rightarrow Y$ quotient , $C:$ locally compact
Hausdorff.\\ $\Rightarrow p\times id :X\times C \rightarrow
Y\times C$
is a quotient map.\\
\begin{pf} Ref. Munkres p.113\end{pf}
\end{pf}

\end{document}
