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\begin{document}
 \parindent=0cm
  \section*{Lifting Property  }

\begin{thm}\textit{\textbf{(Uniqueness of lifting)}}\\
\textcolor{blue}{ÁÖ¾îÁø covering
p:$(\widetilde{X},\widetilde{x_0})\rightarrow(X,x_0 )$¿¡ ´ëÇØ Y°¡
connectedÀÌ°í $f:(Y,y_0 )\rightarrow(X,x_0 )$°¡ lifting
$\widetilde{f} : (Y,y_0 ) \rightarrow
(\widetilde{X},\widetilde{x_0 })$À» °¡Áö¸é, ÀÌ´Â
uniqueÇÏ´Ù.}\end{thm}

\framebox{\hspace*{1em}\parbox[b]{14cm}{
$\hspace*{6em}(\widetilde{X},\widetilde{x_0})$\\
$\hspace*{2.3em}\exists
\widetilde{f}\nearrow\hspace{3em}\downarrow p$ : covering
$\hspace{1em}\Rightarrow\hspace{1em}\widetilde{f}$,
a lifitng of $f$, is unique if it exists.\\
$(Y,y_0)\hspace{1em}\overset{f}{\rightarrow}\hspace{1em}(X,x_0)$
}}\\

\begin{proof}\\
$\widetilde{f} , \widetilde{f}'$¸¦  fÀÇ liftingÀ̶ó ÇÏÀÚ.\\

A := $\{y \in Y| \widetilde{f}(y) = \widetilde{f} '(y)\}\ni y_0$\\

A is open : \\
$y\in A$¿¡ ´ëÇØ f($y$)ÀÇ evenly coveredµÈ open neighborhood U¸¦
ãÀÚ.\\
±×·¯¸é $p^{-1} (U) = \coprod_{\alp} \widetilde{U_{\alp}}$ÀÌ°í,
$\exists \widetilde{U_{\alp}} \ni \widetilde{f} (y) =
\widetilde{f} '(y)$ÀÌ´Ù.\\
±×·¯¸é $y\in \widetilde{f}^{-1} (\widetilde{U_{\alp}}) \cap
\widetilde{f}'^{-1}
(\widetilde{U_{\alp}}) \subset A$À̹ǷΠA°¡ openÀÌ´Ù. \\

¸¶Âù°¡Áö·Î $A^c = \{y \in Y|\widetilde{f}(y)\neq\widetilde{f}
'(y)\} $µµ openÀÌ´Ù.\\

±×·±µ¥ Y°¡ connectedÀ̹ǷΠ$A^c = \phi$ÀÌ°í À̷μ­ uniqueness of
liftingÀÌ Áõ¸íµÇ¾ú´Ù.
\end{proof}

\begin{lem}\textit{\textbf{(Lebesgue Covering lemma)}}\\
\textcolor{blue}{In a compact metric space, every open cover
$\mathcal{ U}$ has a Lebesgue number, i.e. , $\exists \epsilon
>0$(depending on $\mathcal {U})$ s.t. $\forall$ A $\subset X$ with
diam(A)$<\epsilon$, $\exists$ U $\in \mathcal{U}$ s.t. A$\subset
U$.}\end{lem}
\begin{proof}\\
D¸¦ $\{$diam(A) $\in \mathbb{R} |$ A is not contained in any
$U\in\mathcal{U}\}$¶ó°í µÎÀÚ. ÀÌ·± A¸¦ "big set"À̶ó ºÎ¸£ÀÚ.

±×·¯¸é $\epsilon $=inf D $>$0ÀÓÀ» º¸À̸é Áõ¸íÀÌ ³¡³­´Ù.

¸¸¾à inf D°¡ 0 À̶ó¸é $\forall n, \exists A_n :$ big set s.t.
diam($A_n) < \frac{1}{n}$ ÀÌ°í, $x_n \in A_n$ ÀÎ $\{x_n\}$À» ãÀ»
¼ö ÀÖ´Ù.

±×·¯¸é X °¡ compactÇϹǷΠ$x_n \rightarrow x$(by passing to a
subsequence)ÀÎ $x$°¡ Á¸ÀçÇÑ´Ù.

±×·¯¸é $x$¸¦ Æ÷ÇÔÇÏ´Â U$\subset \mathcal{U}$¿¡ Æ÷ÇԵǴ $A_n$(nÀÌ
ÃæºÐÈ÷ Ŭ ¶§)µéÀ» ãÀ» ¼ö ÀÖ°í ÀÌ´Â big setÀÇ Á¤ÀÇ¿¡ ¸ð¼øµÈ´Ù.\end{proof}\\

\begin{thm}\textit{\textbf{
(Unique path lifting property)}}\\
\textcolor{blue}{Let $p: \widetilde{X} \rightarrow X$ be a
covering map and let $\alp : I \rightarrow X$ be a path with
$\alp(0)=x_0 \in X $ and $ p(\widetilde{x_0})=x_0$. Then $\alp$
has a unique path lifting $\widetilde{\alp}:I\rightarrow X$ with
$\widetilde{\alp}(0)=\widetilde{x_0}$ i.e., $p\circ
\widetilde{\alp}(t)=\alp (t).\,\,\forall \,\,t \in I.$}\end{thm}
\begin{proof}\\\textit{\textbf{(Existence)}}\\ For each t, $\alp(t)\in X$ has an
open neighborhood $U_t$ which is evenly covered by
$\displaystyle{\coprod_{a \in A}}V_{t,a}$ .  Since I=[0,1] is
compact, we can choose a Lebesgue number $\epsilon > 0$ for a
cover $\{\alp^{-1}(U_t)| t
\in I \}$ of I. Choose a partition of $\,\,$ I,\\
$\,\,0=t_0<t_1<...<t_{n+1}=1\,\,$ so that
$t_{i+1}-t_{i}<\epsilon,\,\,\,\,i=1,...n$.

 Then note that $\alp[t_i,t_{i+1}] \subset U_t$ for some t and we
 lift $\alp|_{[t_i,t_{i+1}]}$ inductively :

 Suppose $\alp|_{[t_0,t_{i}]}$ is already lifted (note that the
 initial point $x_0$ is lifted to $\widetilde{x_0}$). Then $\alp[t_i,t_{i+1}]\subset
 U_t$ for some t and $p^{-1}(U_t)=\displaystyle{\coprod_{a \in
 A}}V_{t,a}$ and there exists a unique $a \in A$ such that
 $\tilde{\alp}(t_i)\in V_{t,a}\,.$\\And since $p|_{V_{t,a}}:V_{t,a}\rightarrow
 U_t\,\,$is homeomorphism we can lift $\alp|_{[t_i,t_{i+1}]}$ using $(p|_{V_{t,a}})^{-1}$ and the proof is
 completed.\\

\textit{\textbf{(Uniqueness)}}\\ The uniqueness follows from the
uniqueness property of general lifting.
\end{proof}\\

\begin{thm}
\textit{\textbf{(Covering homotopy
property)}}\\\textcolor{blue}{Let
$p:(\widetilde{X},\widetilde{x_0})\rightarrow (X,x_0)$ be a
covering.\\
Let $f :(Y,y_0)\rightarrow (X,x_0)$ and
$\widetilde{f}:(Y,y_0)\rightarrow
(\widetilde{X},\widetilde{x_0})\,\,\,$ be a lifting of f. \\
Then for a homotopy F : $Y\times I \rightarrow X$ s.t.
$F|_{Y\times \{0\}} =f $, there exists a unique lifting
$\widetilde{F}$ of F s.t. $\widetilde{F} |_{Y\times \{0\}} =
\widetilde{f}$.}
\end{thm}

\framebox{\hspace*{1em}\parbox[b]{14cm}{
$\hspace*{6em}(\widetilde{X},\widetilde{x_0})\hspace{13em}\widetilde{X}$\\
$\hspace*{2.3em}\exists
\widetilde{f}\nearrow\hspace{3em}\downarrow
p\hspace{4em}\Rightarrow\hspace{3em}\exists ! \widetilde{F}$
\textcolor{blue}{$\nearrow$}$\hspace{2em}\downarrow p\hspace{1em}$
s.t. $\widetilde{F}
|_{Y\times \{0\}} = \widetilde{f}$\\
$(Y,y_0)\hspace{1em}\overset{f}{\rightarrow}\hspace{1em}(X,x_0)\hspace{7em}Y\times
I \hspace{1em}\overset{F}{\rightarrow}\hspace{1em} X $}}\\
\newpage
\begin{proof}\\Step 1 . Lift F on $U_y \times I $ for each $y\in
Y$.\\
¿©±â¼­ $y$ÀÇ neighborhood $U_y$´Â IÀÇ compactness¸¦ ÀÌ¿ëÇÏ¿©
constructÇÑ´Ù.

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1)$f(y\times I)$ À§¿¡¼­ evenly cover µÇ´Â covering $\{W_t\}$À» Àâ´Â´Ù.\\
2) ÀÌ ¶§ $\{f^{-1}(W_t )\}$´Â $y\times I$ÀÇ coveringÀÌ µÇ°í °¢ Á¡
(y,t)¸¶´Ù $f^{-1}(W_t)$¿¡ Æ÷ÇԵǴ product neighborhood $U_t
\times V_t $¸¦ ÀâÀ¸¸é IÀÇ
compactness¿¡ ÀÇÇØ À̵é Áß À¯ÇÑ°³·Î $y\times I$¸¦ coverÇÒ ¼ö ÀÖ´Ù. \\
3) ÀÌ finiteÇÑ product neighborhoodµé¿¡¼­ $U_t$µéÀÇ intersectionÀ»
$U_y$¶ó µÐ´Ù.\\
4) ´ÙÀ½¿¡ $U_y \times I$¸¦ liftÇϱâ À§ÇÏ¿© I¿¡ Àû´çÈ÷ partitionÀ»
ÁÖ¾î °¢ $U_y \times [t_i , t_{i+1}]$ÀÌ 2)¿¡¼­ ÀâÀº finite product
neighborhood¿¡ Æ÷ÇԵǵµ·Ï ÇÑ´Ù.\\
5) path lifting ¶§¿Í ¸¶Âù°¡Áö·Î inductiveÇÏ°Ô liftingÇÑ´Ù.
\\

Step 2. Piece together $\widetilde{F}|_{U_y \times I}$
to get $\widetilde{F}$ on $Y\times I$.\\
Step1¿¡¼­ °¢ $y\in Y$¿¡ ´ëÇØ $U_y \times I$»ó¿¡¼­
$\widetilde{F}$¸¦ Á¤ÀÇÇߴµ¥ À̵éÀÌ °ãÄ¡´Â
ºÎºÐ¿¡¼­ ÀÏÄ¡ÇÔÀ» º¸À̸é $Y\times I$¿¡¼­ $\widetilde{F}$°¡ Àß Á¤ÀǵÊÀ» ¾Ë ¼ö ÀÖ´Ù.\\
Áï, $z \in U_y \cap U_y'$¿¡ ´ëÇØ $\widetilde{F}|_{U_y \times I}
(z,t) =
\widetilde{F}|_{U_y' \times I} (z,t)$ÀÓÀ» º¸¿©¾ß ÇÑ´Ù.\\
ÀÌ°ÍÀº $U_y$¿¡¼­ º¸³ª $U_y'$ ¿¡¼­ º¸³ª $ y\times I$ÀÇ liftingÀÌ°í
°°Àº initial point¸¦ °¡Áö°í ÀÖÀ¸¹Ç·Î uniqueness of lifting¿¡ ÀÇÇØ
ÀÏÄ¡µÊÀ» ¾Ë ¼ö ÀÖ´Ù .\end{proof}
\\
\begin{cor}\textit{\textcolor{blue}{$\alp \sim \beta \Rightarrow \widetilde{\alp}\sim
\widetilde{\beta}$.}}\\\textcolor{blue}{¿©±â¼­ $\alp,\beta$´Â XÀÇ
pathµéÀÌ°í $\widetilde{\alp}$,$\widetilde{\beta}$´Â °°Àº initial
point¸¦ °¡Áö´Â lifting µéÀÌ´Ù. µû¶ó¼­ $\widetilde{\alp}$¿Í
$\widetilde{\bet}$´Â °°Àº terminal point¸¦ °®´Â´Ù.}
\end{cor}
\newpage

\textbf{ Some Consquences}\\

\textbf{1.} \textit{\textcolor{blue}{$\pi_1(S^1)\cong \textbf{Z}$.}}\\

\begin{proof}
Let $p : \textbf{R}\rightarrow S^1$ be a covering map given by
$p(x)=e^{2\pi ix}$. \\If $[\alp]\in \pi(S^1,1)$, then $\alp$ can
be lifted uniquely to $\widetilde{\alp}:I\rightarrow \textbf{R}$
with $\widetilde{\alp}(0)=0$.\\Define
$\phi:\pi(S^1,1)\rightarrow\textbf{Z}\,\,$ by $\,\,[\alp] \mapsto
\widetilde{\alp}(1)$. $\phi $ is well-defined by µû¸§Á¤¸® 5.\\

ÀÌ $\phi$°¡ isomorphismÀÓÀ» º¸À̱â À§ÇØ ¸ÕÀú  homomorphismÀÓÀ»
º¸ÀÌÀÚ.

$\phi([\alp][\beta])=\phi([\alp*\beta])=\widetilde{\alp*\beta}(1)$ÀÌ°í
ÀÌ°ÍÀÌ $\widetilde{\alp}(1)+\widetilde{\beta}(1)$ÀÓÀ» º¸À̱â À§ÇØ
$\tau$¸¦ ´ÙÀ½°ú °°ÀÌ Á¤ÀÇÇÏÀÚ. \\$\tau : \textbf{R}\rightarrow
\textbf{R}$ be a translation given by
$\tau(x)=x+\widetilde{\alp}(1).$

ÀÌ ¶§,  $\widetilde{\alp*\beta}=\widetilde{\alp}*(\tau\circ
\widetilde{\beta})$ÀÌ ¼º¸³ÇÑ´Ù. ¸ÕÀú $\widetilde{\alp}*(\tau\circ
\widetilde{\beta})$´Â Àß Á¤ÀǵǾú´ÂÁö Áï
$\widetilde{\alp}(1)=(\tau\circ \widetilde{\beta})(0)$ ÀÎÁö
»ìÆ캸ÀÚ. $\widetilde{\alp}$¿Í $\widetilde{\bet}$´Â µÑ ´Ù 0¿¡¼­
Ãâ¹ßÇÏ´Â pathÀÌ°í  $\tau$´Â $\widetilde{\alp(1)}$¸¸Å­ translation
½ÃÄÑÁÖ´Â mapÀ̹ǷÎ,  $\widetilde{\beta}(0)$´Â $\tau$¿¡ ÀÇÇØ
$\widetilde{\alp}(1)$À¸·Î ¿Å°ÜÁö°í $\widetilde{\alp}*(\tau\circ
\widetilde{\beta} )$´Â Àß Á¤ÀǵȴÙ.\\´ÙÀ½À¸·Î
$\widetilde{\alp}*(\tau\circ \widetilde{\beta})$ °¡ $\alp*\beta$ÀÇ
lifting ÀÓÀ» º¸ÀÌÀÚ.

$p\circ (\widetilde{\alp}*(\tau \circ \widetilde{\beta}))=(p\circ
\widetilde{\alp} )*(p\circ (\tau \circ \widetilde{\beta}))=(p\circ
\widetilde{\alp} )*(p\circ \widetilde{\beta})=\alp*\beta$

µû¶ó¼­ $\widetilde{\alp*\beta}=\widetilde{\alp}*(\tau\circ
\widetilde{\beta})$°¡ ¸¸Á·ÇÏ°í ´ÙÀ½ÀÌ ¼º¸³ÇÑ´Ù.

$\widetilde{\alp*\beta}(1)=\widetilde{\alp}*(\tau\circ
\widetilde{\beta})(1)=(\tau\circ
\widetilde{\beta})(1)=\tau(\widetilde{\beta}(1))=\widetilde{\alp}(1)+\widetilde{\beta}(1)$.\\

´ÙÀ½À¸·Î $\phi$ °¡ 1-1 ÀÓÀ» º¸ÀÌÀÚ.\\ ¸¸ÀÏ $\phi([\alp])=0$ À̶ó¸é
$\widetilde{\alp}(1)=0$ ÀÌ µÇ¾î $\widetilde{\alp}$ ´Â 0À» base
point·Î ÇÏ´Â loop°¡ µÈ´Ù. Áï $[\widetilde{\alp}]\in
\pi_1(\textbf{R},0)=0$À̹ǷÎ\\
$0=p_{\sharp}[\widetilde{\alp}]=[p\circ
\widetilde{\alp}]=[\alp]\in \pi(S^1,1)$ .  Áï
$[\alp]=0$ÀÌ´Ù.\\

¸¶Áö¸·À¸·Î $\phi$°¡ ontoÀÓÀ» º¸À̱â À§ÇØ $\forall\,\,n \in
\textbf{Z}$¿¡ ´ëÇØ $ \widetilde{\alp}(t)=nt\,\,$,$\,\,\alp=p\circ
\widetilde{\alp}$ ·Î ÀâÀ¸¸é,
$\,\,\phi([\alp])=\widetilde{\alp}(1)=n$ ÀÌ µÈ´Ù. µû¶ó¼­ $\phi$´Â
ontoÀÌ´Ù.

\end{proof}\\

\textbf{2.} \textcolor{blue}{Let $\widetilde{X}$ be a path-
connected space and $p: (\widetilde{X},\widetilde{x_0})
\rightarrow (X,x_0 )$ be a covering. Then $p_{\sharp} : \pi_1
(\widetilde{X},\widetilde{x_0})\rightarrow\pi_1(X,x_0)$ is
injective.}\\

\begin{proof} Clear from Covering homotopy property(µû¸§Á¤¸® 5)
\end{proof}\\
\newpage
\textbf{3.}\textcolor{blue}{Let $\widetilde{X}$ be a path-
connected space and $p: (\widetilde{X},\widetilde{x_0})
\rightarrow (X,x_0 )$ be a covering. Then $p_{\sharp}\pi_1
(\widetilde{X},\widetilde{x_0})$ and $p_{\sharp}\pi_1
(\widetilde{X},\widetilde{x_0}')$ are conjugate in $\pi_1 (X,x_0 )
$.}\\

\begin{proof}$\hspace*{1em} \widetilde{x_0}$¿Í $\widetilde{x_0}'$À» ¿¬°áÇÏ´Â path $\tau$¸¦ ÀâÀ¸¸é ´ÙÀ½ diagram
ÀÌ commuteÇÑ´Ù.\\

 $\hspace*{5em}\pi_1
(\widetilde{X},\widetilde{x_o})\hspace{3em}\overset{\phi
_\tau}{\longrightarrow}\hspace{3em} \pi_1
(\widetilde{X},\widetilde{x_o}')$\\
$\hspace*{7em}\downarrow p_{\sharp} \hspace{9em}\downarrow
p_{\sharp}$\\
$\hspace*{5em}\pi_1
(X,x_o)\hspace{2em}\overset{conjugate\hspace{1mm} by [p\cdot
\tau]^{-1}}{\longrightarrow}\hspace{1em} \pi_1 (X,x_o)$\end{proof}\\

\begin{thm}
\textit{\textbf{(General lifting theorem)}} \\
\textcolor{blue}{Let $p:(\widetilde{X},\widetilde{x_0})\rightarrow
(X,x_0)$ be a covering and  $Y$ be a path-connected and locally
path-connected space.
Let $f :(Y,y_0)\rightarrow (X,x_0)$. Then\\
$\exists \,\, \widetilde{f}:(Y,y_0)\rightarrow
(\widetilde{X},\widetilde{x_0})\,\,\,$ , a lifting of f
$\,\,\,\Leftrightarrow\,\,\,$ $f_{\sharp}\pi_1(Y,y_0)\subset
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$.\\In this case
 $\widetilde{f}$is unique.}
\end{thm}
\framebox{\hspace*{1em}\parbox[b]{14cm}{
$\hspace{6em}(\widetilde{X},\widetilde{x_0})$

$\hspace{2.3em}\exists \widetilde{f}\nearrow\hspace{3em}\downarrow
p\hspace{4em}\Leftrightarrow\hspace{4em}$
$f_{\sharp}\pi_1(Y,y_0)\subseteq
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$

$(Y,y_0)\hspace{1em}\rightarrow\hspace{1em}(X,x_0)$

$\hspace{4em}f$}}\\

\begin{proof}\\
($\Rightarrow$)

$f=p\circ \widetilde{f}$ À̹ǷΠ$f_{\sharp}=p_{\sharp}\circ
\widetilde{f}_{\sharp}$ ÀÌ°í
$f_{\sharp}\pi_1(Y,y_0)=p_{\sharp}(\widetilde{f_{\sharp}}\pi_1(Y,y_0))\subseteq
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0}) $.\\

($\Leftarrow$)

ÀÓÀÇÀÇ $y\in Y$¿¡ ´ëÇØ $\widetilde{f}(y)$¸¦ ´ÙÀ½°ú °°ÀÌ Á¤ÀÇÇÏÀÚ.
$y_0$¿Í $y$ »çÀÌÀÇ path $\rho$ ¸¦ ÀâÀº ÈÄ  $f\circ \rho$ÀÇ
$\widetilde{x_0}$¿¡¼­ ½ÃÀÛÇÏ´Â lifting $\widetilde{f\circ \rho}$
¿¡ ´ëÇØ $\widetilde{f}(y)=\widetilde{f\circ \rho}(1)$ ·Î Á¤ÀÇÇϸé
ÀÌ°ÍÀÌ ¹Ù·Î ¿øÇÏ´Â $\widetilde{f}$ °¡ µÈ´Ù.

¸ÕÀú $\widetilde{f}$ °¡ Àß Á¤ÀǵǾú´ÂÁö¸¦ º¸ÀÌÀÚ. Áï, $y_0$¿Í
$y$»çÀÌÀÇ pathÀÇ ¼±Åÿ¡  ¹«°üÇÔÀ» º¸À̱â À§ÇØ, $\sigma$¸¦ ¶Ç´Ù¸¥
path·Î µÎÀÚ. ±×·¯¸é, $f\circ \rho * \overline{f\circ \sigma}$´Â
loop°¡ µÇ°í °¡Á¤¿¡ ÀÇÇØ  $[f\circ \rho * \overline{f\circ
\sigma}]=f_{\sharp}[\rho * \overline{\sigma}]\in
p_{\sharp}\pi_1(\widetilde{X},\widetilde{x_0})$ ÀÌ´Ù.

µû¶ó¼­ $f\circ \rho * \overline{f\circ \sigma}$ÀÇ liftingÀº
loopÀÌ°í, $\widetilde{f\circ \sigma}(1)=\widetilde{f\circ
\rho}(1)$
ÀÌ µÇ¹Ç·Î $\widetilde{f}$´Â Àß Á¤ÀǵǾúÀ½À» ¾Ë ¼ö ÀÖ´Ù.\\

´ÙÀ½À¸·Î $\widetilde{f}$°¡ ¿¬¼ÓÀÓÀ» º¸ÀÌÀÚ. $\widetilde{f}$°¡
¿¬¼ÓÀÓÀ» º¸À̱â À§ÇØ $\forall y\in Y$ ¿¡ ´ëÇØ ¾î¶² ±Ù¹æ $W_y$°¡
À־ $\widetilde{f}|_{W_y}$°¡ ¿¬¼ÓÀ̶ó´Â °ÍÀ» º¸À̸é ÃæºÐÇÏ´Ù.
$x=f(y)$¿¡ ´ëÇØ evenly coverµÇ´Â ±Ù¹æÀ» $U_x$¶ó µÎ¸é $f$°¡
¿¬¼ÓÀÌ°í $Y$°¡ locally path connectedÀ̹ǷΠ$f(W_y)\subset U_x$¸¦
¸¸Á·ÇÏ´Â path connected $W_y$°¡ Á¸ÀçÇÑ´Ù. ±×·¯¸é $\forall z \in
W_y$ ´Â  $y$·ÎºÎÅÍ $W_y$¾È¿¡¼­ path $\tau_z$·Î  ¿¬°áÇÒ ¼ö ÀÖ°í,
$\rho*\tau_z$´Â $y_0$·ÎºÎÅÍ $z$±îÁöÀÇ path¸¦ ÁØ´Ù. µû¶ó¼­
$\widetilde{f}(z)=\widetilde{f\circ (\rho
*\tau_z)}(1)=\widetilde{f\circ \rho}*\widetilde{f\circ
\tau_z}(1)=\widetilde{f\circ \tau_z}(1)$ÀÌ µÇ°í ÀÌ ¶§
$\widetilde{f\circ \tau_z}$´Â $\widetilde{f}(y)$ ¿¡¼­ ½ÃÀÛÇÏ´Â
liftingÀÌ´Ù.  Áï $W_y$¾ÈÀÇ ÀÓÀÇÀÇ Á¡ $z$¸¦ $y_0$¿ÍÀÇ pathÀÇ
³¡Á¡À¸·Î º¸°í ÀÌ path¿¡¼­ $y$¿Í $z$¸¦ ÀÕ´Â ºÎºÐÀ» $f$·Î º¸³½ °ÍÀº
$U_x$¿¡ µé¾î°¨À» ÀÌ¿ëÇؼ­ $U_x$¿¡¼­ÀÇ $V_a$¿ÍÀÇ homeomorphsim
$p^{-1}$¸¦ ¾²ÀÚ.

 $p^{-1}(U_x)=\displaystyle{\coprod_{a\in A}}V_a$ and
$\widetilde{f}(y)\in V_a$ À̶ó°í µÎ¸é   $V_{a}$¿¡¼­ $p^{-1}$´Â
homeomorphismÀ̹ǷΠ$\tau_z$¸¦ ÀÌ¿ëÇؼ­
$\,\,\widetilde{f}|_{W_y}=p|_{V_a}^{-1}\circ f|_{W_y}$ ÀÓÀ» ¾Ë ¼ö
ÀÖ°í, ¶Ç $\,\,\,\widetilde{f}$´Â pathÀÇ ¼±Åÿ¡ ¹«°üÇÏ´Ù´Â
»ç½Ç·ÎºÎÅÍ ±×·¸°Ô µÉ ¼ö ¹Û¿¡ ¾ø´Ù. µû¶ó¼­ $\widetilde{f}|_{W_y}$
´Â ¿¬¼ÓÀÌ´Ù.

¸¶Áö¸·À¸·Î uniqueness ´Â Y°¡ connectedÀ̹ǷΠÁ¤¸® 1¿¡ ÀÇÇØ
Áõ¸íµÈ´Ù.
\end{proof}\\

{\bf Remark.}  This is a generalization of \textit{Unique path
lifting property} and \textit{Covering homotopy property}.\\

Âü°í·Î \textit{Unique path lifting property}´Â $Y$°¡ $I$ÀÎ
°æ¿ìÀÌ°í, $Y\times I$ÀÎ °æ¿ì°¡  \textit{Covering homotopy property}ÀÌ´Ù.\\
\newpage
\textbf{ Review of locally (path-) connected space.}\\

1. $\forall x \in X$ has a (path-) connected neighborhood.
$\Rightarrow$(path-) component is open ( and hence closed).

\begin{proof}$\hspace{1em}$clear\end{proof}\\

2. X : path-connected $\Leftrightarrow$ X : connected and $\forall
x \in X$ has a path-connected neighborhood.

\begin{proof}$\hspace{1em}(\Rightarrow$)clear\\
$\hspace*{3em}(\Leftarrow)$1.¿¡¼­ path component´Â open and closedÀ̹ǷΠÀü°ø°£ÀÌ µÈ´Ù.\end{proof}\\

3. X : locally path connected $\Rightarrow$ path-component =
component

\begin{proof}$\hspace{1em}$ÀϹÝÀûÀ¸·Î´Â $\subset$ÀÌ ¼º¸³Çϴµ¥ component´Â connectedÀ̹ǷΠ2¿¡ ÀÇÇØ =ÀÌ ¼º¸³ÇÑ´Ù.\end{proof}\\

\begin{thm}\textcolor{blue}{X is locally (path-) connected and p:$\widetilde{X} \rightarrow X$
is a covering.\\If $\widetilde{C}$ is a (path-) component of
$\widetilde{X}$, then\\ $(i)$ p($\widetilde{C}$) is a (path-)
component of X.\\ $(ii)$ p$|_{\widetilde{C}} : \widetilde{C}
\rightarrow p(\widetilde{C})$ is a covering.}
\end{thm}

\begin{proof}$\hspace{1em}$ HW $\sharp$ 5
\end{proof}
\\

µû¶ó¼­ covering space theory¿¡¼­ path-connectedÀÎ °æ¿ì·Î
Á¦ÇÑÇÏ¿©µµ »ó°ü¾ø´Ù. ¶ÇÇÑ fundamental groupÀ» µ¿½Ã¿¡ °í·ÁÇϱâ
À§Çؼ­´Â path-connectedÀÏ ÇÊ¿ä°¡ ÀÖ´Ù.\\
´õ¿íÀÌ general lifting theroremÀº covering space theory¿¡¼­ °¡Àå
Áß¿äÇÑ Á¤¸®ÀÌ°í À̸¦ ÀÚÀ¯·Ó°Ô »ç¿ëÇϱâ À§Çؼ­ covering space
theory¿¡¼­ ³ª¿À´Â °ø°£µé¿¡ ´ëÇØ º¸Åë path-connectedness, locally
path-connectedness¸¦ °¡Á¤ÇÑ´Ù.
\end{document}