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\def\ctln{\centerline}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\ssk{\smallskip}
\def\u{\underbar}
\def\ni{\noindent}
\def\hsk{\hskip.01in}

\def\sset{\subseteq}

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\ctln{\bf Math 970 Homework 11}

\msk

\ctln{Due Tuesday, Dec. 9}

\bsk

\item{48.} Show that if $X$ is Hausdorff and 
$f$:$X\rightarrow X$ is continuous, then the fixed 
point set 

\ssk

\ctln{Fix$(f)$ = $\{ x\in X : f(x) = x\}$}

\ssk

\item{} of $f$ is a closed subset of $X$.

\bsk

\item{} A subset $A\subseteq X$ is a \underbar{retract} of $X$ if there is a continuous
map $r$:$X\rightarrow A$ with $r\circ i = Id$, i.e., $r(a)=a$ for all $a\in A$. The map
$r$ is called a \underbar{retraction}.

\bsk

\item{49.} Show that if $X$ is Hausdorff and $A\sset X$ is a retract of $X$, then $A$ is
closed. 

\ssk

\item{} (Hint: show that $A$ is the fixed point set of some map!)

\bsk

\item{50.} Show that if $r: X\rightarrow A$ is a retraction and $a\in A$, then

\ssk

\ctln{$r_*$:$\pi_1(X,a)\rightarrow \pi_1(A,a)$}

\ssk

\item{} is a surjective homomorphism.

\bsk

\item{51.} Show that if $a\in A\subseteq X$, $\pi_1(X,a) = \{1\}$, and 
$f$:\hsk $(A,a)\rightarrow (Y,b)$ is continuous, then if $f$ extends to a continuous 
map $g$:$X\rightarrow Y$ (i.e., $g|_A = f$), then 
$f_*$:$\pi_1(A,a)\rightarrow \pi_1(Y,b)$ is the trivial homomorphism.

\ssk

\item{} (The contrapositive of the last part of this statement sounds stronger....)

\bsk

\item{52.} A space $X$ is \underbar{contractible} if the identity map $I$:$X\rightarrow X$
is homotopic to the constant map $c(x)=x_0$. Show that if $X$ is contractible then
any two maps $f,g$\hsk :\hsk $Y\rightarrow X$ are homotopic. Show that this implies that 
$\pi_1(X,x_0) = \{1\}$.



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\end