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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 5} \vs \\
{\bf due Friday 10/11/2013} \vs \vs
\end{center}  

\noindent
{\bf Problem 1.} Show that every subset of a separable metric space is separable. \vs

\noindent
{\bf Problem 2.} Give an example of a metric space which has a countable perfect subset. \vs

\noindent 
{\bf Problem 3.} Let $X$ be a complete separable metric space. For a set $E \subset X$, we use the notation $E^*$ for the set of all condensation points of $E$, i.e., all points $p \in X$ such that $B(p,r) \cap E$ is uncountable for every $r>0$. \vs 

\begin{enumerate}
\item[(i)]
Show that if $P \subset X$ is perfect, then $P=P^*$. \vs

\item[(ii)]
Show that for any closed set $E \subset X$, the Cantor-Bendixson decomposition $E=P \cup C$ into a perfect set $P$ and an at most countable set $C$ is unique. \vs
\end{enumerate}
(Hint: For (i), use the fact that every perfect set in $X$ is uncountable by Baire's theorem. If $p \in P \sm P^*$, show that $B(p,r) \cap P$ would have an isolated point for some $r>0$, which would be an isolated point of $P$. For (ii), take any decomposition $E=P \cup C$ and use (i) to show that necessarily $P=E^*$ and $C=E \sm E^*$.)  \vs 
  
\noindent
{\bf Problem 4.} 
\begin{enumerate} 
\item[(i)]
Give an explicit example of a differentiable function $f: \RR \to \RR$ with the property $|f'(x)|<1$ for all $x \in \RR$ which nevertheless has no fixed point. \vs
\item[(ii)]
Starting with the number {\tt 0} on my calculator screen, I keep pressing the {\tt COS} key over and over again (my calculator is always set in radians!). What happens and why? \vs
\end{enumerate}

\noindent
{\bf Problem 5.} Let $f$ be a self-map of a complete metric space $X$. Suppose for some $k \geq 2$ the $k$-fold iterate 
$$
f^{\circ k} = \underbrace{f \circ \cdots \circ f}_{k \ \text{times}}:X \to X
$$ 
is a contraction. Show that $f$ has a unique fixed point. (Note that we do not even assume $f$ to be continuous. For example, $f:[0,2] \to [0,2]$ which takes the value $2$ on $[0,1)$ and $1$ on $[1,2]$ is discontinuous, but the iterate $f^{\circ 2}$ is the constant function $1$, clearly a contraction.)    \vs

\noindent
{\bf Problem 6.} Suppose $X$ is a complete metric space. Denote the unique fixed point of a contraction $f:X \to X$ by $p_f$. Show that the assignment $f \mapsto p_f$ is continuous in the following sense: Given $f$ and $\ve>0$, there is a $\delta>0$ such that if $g:X \to X$ is any contraction with $\sup_{x \in X} d(f(x),g(x)) < \delta$, then $d(p_f,p_g)<\ve$. (Hint: Estimate $d(f^{\circ n}(x),g^{\circ n}(x))$ inductively on $n$; the choice $\delta=\ve(1-\lambda)$ will do, where $\lambda \in [0,1)$ is the contraction factor of $f$.) \vs
 





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