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\author{Math 25}

\title{Homework 3 --- due Oct 10th}

\begin{document}

\maketitle
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There are three sections: A,B and C 
(plus a fourth ``optional''
one which won't be graded). 
Please return the three parts
\emph{separately} to the proper CA.

\renewcommand{\thesection}{\Alph{section}}

\section{Problems graded by Jennifer}

\subsection{Distance to a subset} \label{dtas}
Let $(E,d)$ be a metric space, and let
$A$ be a subset of $E$. For $x \in E$, let $d(x,A)=\inf_{a \in A}
d(x,a)$.
\begin{enumerate}
\item Show that $x \mapsto d(x,A)$ is continuous; 
\item Show that $d(x,A)=0$ if and only if $x \in \overline{A}$.
\end{enumerate}

\subsection{Extrema on compact spaces} 
Show that if $E$ is a compact metric space and $f:E \ra
\RR$ is a continuous map, then $f$ admits a maximum on $E$. That is,
there exists $x \in E$ such that $f(x) \geq f(y)$ for any $y \in E$.

Use this to show that if $F_1$, $F_2$ are two disjoint closed subsets
of $E$ then there exists $\mu >0$ such that $d(a_1,a_2) \geq \mu$
whenever $a_1 \in F_1$ and $a_2 \in F_2$.

\subsection{Contracting maps} 
Let $(E,d)$ be a compact metric space and let $f:E \ra
E$ be a continuous map such
that $d(f(x),f(y)) < d(x,y)$ for all $x,y \in E$ with $x \neq y$.

Define the diameter of a closed subset $F$ of $E$ to be $\on{diam}(F)
= \sup_{x,y \in F} d(x,y)$. What are the closed subsets of $E$ whose
diameter is $0$?

\begin{enumerate}
\item Show that there exist $x,y \in F$ such that $\on{diam}(F)=d(x,y)$;
\item Show that $\on{diam}(f(F)) < \on{diam}(F)$ if $\on{diam}(F) \neq 0$; 
\item Use this to show that $f$ has a unique fixed point.
\end{enumerate}

\subsection{Accumulation points II} 
Let $E$ be a compact metric space, let $u=\{u_n\}_n$ be a sequence 
of elements of $E$, and
let $A(u)$ be the set of its accumulation points. Show
that if $d(u_n,u_{n+1}) \ra 0$ as $n \ra +\infty$ then $A(u)$ is
connected.

Hint: if you can write a space $X$ as the disjoint union of
two open sets, then those open sets are also closed.

\section{Problems graded by Benjamin}

\subsection{Uniform continuity}
\begin{enumerate}
\item show that if $f: \RR_{\geq 0} \ra \RR$ is continuous and $f(x)
  \ra \ell$ when $x \ra +\infty$, then $f$ is uniformly continuous;
\item show that if $f: \RR_{\geq 0} \ra \RR$ is uniformly continuous
  then there exist $A,B \in \RR$ such that $|f(x)| \leq Ax+B$;
\item
Which of the following functions from $\RR_{\geq 0}$ to $\RR$ are uniformly
continuous? 
\[ x \mapsto e^x \qquad x \mapsto \begin{cases} x \sin(1/x) & \text{if $x
    \neq 0$} \\ 0 & \text{if $x=0$} \end{cases} \qquad x \mapsto \sqrt{x}. \]
\end{enumerate}

\subsection{} Show that if $F$ is a closed subset of $[0,1]$ then
there exists a continuous function $f:[0,1] \ra \RR$ such that
$f^{-1}(0)=F$.

\subsection{} Show that if $f: \RR \ra \RR$ is bijective and
continuous, then $f$ is a homeomorphism (this is obvious if you draw
a picture, but you need to give a rigorous proof).

\section{Problems graded by Inna}

\subsection{} Let $E$ be a compact metric space, and for $n \geq 1$
let $F_n$ be a closed subset of $E$, such that 
$F_n$ is non-empty and $F_{n+1} \subset
F_n$. Show that $\cap_{n=1}^{+\infty} F_n$ is non-empty.

Give an example to show that this can fail if $E$ is not compact.

\subsection{} Show that if $E$ is compact, then it contains a dense
denumerable subset (for example, $[0,1]$ contains $[0,1] \cap \QQ$).

\subsection{The Cantor Set} \label{cantor}
If $I \subset [0,1]$ is a finite union of disjoint
  closed intervals $[a_i,b_i]$, let $T(I)$ be the union of the
  intervals $[a_i,a_i+(b_i-a_i)/3]$ and $[a_i+2(b_i-a_i)/3,b_i]$ (one
  ``removes the middle third'' from each interval in $I$ to get $T(I)$).

Let $K_0=[0,1]$ and $K_n=T(K_{n-1})$ for $n \geq 1$. Define $K =
  \cap_{n=0}^{+\infty} K_n$.

\begin{enumerate}
\item Show that $K$ is compact;
\item Show that $[0,1] \setminus K$ is a coutable union of intervals  
the sum of whose lengths is $1$;
\item Show that the connected component of each point $x$ of $K$ is the
  point $x$ itself;
\item Show that $K$ is also the set of $\alpha \in [0,1]$ such that one
  can write $\alpha=\sum_{i=0}^{+\infty} \alpha_i 3^{-i}$ with
  $\alpha_i \in \{0,2\}$ for every $i$;
\item Show that $K$ is not countable.
\end{enumerate}

One can also show that if $X$ is  \emph{any} compact metric space, then
there is a surjective map $f : K \ra X$ so that $K$ is some
``universal'' compact space.

\section{Optional problems}

We will most likely prove the following theorems in class 
later in
the year, but you can try your hand at them now. You can also try
proving the assertion above, that if $X$ is  any compact metric space, then
there is a surjective map $f : K \ra X$.

\subsection{Borel-Lebesgue's theorem} Let $E$ be a metric space. Show that the
following three properties are equivalent:
\begin{enumerate}
\item $E$ is compact;
\item if $\{U_i\}_{i \in I}$ is any family of open subsets of $E$ such
  that $E=\cup_{i \in I} U_i$ then there exists a finite subset $J
  \subset I$ such that $E=\cup_{j \in J} U_j$;
\item if $\{F_i\}_{i \in I}$ is any family of closed subsets of $E$ such
  that $\cap_{i \in I} F_i = \varnothing$ then there exists a finite subset $J
  \subset I$ such that $\cap_{j \in J} F_j = \varnothing$.
\end{enumerate}

\subsection{Baire's theorem} Let $E$ be a complete metric space and let
$\{U_i\}_{i \in I}$ be a countable family of dense open
subsets of $E$. Show that $\cap_{i \in I} U_i$ is still dense in $E$
(in general it won't be open!).

Show that this can fail if $I$ is not countable.
\end{document}