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\author{Math 25}

\title{Homework 2 --- due Oct 3rd}

\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{} Give an example of a metric space $(E,d)$ and infinitely
many open sets $\{U_i \subset E\}_{i \in I}$ such that $\cap_{i \in I}
U_i$ is not open. You could consider suitable intervals in $\RR$ for example.

\subsection{}
Let $X$ be a subset of a metric space $(E,d)$. Show that:
\[ \overset{\circ}{\overline{X}} =
\overset{\circ}{\overline{\overset{\circ}{\overline{X}}}} 
\quad\text{and that}\quad \overline{\overset{\circ}{X}} =
\overline{\overset{\circ}{\overline{\overset{\circ}{X}}}}. \]
Give an example of a subset $X$ of $\RR$ such that the following sets:
\[ X,\ \overline{X},\ \overset{\circ}{X},\ 
\overset{\circ}{\overline{X}},\
\overline{\overset{\circ}{X}},\
\overline{\overset{\circ}{\overline{X}}},\
\overset{\circ}{\overline{\overset{\circ}{X}}},
\]
are pairwise distinct.

\subsection{Induced topology} Let $(E,d)$ be a metric space, and let
$X$ be a subset of $E$ so that $(X,d)$ is itself a topological space
with the induced distance. Show that $U \subset X$ is open in $X$ if
and only if there exists an open set $V$ of $E$ such that $U= X \cap
V$. 

Be careful: if $U \subset X$ is open in $X$, it is not necessarily
open in $E$.

\subsection{Product topology} Let $(E_1,d_1)$ and $(E_2,d_2)$ be two
metric spaces. Define a distance $d$ on $E_1 \times
E_2$ by \[ d((x_1, x_2),(y_1,y_2)) = d_1(x_1,y_1) + d_2(x_2,y_2). \] 
\begin{enumerate}
\item
Show that $U \subset E_1 \times E_2$ is open if and only if for 
each point $(u_1,u_2) \in U$, there exist $U_1$ and $U_2$ open in
$E_1$ and $E_2$ with $(u_1,u_2) \in U_1 \times U_2$ and $U_1 \times
U_2 \subset U$.

\item
If $X$ is another metric space, show that $(f_1,f_2) : X \ra E_1
\times E_2$ is continuous if and only if $f_1$ and $f_2$ both are.
\end{enumerate}

\section{Problems graded by Benjamin}

\subsection{The $p$-adic topology}\label{padic}
Given $a \in \QQ^*$ there is a unique $v=v(a) \in \ZZ$ such that 
one can write $a = p^v n/d$ with $n,d \in \ZZ$ relatively prime to $p$. 
For $x,y \in \QQ$ define $d_p(x,y)=p^{-v(x-y)}$ if $x \neq y$ and
$d(x,x)=0$.  

\begin{enumerate}
\item
Show that $(\QQ,d_p)$
is a metric space and that actually $d(x,z) \leq
\max(d(x,y),d(y,z))$;
\item show that if $x \in B(a,r)$, then $B(x,r)=B(a,r)$ so that 
any point of the ball $B(a,r)$ is a ``center'' of that ball;
\item show that given two balls $B(a_1,r_1)$ and $B(a_2,r_2)$, then
either they're disjoint or one is contained in the other; 
\end{enumerate}
This goes to show that  metric spaces can be quite strange.

\subsection{Accumulation points}\label{accpts}
If $u=\{u_n\}_{n \geq 1}$ is a
sequence of elements of a metric space $(E,d)$, let $A(u)$ be
the set of its accumulation points. Show that $A(u)$ is closed.

%Hint: it will be helpful to consider $F_n = \overline{\{u_k\}_{k \geq n}}$.

\subsection{} Let $f: [0,1] \ra \RR$ be defined by
$f(x)=\sqrt{x}$. Show that $f$ is uniformly continuous.

\subsection{}
We say that a function $f:X \ra Y$ is $K$-Lipshitz if $d_Y(f(x),f(x'))
\leq K \cdot d_X(x,x')$ for all $x,x' \in X$. 
Check that such a function is uniformly continuous.

Show that for every $0 < a <1$, 
there exists $K_a$ such that 
$f(x)=\sqrt{x}$ is $K_a$-Lipshitz  
on $[a,1]$. What happens if $a=0$?

\section{Problems graded by Inna}

\subsection{} We say that a function $f: X \ra Y$ is continuous at a point
$x$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that
if $d(x,x') < \delta$, then $d(f(x),f(x')) < \epsilon$. This way, 
$f: X \ra Y$ is continuous if and only if it is continuous at every $x \in X$.

\begin{enumerate}
\item Show that $f$ is continuous at $x$ if and only if for every sequence
$\{x_n\}_n$ which converges to $x$, we have $f(x_n) \ra f(x)$;

\item
If $x \in \QQ$, one can write $x=p/q$ in a unique way with
$\on{gcd}(p,q)=1$ and $q \geq 1$ (if $x=0$, set $p=0$ and $q=1$). 
Let $f:[0,1] \ra \RR$ be defined by \[ f(x)= \begin{cases}
0 & \text{if $x \in \RR \setminus \QQ$,} \\
1/q & \text{if $x =p/q \in \QQ$.} \end{cases}
\] At what points $x$ of $[0,1]$ is $f$ continuous?
\end{enumerate}

\subsection{} Let $\RR^2$ be the plane with the usual distance and let
$x_1,\cdots,x_n$ be finitely many points. Show that $\RR^2 \setminus
\{ x_1,\cdots,x_n \}$ is connected.

\subsection{} This problem has been removed and will be in the third set.

\subsection{} Show that a circle is not homeomorphic to an interval of
$\RR$. Show also that a ``figure eight'' is not homeomorphic to a circle.

Hint: think about connectedness.

\section{Supplementary problems}

If you get bored, do the following.

\subsection{} In exercise \ref{padic}, show that $(\QQ,d_p)$ is not complete.

\subsection{} Show that if $F$ is a closed subset of 
a metric space 
$E$ and if $f:F
\ra [0,1]$ is a continuous function, then there exists a continuous
function $\tilde{f} : E \ra [0,1]$ which coincides with $f$ on $F$.
\end{document}