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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due June 8 (Friday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 453 $|$ Real Analysis I $|$ Summer
    I 2012 $|$ M.~Macauley}}


\begin{document}

$\;$ 

\noindent\emph{Read}: Rudin, Chapter 2, pages 41--43. \\

\begin{enumerate}


\item Regard $\Q$, the set of all rational numbers, as a metric space
  with $d(p,q)=|p-q|$. Let $E$ be the set of all $p\in\Q$ such that
  $2<p^2<3$. Show that $E$ is closed and bounded in $\Q$, but that $E$
  is not compact. Is $E$ open in $\Q$? [\emph{Hint}: Use Theorems 2.30
    and 2.33 in Rudin.] \\



\item Let $E$ be the set of all $x\in[0,1]$ whose decimal expansion
  contains only the digits $4$ and $7$. Is $E$ countable? Is $E$ dense
  in $[0,1]$? Is $E$ compact? Is $E$ perfect? Prove all of your
  claims. [A set $E$ is \emph{perfect} if it is closed, and every
    point is a limit point of $E$.] \\



\item Consider the set $S=\Q\cap[0,1]$, which we know is countable,
  Enumerate $S$ via a function $f\colon\N\to\Q$, so that
  $S=\{f(1),f(2),\ldots\}$. Define the set $P\subset S$ 
  \[
  P=\{0.d_1d_2d_3\ldots \in S: d_i\neq f(n)_n\},
  \]
  where $0.d_1d_2d_3\ldots$ is the base-10 decimal representation of a
  number (assuming it does not end in an infinite string of $9$s), and
  $f(n)_n$ is the $n^{\rm th}$ digit of the decimal representation of
  $f(n)$. By construction, $P\cap\Q=\emptyset$. Prove that $P$ is
  perfect, and deduce that there are nonempty perfect sets in $\R$
  that contain no rational number? \\



\item  Let $X$ be a metric space. 
  \begin{enumerate}
  \item If $A$ and $B$ are disjoint closed sets in $X$, prove that
    they are separated.
  \item Prove the same for disjoint open sets.
  \item Fix $p\in X$, $\delta>0$, and define $A$ to be the set of all
    $q\in X$ for which $d(p,q)<\delta$. Define $B$ similarly, but with
    $>$ in place of $<$. Prove that $A$ and $B$ are separated. 
  \item Prove that every connected metric space with at least two
    points is uncountable. [\emph{Hint}: Use (c).] \\
  \end{enumerate}



\item Are closures and interiors of connected sets always connected?
  Prove or disprove each assertation.

\end{enumerate}

\end{document}
