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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due June 5 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 453 $|$ Real Analysis I $|$ Summer
    I 2012 $|$ M.~Macauley}}


\begin{document}

$\;$ 

\noindent\emph{Read}: Rudin, Chapter 2, pages 36--40. \\

\noindent Throughout, let $X$ be a metric space. Recall that a set
$E\subset X$ is \emph{dense} if every point of $X$ is either in $E$ or
is a limit point of $E$. Equivalently, $E$ is dense in $X$ if
$\bar{E}=X$. A metric space is called \emph{separable} if it contains
a countable dense subset. \\

\noindent A collection $\{V_\alpha\}$ of open subsets of $X$ is said to be
a \emph{basis} for $X$ if the following is true: \\

\hspace{0.33in}
\begin{minipage}{0.75\textwidth}
  \vspace{0.05in} For every $x\in X$ and every open set $G\subset X$
  such that $x\in G$, we have $x\in V_\alpha\subset G$ for some
  $\alpha$. (In other words, every open set in $X$ is the union of a
  subcolection of $\{V_\alpha\}$.)
  \vspace{0.05in}
\end{minipage}


\begin{enumerate}



\item Show that $\R^k$ is separable. [\emph{Hint}: Consider points
  which only have rational coordinates -- prove this set is dense.] \\



\item Let $X$ be a separable metric space.
  \begin{enumerate}
  \item Prove that $X$ has a \emph{countable} basis. [\emph{Hint}: Take all
    neighborhoods with rational radius and center in some countable
    dense subset of $X$.]
  \item Prove that every open cover of $X$ has a \emph{countable} subcover.
  \item Deduce that every open set in $\R$ is the union of an at most
    countable collection of disjoint open segments. \\
  \end{enumerate}
  

  
\item Let $X$ be a metric space in which every infinite subset has a
  limit point. 
  \begin{enumerate}
  \item Show that for any $\delta>0$, $X$ can be covered by finitely
    many neighborhoods of radius $\delta$. [\emph{Hint}: Pick the
      centers inductively: Start with any $x_1\in X$. Having chosen
      $x_1,\dots,x_j\in X$, choose $x_{j+1}\in X$, if possible, so
      that $d(x_i,x_{j+1})\geq\delta$ for $i=1,\dots,j$. Show that
      this process must stop after a finite number of steps.]
  \item Prove that $X$ is separable. [\emph{Hint}: Take $\delta_n=1/n$
    for $n=1,2,3\dots$ and consider the centers of the corresponding
    neighborhoods. Show that this set is dense in $X$.] \\
  \end{enumerate}
   


\item Theorem 2.37 in Rudin says that every infinite subset of a
  compact set has a limit point. Prove the converse: If $X$ is a
  metric space in which every infinite subset has a limit point, then
  $X$ is compact. [\emph{Hint}: By Problems 2 and 3, $X$ is separable,
    and hence every open cover of $X$ has a \emph{countable} subcover
    $\{G_n\}$, $n=1,2,3,\dots$. If no finite subcollection of
    $\{G_n\}$ covers $X$, then the complement $F_n$ of
    $G_1\cup\cdots\cup G_n$ is nonempty for each $n$, but $\cap F_n$
    is empty. If $E$ is a set which contains a point from each $F_n$,
    consider a limit point of $E$, and obtain a contradiction.] \\



\item Let $X$ be a compact metric space. Prove that $X$ is
  separable, and deduce that $X$ has a countable basis.


\end{enumerate}

\end{document}
