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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due June 1 (Friday) }}
\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. \\

\begin{enumerate}


\item Let $A_1$, $A_2$, $A_3$,\dots be subsets of a metric space.
  \begin{enumerate}
  \item If $\disp B_n=\bigcup_{i=1}^n A_i$, prove that
    $\disp\bar{B}_n=\bigcup_{i=1}^n\bar{A}_i$, for $n=1,2,3,\dots$.
  \item If $\disp B=\bigcup_{i=1}^\infty A_i$, prove that
    $\disp\bar{B}\supset\bigcup_{i=1}^\infty\bar{A}_i$. Show, by an
    example, that this inclusion can be proper. \\
  \end{enumerate}
  
 

\item Is every point of every open set $E\subset\R^2$ a limit point of
  $E$? Answer the same question for closed sets in $\R^2$. \\
  
  
  
\item Let $E^\circ$ denote the set of all interior points of a set
  $E$, which we call the \emph{interior} of $E$. 
  \begin{enumerate}
  \item Prove that $E^\circ$ is always open.
  \item Prove that $E$ is open if and only if $E^\circ=E$.
  \item If $G\subset E$ and $G$ is open, prove that $G\subset E^\circ$.
  \item Prove that the complement of $E^\circ$ is the closure of the
    complement of $E$.
  \item Do $E$ and $\bar{E}$ always have the same interiors? Prove or
    disprove.
  \item Do $E$ and $E^\circ$ always have the same closures? Prove or
    disprove. \\
  \end{enumerate}
  
  

\item Let $X$ be an infinite set. For $p,q\in X$, define 
  \[
  d(p,q)=\left\{\begin{array}{lll}1 && \mbox{if $p\neq q$} \\
  0 && \mbox{if $p=q$.}\end{array}\right.
  \]
  Prove that this is a metric. Which subsets of the resulting metric
  space are open? Which are closed? Which are compact? Prove all your
  claims. \\ 
  

  
\item Let $K=\{1/n:n\in\N\}\cup\{0\}$. Prove that $K$ is compact. Is
  the set $L=\{1/n:n\in\N\}$ compact? Prove or disprove.



\end{enumerate}

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