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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due September 12 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 4560/6560 $|$ Topology $|$ Fall
    2013 $|$ M.~Macauley}}



\begin{document}

$\;$

\noindent Throughout, a metric space $X$ is endowed with a distance
function $d\colon X\times X\to\R$, even if not explicitly
mentioned. Also, if the distance function for $\R^k$ is not specified,
it is assumed to be Euclidean. \\

\begin{enumerate}

% From Rudin
\item Let $A_1$, $A_2$, $A_3$,\dots be subsets of a metric space
  $(X,d)$. The \emph{closure} of $A_i$, denoted $\bar{A}_i$, is
  defined to be the union of $A_i$ with its limit points. It is
  elementary to show that the closure of a set is indeed closed.
  \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
    $\disp\bar{B}\supseteq\bigcup_{i=1}^\infty\bar{A}_i$. Show, by an
    example, that this inclusion can be proper. \\
  \end{enumerate}
  
  
  
  % From Rudin
\item Let $E^\circ$ denote the set of all interior points of a set
  $E\subseteq X$, 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 $U\subseteq E$ and $U$ is open, prove that $U\subseteq E^\circ$.
  \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}
  
  
  
  % Mendelson Exercise 1.7.6
\item Let $(Y,d_Y)$ be a subspace of the metric space $(X,d_X)$.
  \begin{enumerate}
  \item Prove that a subset $V\subseteq Y$ is an open set of $Y$
    if and only if there is an open subset $U$ of $X$ such that
    $V=Y\cap U$.
  \item Prove that a subset $G\subseteq Y$ is a closed set of $Y$ if
    and only if there is an closed subset $F$ of $X$ such that
    $G=Y\cap F$.
  \item For a point $y\in Y$, prove that a subset $N\subseteq Y$ is a
    neighborhood of $y$ if and only if there is a neighborhood $M$ of
    $y$ in $X$ such that $N=Y\cap M$. \\
  \end{enumerate}



    % Mendelson Exercise 1.7.2
  \item Two metric spaces $(X,d_X)$ and $(Y,d_Y)$ are
    \emph{topologically equivalent} if there are inverse functions
    $f\colon X\to Y$ and $g\colon Y\to X$ that are continuous.
    \begin{enumerate}
    \item Prove that the open interval $(-\pi/2,\pi/2)$ is
      topologically equivalent to $\R$.
    \item Prove that any two open intervals are topologically
      equivalent to each other, and hence to $\R$.
    \end{enumerate}


\end{enumerate}

\end{document}
