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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due September 19 (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 $d\colon X\times X\to\R$, and a topological space $X$ is
endowed with a topology $\tau$, even if not explicitly
mentioned. \\

\begin{enumerate}

% Mendelson, Exercise 3.4.8.
\item A subset $A$ of a topological space $(X,\tau)$ is said to be
  \emph{dense} in $X$ if $\overline{A}=X$. Prove that if for each open
  set $U$, we have $A\cap U\neq\emptyset$, then $A$ is dense in $X$. \\



% Mendelson, Exercise 3.4.7, 12, 13
\item Let $A$ be a subset of a topological space $(X,\tau)$.  The
  \emph{boundary} of $A$ is the set $\partial
  A:=\overline{A}\cap\overline{A^c}$.
  \begin{enumerate}
    \item Prove that $\partial A=\emptyset$ if and only if $A$ is open
      and closed.
    \item Prove that $\overline{A}=A\cup\partial A$.
    \item Prove that $A$ is closed if and only if $\partial A\subseteq A$.
    \item Prove that $A$ is open if and only if $\partial A\subseteq
      A^c$. \\
  \end{enumerate}



\item A subset $E$ of a topological space $X$ is called a $G_\delta$
  if there is a sequence $U_1,U_2,\dots$ of open sets such that
  $\displaystyle E=\bigcap_{i=1}^\infty U_i$.
  \begin{enumerate}
    \item Show that if $f$ is a continuous function from $X$ to the
      real line with the usual topology, then the set $Z=\{x\in X\colon
      f(x)=0\}$ is closed and is a $G_\delta$.
    \item Show that in a metric space, every closed set is a $G_\delta$. 
    \item Show that the conclusion of Part~(b) fails in general for
      topological spaces. \\
  \end{enumerate}



%\item Let $S$ be a nonempty set and $X=2^S$, the power set of $S$. Define 
%  \[
%  \tau=\{U\subseteq S:|S\setminus U|<\infty\}\cup\{\emptyset\}.
%  \]
%  \begin{enumerate}
%    \item Show that $\tau$ is a topology on $X$. 
%    \item Define $f\colon X\times X\to X$ by $f(A,B)=A\cap B$. Prove
%      that $f$ is continuous.
%    \item Define $g\colon X\times X\to X$ by $g(A,B)=A\cup
%      B$. Formulate (with proof) a necessary and sufficient condition
%      on $S$ that ensures that $g$ is continuous.
%  \end{enumerate}


\end{enumerate}

\end{document}
