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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due September 5 (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}

  % Mendelson Exercise 1.4.6
\item Let $a$ and $b$ be distinct points in a metric space $X$. Prove
  that there are neighborhoods $N_a$ and $N_b$ of $a$ and $b$
  respectively such that $N_a\cap N_b=\emptyset$. A topological space
  with this property is said to be \emph{Hausdorff}; thus this
  exercise shows that metric spaces are Hausdorff. \\



  % Mendelson Exercise 1.5.6-8
\item Let $(X,d)$ be a metric space. The distance between a point
  $x\in X$ and a nonempty set $A\subseteq X$ is defined by 
  \[
  d(x,A)=\inf\{d(x,a)\mid a\in A\}\,.
  \]
  \begin{enumerate}
    \item Prove that $d(x,A)\leq d(x,y)+d(y,A)$.
    \item Prove that the function $f\colon X\to\R$ defined by
      $f(x)=d(x,A)$ is continuous.
    \item Prove that $d(x,A)=0$ if and only if every neighborhood of
      $x$ contains some point in $A$. \\
  \end{enumerate}



% Rudin Exercise 2.8
\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 $\R^2$. What are
  the answers to these questions in $\R^2$ under the discrete metric? \\



% Mendelson Exercise 1.6.3-4
\item Let $(X,d_1)$ and $(Y,d_2)$ be metric spaces and $f\colon X\to
  Y$ be continuous. Define a distance function $d$ on $X\times Y$ in
  the standard manner:
  \[
  d((x,y),(x',y'))=\max\{d_1(x,x'),d_2(y,y')\}\,.
  \]
  \begin{enumerate}
    \item Prove that the graph $\Gamma_f:=\{(x,f(x))\mid x\in X\}$ is
      a closed subset of $(X\times Y,d)$.
    \item Show by example that a function whose graph is closed need
      not be continuous. \\
  \end{enumerate}



\end{enumerate}

\end{document}
