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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due August 29 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 4560/6560 $|$ Topology $|$ Fall
    2013 $|$ M.~Macauley}}



\begin{document}

$\;$

\begin{enumerate}


\item Define a function on $\N$ by setting $d(x,y)=0$ if $x=y$, and
  $d(x,y)=3^{-k}$ otherwise, where $3^k$ is the highest power of $3$
  dividing $|x-y|$. Prove that $(\N,d)$ is a metric space. \\


 
\item Fix $x_0$ in a metric space $(X,d)$ and define $\rho\colon
  X\to\R$ by $\rho(x)=d(x,x_0)$. Show that $\rho$ is continuous. \\



\item Let $X=\R$, and let $d(x,y)=|x-y|$ be the usual Euclidean
  metric, and let $d'\colon X\times X\to\R$ be the \emph{discrete
    metric}, defined by
  \[
  d'(x,y)=\begin{cases}0 & x=y \\ 1 & x\neq y.\end{cases}
  \]
  \begin{enumerate}
    \item Prove or disprove: the identity map
      $i\colon(\R,d)\to(\R,d')$ is continuous.
    \item Prove or disprove: the identity map
      $i'\colon(\R,d')\to(\R,d)$ is continuous.
  \end{enumerate}
  Use the $(\epsilon,\delta)$-definition of continuity. The ``inverse
  image of open sets'' characterization, which we have not seen yet,
  will almost trivialize this question. \\



\item Let $(X,d)$, $(Y,d')$, and $(Z,d'')$ be metric spaces. A
  function $f\colon X\to Y$ is \emph{uniformly continuous} if for all
  $\epsilon>0$, there is some $\delta>0$ such that $d(x,y)<\delta$
  implies $d'(f(x),f(y))<\epsilon$. This is a stronger condition than
  ordinary continuity because $\delta$ does not depend on $x\in X$.
  \begin{enumerate}
    \item Give an example of a function that is continuous but not
      uniformly continuous.
    \item Prove that if $f\colon X\to Y$ and $g\colon Y\to Z$ are
      uniformly continous, then $g\circ f\colon X\to Z$ is uniformly
      continuous.
  \end{enumerate}

\end{enumerate}

\end{document}
