\documentclass[minion]{homework651}
\input{hwextras}

% A macro for a decorated right arrow (indicating the kind of convergence)
\newcommand\converges[1]{\mathrel{\mathop{\longrightarrow}\limits_{#1}}}

\doclabel{Math F651: Homework 1}
\docdate{Due: January 25, 2023}

\begin{document}
\begin{problems}
\problem Prove that every ball $B_r(x)$ in a metric space $(X,d)$ is an open set.

\problem Let $V$ be a subset of a metric space $(X,d)$.  The set of limit
points of $V$ are those points $x$ that can be written as a the limit of a 
sequence of points in $V$.  Show that a set $V\subseteq X$ is closed if and only if
it contains its limit points.

\problem Let $d_1$ and $d_2$ be two metrics on a set $X$.  Show that the following
conditions are equivalent.
\begin{subproblems}
\item For every sequence $\{ p_i\}_{i=1}^\infty$, if $p_i\converges{d_2} p$ 
then $p_i\converges{d_1} p$.
\item For every function $f:X\rightarrow \Reals$, if $f$ is continuous with 
respect to $d_1$ then $f$ is continuous with respect to $d_2$.
\item For every set $V$, if $V$ is closed with respect to $d_1$ then
$V$ is closed with respect to $d_2$.
\item For every set $U$, if $U$ is open with respect to $d_1$ then
$U$ is open with respect to $d_2$.
\end{subproblems}

{\it Hint:} You might want to show $\;{\rm a)} \iff {\rm b)}\;$ and $\;{\rm a)} \Longrightarrow {\rm c)} \Longrightarrow {\rm d)} 
\Longrightarrow {\rm a)}$.

\problem Lee, Problem 2-1

\problem Lee, Exercise (Not Problem) 2.6


\end{problems}

\end{document}
