% HMC Math dept HW template example
% v0.04 by Eric J. Malm, 10 Mar 2005
\documentclass[12pt,letterpaper,boxed]{hmcpset}

% set 1-inch margins in the document
\usepackage[margin=1in]{geometry}

% include this if you want to import graphics files with /includegraphics
\usepackage{graphicx}

% include this to have lists enumerated by letters
\renewcommand{\labelenumi}{(\alph{enumi})}

% info for header block in upper right hand corner
\name{}
\class{Math 315}
\assignment{Homework \#11}
\duedate{5/16/2017}

% macros
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\II}{\mathbb{I}}

\begin{document}

\problemlist{Problem A; Abbott: 3.4.5, 3.4.7}

\begin{problem}[A]
\begin{enumerate}
\item Find an example of a disconnected set whose closure is connected.
\item If $A$ is connected, is $\overline{A}$ necessarily connected?
\end{enumerate}
\end{problem}

%\begin{solution}
%
%\end{solution}

\newpage

\begin{problem}[3.4.5]
Let $A$ and $B$ be nonempty subsets of $\RR$. Show that if there exist disjoint open sets $U$ and $V$ with $A \subseteq U$ and $B \subseteq V$, then $A$ and $B$ are separated.
\end{problem}

%\begin{solution}
%
%\end{solution}

\newpage

\begin{problem}[3.4.7]
A set $E$ is \emph{totally disconnected} if, given any two distinct points $x, y \in E$, there exist separated sets $A$ and $B$ with $x \in A$, $y \in B$, and $E = A \cup B$.
\begin{enumerate}
\item Show that $\QQ$ is totally disconnected.
\item Is the set of irrational numbers totally disconnected?
\end{enumerate}
\end{problem}

%\begin{solution}
%
%\end{solution}


\end{document}
