\documentclass{article}
\begin{document}
\title{Topology M.A. Comprehensive Exam}
\author{K. Lesh \and G. Martin}
\date{July 24, 1999}
\maketitle
\noindent
This exam has been checked carefully for errors. If you find what
you believe to be an error in a question, report this to the proctor.
If the proctor's interpretation still seems unsatisfactory to you, you may
alter the question so that in your view it is correctly stated, but
not in such a way that it becomes trivial.
\bigskip
Do 4 of the following 7 problems.
\begin{enumerate}
\item
Is $\{{1\over n}\}_{n \in \mbox{\bf Z}_+}$ a closed subset of $[0,1]$ ?
Is it a closed subset of $(0,1)$ ? Carefully justify your answer.
%\item Prove that none of the spaces $[0,1]$, $[0,1)$, and $(0,1)$ is
%homeomorphic to any of the others.
\item Suppose that $X$ is metric space with metric $\delta\colon
X\times X \to X$. Let $c>0$ be fixed, and define a function
$\delta_{c}: X \times X \to X$ by
\[
\delta_c(x,y)=\left\{\begin{array}{ll}
\delta(x,y) & \delta(x,y)