\documentclass[minion]{homework}
\usepackage{cmacros}
%\usepackage{times}

\doclabel{Math 651: Homework 3}
\docdate{Due: February 20, 2009}
\def\calB{\mathcal{B}}
\begin{document}

\begin{aproblems}

\hproblem Munkres 17.9

\hproblem Munkres 17.13

\hproblem Munkres 17.14

\hproblem Munkres 17.17

\hproblem Munkres 17.19

\hproblem A set $A\subseteq X$ is said to be {\it dense} in $X$ if $\bar{A}=X$.
A topological space is {\it second countable} if it admits a countable basis.
\begin{subproblems}
\subprob 
Suppose $X$ is second countable.  Prove that it has a countable dense subset.
\subprob Prove that a metric space is second countable if and only if it has
         a countable dense subset.
\end{subproblems}

\hproblem A topological space $X$ is said to be locally Euclidean (of dimension $n$)
if for every $x\in X$ there is an open set $U$ that is homeomorphic to an open subset
of $\Reals^n$.  A topological $n$-manifold is a topological space that is 
locally Euclidean of dimension $n$, Hausdorff, and second countable.

Show that the product of an $n$-manifold $N$ and an $m$-manifold $M$ is an $n+m$ manifold.
\end{aproblems}

\end{document}