\documentclass[minion]{homework}
\usepackage{cmacros}
\usepackage{graphicx}
\usepackage[all,cmtip]{xy}
\usepackage{wf}
\def\net<#1>{\left<#1\right>}

\newcommand{\bbB}{\mathbb{B}}
\doclabel{Math F651: Homework 9}
\docdate{Due: April 6, 2011}
\begin{document}
\begin{aproblems}

\aproblem Show that a topological space in $X$ is Hausdorff if and only if every 
convergent net in $X$ has exactly one limit.

\aproblem Consider the product space $X\times Y$.  Find (and prove) a condition
in terms of coordinate functions that characterizes convergence of nets in the product.
Does your condition also work for an arbitrary product?

\aproblem Munkres 26.8

\aproblem Midterm 8b


\aproblem Show that if $p$ and $q$ are elements of the interior of the closed unit ball 
$$
\bbB^n=\{x\in \Reals^n:|x|\le 1\},
$$
then there is a homeomorphism $\phi:\bbB^n\ra \bbB^n$ such that $\phi(p)=q$ and such that $\phi(x)=x$ for all $x$
with $|x|=1$.  Be as rigorous as you can, but avoid writing a tome.

\aproblem Show that the homeomorphism group of a connected
manifold acts transitively.  In other words, show that if $M$ is a 
connected manifold, then for any two points $p$ and $q$ in $M$
there is a homeomorphism $\psi:M\ra M$ such that $\psi(p)=q$.

\end{aproblems}


\end{document}

