\documentclass[minion]{homework}
\usepackage{cmacros}
\usepackage{graphicx}
\usepackage{wf}
\usepackage[all,cmtip]{xy}

\doclabel{Math F651: Homework 10}
\docdate{Due: April 15, 2009}

\begin{document}
\begin{aproblems}

\hproblem Munkres p.146 Number 4. (i.e. Problem 4 in the Supplementary Exercises on Topological Groups).

\hproblem
Suppose a topological group $G$ acts continuously on 
a topological space $X$.  Show that the quotient
map $\pi: X \ra X/G$ is an open map.

\hproblem
Suppose $G$ is a topological group and $H$ is a
closed subgroup of $G$.
Show that if $H$ is a normal subgroup of $G$, then
$G/H$ is a topological group.

\hproblem Munkres 26.13

\hproblem Let $G$ be a topological group acting continuously on a topological space
$X$. The orbit relation of $G$ is 
$$
\{x,y: x=gy \quad\text{for some $g\in G$}\}.
$$
\begin{subproblems}
	\item Draw a diagram of the orbit relation of the $\Ints$ action on $\Reals$.
	\item Show that $X/G$ is Hausdorff if and only if the orbit relation is closed.
	\item Quickly prove that the orbit relation of the $\Ints$ action on $\Reals$ is closed
	      to conclude that $\Reals/\Ints$ is Hausdorff.
\end{subproblems}

\end{aproblems}
\end{document}
