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

\doclabel{Math 651: Homework 5}
\docdate{Due: February 27, 2009}
\def\calB{\mathcal{B}}
\begin{document}

\begin{aproblems}
\hproblem Let $\pi:X\ra Y$ be a surjective continuous map.  Show that $\pi$ is a quotient map
if and only if it takes saturated closed sets to saturated closed sets.

\hproblem\label{p1} Let $\pi:X\ra Y$ be a quotient map.  Show that if $A$ is a saturated open set
or a saturated closed set, then $\left.\pi\right|_A:A\ra \pi(A)$ is a quotient map.

\hproblem 
Let $X$ be the line with two zeros. That is, $X$ is the
quotient space of $\{0,1\}\times\Reals$ given by the
equivalence relation $(0,x)\sim (1,x)$ if $x\neq 0$.
Show that $X$ is locally Euclidean and second countable,
but not Hausdorff.

\hproblem\label{p:countable-cover}
Suppose $X$ is a second countable space.  Let $\{U_\alpha\}_{\alpha\in I}$ be 
a collection of open sets in $X$ such that $\cup_\alpha U_\alpha=X$.  (We call
$\{U_\alpha\}_{\alpha\in I}$ an \define{open cover} of $X$.)  Prove that there exists
a countable sub-collection $\{U_{\alpha_i}\}$ such that $\cup_i U_{\alpha_i}=X$. (We say
that every open cover of a second countable space has a countable subcover).

\hproblem Suppose $\pi$ is a quotient map from a second countable space $X$ to a locally Euclidean
space $Y$. Prove that $Y$ is second countable. \Hint \pref{p:countable-cover} might be handy.

\hproblem (The Characteristic Property of the Quotient Topology 
is characteristic.) Suppose $\pi:X\ra Y$ is a surjective function.
Suppose that for any topological space $Z$ that a function $f:Y\ra Z$ is 
continuous if and only if $f\circ\pi$ is.  Show that $Y$ has the quotient topology
induced by $\pi$.
$$
\xymatrix{ X \ar[d]_{\pi} \ar[rd]^{f\circ\pi} \\
Y\ar[r]_{f} & Z\\}
$$

\begin{wrapfigure}{r}{2in}
				\includegraphics{hw5f1}
				\caption{Gluing a half sphere.}
\end{wrapfigure}
\hproblem Show that the upper half sphere $S^{n,+}$
with antipodal points on $\partial S^{n,+}$ identified is homeomorphic to $\RPn$.

%\hproblem (Munkres 17.7) Let $\Reals^\infty$ be the set of real sequences that are eventually zero. What
%is the closure of $\Reals^\infty$ in the product and box topologies of $\Reals^\omega$?

\end{aproblems}
\end{document}
