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

\doclabel{Math F651: Homework 5}
\docdate{Due: March 2, 2011}
\begin{document}
\begin{aproblems}

    \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 Show that $\RPn$ is an $n$-manifold. You are free to use any facts we have proved in
    class or on the homeworks in your proof. There are several approaches to this problem; 
    try to find the shortest proof you can.

    \hproblem Munkres 22.4: No rigor, please!

    \hproblem Munkres 19.7

\end{aproblems}


\end{document}

