\documentclass[minion]{homework}
\usepackage{cmacros}
\doclabel{Math F651: Homework 5}
\docdate{Due: February 23, 2011}
\begin{document}
\begin{aproblems}

    \hproblem Show that if $X$ is a first countable space, and if convergent sequences in $X$ have unique limits, then $X$ is Hausdorff.
    
    \hproblem Let $X$, $Y$, and $Z$ be topological spaces.  Show that $(X\times Y)\times Z$ is homeomorphic to $X\times Y\times Z$.
    
    \hproblem Show that the product of an $n$-manifold $N$ and an $m$-manifold $M$ is an $n+m$ manifold.

    \hproblem Munkres 17.13

    \hproblem Munkres 17.9

    \hproblem Show that a topological space is a 0-manifold if and only if it is at most countable and has the discrete topology.

    \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 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 22.3

    \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.


\end{aproblems}


\end{document}


