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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due October 3 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 4560/6560 $|$ Topology $|$ Fall
    2013 $|$ M.~Macauley}}



\begin{document}

$\;$

\noindent Throughout, a topological space $X$ is endowed with a
topology $\tau$, even if not explicitly mentioned. \\

\begin{enumerate}

% f02.3
\item A topological space is \emph{normal} if for any two disjoint
  closed sets $F_1,F_2$, there exists disjoint open sets $U_1,U_2$
  such that $F_1\subseteq U_1$ and $F_2\subseteq U_2$. Suppose $X$ is
  Hausdorff and normal and let $F$ be a closed set. Define an
  equivalence relation on $X$ by $x\sim y$ whenever both $x$ and $y$
  belong to $F$. Prove that the quotient space $X/\!\sim$ is
  Hausdorff. \\



% m22.2
\item 
  \begin{enumerate}
  \item Let $p\colon X\to Y$ be a continuous map. Show that if there
    is a continuous map $f\colon Y\to X$ such that $p\circ f=1_Y$, the
    identity map on $Y$, then $p$ is a quotient map.
  \item If $A\subset X$, a \emph{retraction} of $X$ onto $A$ is a
    continuous map $r\colon X\to A$ such that $r(a)=a$ for each $a\in
    A$. Show that a retraction is a quotient map. \\
  \end{enumerate}



\item A map $f\colon X\to Y$ of topological spaces is called an
  \emph{open (closed) map} if the image of every open (closed) set in
  $X$ is again open (closed). Show that if a continuous surjective map
  $f\colon X\to Y$ is either open or closed, then it is a quotient map. \\



\item Consider the following commutative diagram, i.e., $h\circ g=f$.
  \[
  \xymatrix{X\ar[rr]^f\ar[dr]_g && Z \\ & Y\ar[ur]_h &}
  \]
  Prove that if $f$ and $g$ are quotient maps and $h$ is a bijection,
  then $h$ must be a homeomorphism. \\



\item Use the previous problem(s) to establish the following
  homeomorphisms. Prove all of your claims. 
  \begin{enumerate}
  \item Let $X=[0,1]$ and let $\sim$ be the equivalence relation that
    such that $x\sim x$ and $0\sim 1$. Prove that $X/\!\sim$ is
    homeomorphic to $S^1$. [\emph{Hint}: Can you define a continuous
      surjective map $f\colon[0,1]\to S^1$?]
  \item Consider the disjoint union $X=D_1\sqcup D_2$ of two unit
    disks. Identify $(x_1,y_1)\sim(x_2,y_2)$ if $x_1=x_2$ and
    $y_1=y_2$ where $(x_1,y_1)\in\partial D_1$ and
    $(x_2,y_2)\in\partial D_2$. Show that $X/\!\sim$ is homeomorphic
    to the unit $2$-sphere $S^2$ in $\R^3$.
  \item Consider the unit square $X:=[0,1]\times[0,1]$ in
    $\R^2$. Identify $(0,y)\sim(1,y)$ and $(x,0)\sim(x,1)$ for all
    $x,y\in[0,1]$ The quotient space $X/\!\sim$ is called a
    \emph{torus}. Show that it is homeomorphic to $S^1\times S^1$.
  \end{enumerate}

\end{enumerate}

\end{document}
