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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due October 31 (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}

%
\item A collection $\{A_\alpha\}$ of subsets of $X$ satisfies the
  \emph{finite intersection property} if $\ds\bigcap_{i=1}^n
  A_{\alpha_i}\neq\emptyset$ for any finite subcollection.
  \begin{enumerate}
  \item Prove Cantor's ``finite intersection lemma'': Suppose
    $\{K_\alpha\}$ is a collection of compact sets of a Hausdorff
    space $X$. If $\ds\bigcap_{i=1}^n K_{\alpha_i}\neq\emptyset$ for
    any finite subcollection, then $\ds\bigcap_\alpha
    K_\alpha\neq\emptyset$.
    
  \item Prove that $X$ is compact if and only if every collection of
    closed sets $\{F_\alpha\}$ satisfying the finite intersection
    property must also satify $\ds\bigcap_{\alpha\in I}
    F_\alpha\neq\emptyset$. \\
  \end{enumerate}
  


%f98.6
\item On HW 2, you proved that if a function $f\colon X\to Y$ between
  metric spaces is continuous, then its graph
  \[
  \Gamma_f:=\{(x,f(x))\mid x\in X\}
  \]
  is a closed subset of $X\times Y$. Now, suppose $f\colon X\to Y$ is
  a map between topological spaces, and $Y$ is Hausdorff.
  \begin{enumerate}
  \item Show that if $f$ is continuous, then the graph $\Gamma_f$ is
    closed in $X\times Y$.
  \item Show that the conclusion of Part~(a) may fail of $Y$ is not
    Hausdorff. 
  \item Show that if $X$ and $Y$ are both compact and Hausdorff, then
    the converse to Part~(a) holds. \\
  \end{enumerate}



\item Let $f\colon X\to Y$ be a continuous mapping of a compact space
  $X$ onto a Hausdorff space $Y$. Prove that $f$ is a closed map, and
  hence a quotient map. \\


\item Suppose $X$ is a Hausdorff space and $q:X\rightarrow Y$ is a
  quotient map. Further suppose that $q$ is a closed map and that
  $q^{-1}(y)$ is compact for all $y\in Y$. Prove that $Y$ is
  Hausdorff.
    

\end{enumerate}

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