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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due October 25 (Friday)
}} \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}

%f01.1/m26.9
\item If $A\times B$ is a compact subset of a product space $X\times
  Y$ contained in an open set $W\subseteq X\times Y$, prove that there
  are open sets $U\subseteq X$ and $V\subseteq Y$ such that $A\times
  B\subseteq U\times V\subseteq W$. \\


%f99.5/m26.5
\item Let $A,B$ be disjoint compact subspaces of a Hausdorff space
  $X$. Prove that there are disjoint open sets $U,V$ with $A\subseteq
  U$ and $B\subseteq V$. \\
% Would be a good Part(b)
%\item Suppose $X$ is a Hausdorff space, $q:X\rightarrow Y$ is a
%  surjection, and the set $Y$ is given the quotient topology.
%  Further suppose that, with these topologies, $q$ is a closed map
%  and for each $y\in Y$, $q^{-1}(y)$ is compact. Prove that $Y$ is
%  Hausdorff.


%s01.5 
\item Suppose $C_n$ is a compact connected subspace of a Hausdorff
  space $X$ and $C_n\supseteq C_{n+1}$ for each $n\in\N$. Prove that
  $\displaystyle\bigcap_{n=1}^\infty C_n$ is connected. \\


\end{enumerate}

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