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\def\ctln{\centerline}
\def\msk{\medskip}
\def\bsk{\bigskip}
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\ctln{\bf Math 970 Homework \# 7}

\msk

\ctln{{\bf Due: } Never (unless you want to turn it in...)}

\bsk


\item{30.} Show that if $X_\alpha, \alpha\in I$ are all path-connected, then so is
$\displaystyle \prod_{\alpha\in I} X_\alpha$, if we use the product topology.

\bsk

\item{31.} Show that if $A_\alpha\subseteq X, \alpha\in I$ are all path-connected, and
$\displaystyle\bigcap_{\alpha\in I}A_\alpha\neq\emptyset$, then 
$\displaystyle\bigcup_{\alpha\in I}A_\alpha$ is path-connected.

\bsk

\item{32.} Show that if $C\subseteq{\Bbb R}^3$ is countable, then ${\Bbb R}^3\setminus C$
is path-connected. (Hint: a plane in ${\Bbb R}^3$ will hit $C$ in how many points?)


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\end