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\begin{center}\begin{large}\textbf{Math 131 - Problem Set 9\\
Due Thursday, Nov 15}\end{large}\end{center}



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\noindent From Munkres: 55.1, 55.2, 57.1, 57.2, 57.3

\begin{enumerate}

\item Show that if $S^1$ is covered by two closed sets $A$ and $B$, then one of them contains a pair of
antipodal points. (Hint: Consider the function $f(x)=d(x,A)-d(x,B)$.)

\item 

\begin{enumerate}

\item Prove that if $S^2$ is covered by 3 closed sets, then one of them contains a pair of antipodal points.

\item (Not to be graded -- just for fun!) Show that (a) is not true if we allow 4 closed sets.

\end{enumerate}

\end{enumerate}





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