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\begin{center}\begin{large}\textbf{Math 131 - Problem Set 3\\
Due Tuesday, Sept 25}\end{large}\end{center}



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\noindent From Munkres: 17.11, 17.13, 17.20, 19.6, 19.7, 20.4 


\begin{enumerate} 

\item The \emph{Zariski topology} on $\R^2$ is defined to be the topology with basis 
$$U_f = \{(x,y) \in \R^2 \mid f(x,y) \neq 0\}$$ 
where $f$ ranges over all polynomials $f \in \R[x,y]$.

\begin{enumerate}

\item Show that the subsets $U_f \subset \R^2$ are in fact a basis.
\item Show that this topology is not Hausdorff.
\item If $\R \subset \R^2$ is any line, show that the subspace topology on $\R$ induced by the Zariski topology is the finite complement topology.
\item Let $$\Phi = \{(x,y) \in \R^2 \mid y = \sin x \}.$$ What is the closure of $\Phi$ in the Zariski topology?

\end{enumerate}

\end{enumerate}

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