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\title{Problem Set 6: Compactness and The Zariski topology}
\date{Due: Thursday, April 7}
\author{Your name:}





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\maketitle

Read the notes on the Zariski topology posted on the web page.

\begin{prob}[8]
Did you do your reading assignment?
\end{prob}

\begin{prob}[24] \noindent
\begin{enumerate}
	\item Let $R$ be a commutative ring with unit ($1 \in R$). Prove that $\spec(R)$ is compact.
	\item When is $\spec(R)$ Hausdorff?
\end{enumerate}
Consider the map $$i : \R^n \to \spec(\R[x_1, \ldots, x_n])$$ that is defined by taking the $n$-tuple $(r_1, \ldots, r_n)$ to the ideal generated by the degree one polynomials $$x_1 - r_1, \ldots , x_n - r_n.$$
\begin{enumerate}
	\addtocounter{enumi}{2}
	\item Prove that the map $i$ is injective. Identifying $\R^n$ with its image under the map, it acquires a topology as a subspace of $\spec(\R[x_1, \ldots, x_n])$. Denote $\R^n$ with this topology by $\A^n$
	\item What is the closure of $\left\{(n, \frac{1}{n}) : n \in \N \setminus \{0\} \right\} \subset \A^n$?
\end{enumerate}
\end{prob}


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