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\chapter{Determine if these sets are open and/or closed.}



Let $\eucal B$ be the set of ``open rectangles" in $\bbr$:
\begin{align*}
\eucal B = \{(a,b) \times (c,d) \subseteq \bbr^2: (a,b) \subseteq \bbr, (c,d) \subseteq \bbr\}.
\end{align*}
Here are two examples of elements in $\eucal B$: $B = (B_1,B_2) \times (B_3,B_4)$ and $C = (C_1,C_2) \times (C_3,C_4)$.

\begin{center}
\begin{tikzpicture}
\draw (-2.5,0) -- (2.5,0)
	(0,-2.5) -- (0,2.5);
\foreach \x/\y/\z/\w/\l/\c in {-1/1/-.3/2.1/B/red,-2.1/-1.9/1.1/-.4/C/green} {
	\draw[dashed,fill opacity=.4,,fill=\c] (\x,\y) rectangle (\z,\w);
	\draw (\x/2+\z/2,\y/2+\w/2) node {$\l$};
	\draw (\x,-.1) -- (\x,.1) node[above,font=\footnotesize] {$\l_1$}
	        	(\z,-.1) -- (\z,.1) node[above,font=\footnotesize] {$\l_2$}
		(-.1,\y) -- (.1,\y) node[right,font=\footnotesize] {$\l_3$}
		(-.1,\w) -- (.1,\w) node[right,font=\footnotesize] {$\l_4$};
}

\end{tikzpicture}
\end{center}

The set $\eucal B$ is a basis for a topology on $\bbr^2$. (You do not have to prove that $\eucal B$ is a basis.) 

Consider the following sets.

\begin{enumerate}
\item The interior of the unit circle: $U = \{(x,y) \in \bbr^2 : x^2 + y^2 < 1\}$.
\item The integer lattice $\bbz^2$: $U = \{(x,y) \in \bbr^2 : x \in \bbz, y \in \bbz\}$.
\end{enumerate}

For each set $U$, complete the following.

\begin{enumerate}[label={\alph*.}]
\item Determine if $U$ is open or not open. Prove your claim.
\item Determine if $U$ is closed or not closed. Prove your claim.
\end{enumerate}

In the end, you should have four short proofs.

\section{First draft}
Due Monday, April 4 at 6:00 PM.

\begin{enumerate}
\item The interior of the unit circle: $U = \{(x,y) \in \bbr^2 : x^2 + y^2 < 1\}$.
\begin{enumerate}
\item Claim: $U$ is (pick one: open/not open).
\begin{proof}

\end{proof}

\item Claim: $U$ is (pick one: closed/not closed).
\begin{proof}

\end{proof}
\end{enumerate}

\item The integer lattice $\bbz^2$: $U = \{(x,y) \in \bbr^2 : x \in \bbz, y \in \bbz\}$.
\begin{enumerate}
\item Claim: $U$ is (pick one: open/not open).
\begin{proof}

\end{proof}

\item Claim: $U$ is (pick one: closed/not closed).
\begin{proof}

\end{proof}
\end{enumerate}
\end{enumerate}