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\begin{center}{\bf Homework 7, due Thursday, November 1, 2012}\end{center}

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Do any 4 of the 5 problems. 
Each problem is worth 25 points. Solutions will be graded for correctness, clarity and style.

\begin{itemize}

\item[(1)] Let $\Gamma\subset{\bf R}^2$ be the graph of a function 
$f:{\bf R}\to{\bf R}$. If ${\bf R}$ and ${\bf R}^2$ are given the standard topologies 
(and as we now know, the standard topology on ${\bf R}^2$ is the same as the
product topology)
and if $f$ is continuous with respect to the standard topologies, show that $\Gamma$ is closed.
[Hint: Define $F:{\bf R}^2\to {\bf R}$ by $F((a,b))=f(a)-b$. Show that $F$ is continuous
and that $\Gamma=F^{-1}(\{0\})$. You may assume that $+:{\bf R}^2\to {\bf R}$, defined as
$+((a,b))\mapsto a+b$, is continuous, and that $-:{\bf R}\to {\bf R}$, defined as
$-(a)=-a$, is continuous.]
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\item[(2)] Let $\pi_2:{\bf R}^2\to{\bf R}$ be defined by $\pi_2((x,y))=y$. 
If ${\bf R}$ and ${\bf R}^2$ are given the standard topologies, show that
$\pi_2$ is not a closed map (i.e., give an example of a closed subset $C\subseteq {\bf R}^2$
such that $\pi_2(C)$ is not closed).
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\item[(3)] Give an example of a relation $R$ on a set $S$ which is reflexive and symmetric but not transitive.
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\item[(4)] Give an example of a relation $R$ on a set $S$ which is reflexive and transitive but not symmetric.
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\item[(5)] Give an example of a relation $R$ on a set $S$ which is transitive and symmetric but not reflexive.

\end{itemize}

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