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\begin{center}{\bf Homework 4, due Thursday, September 20, 2012}\end{center}

Given sets $X$ and $Y$, define the product $X\times Y$ to be the set
$X\times Y=\{(x,y):x\in X,y\in Y\}$, and define the projection maps
$\pi_X:X\times Y\to X$ and $\pi_Y:X\times Y\to Y$ by
$\pi_X((x,y))=x$ and $\pi_Y((x,y))=y$ for all $(x,y)\in X\times Y$.
\vskip\baselineskip

Now assume ${\mathcal T}_X$ is a topology on $X$ and 
${\mathcal T}_Y$ is a topology on $Y$. 
Let ${\mathcal B}_{X\times Y}$ be the collection of all subsets of $X\times Y$ of the form
$U\times V$, where $U\subseteq X$ is open in $X$ (i.e., $U\in {\mathcal T}_X$) 
and $V\subseteq Y$ is open in $Y$ (i.e., $V\in {\mathcal T}_Y$).
Let ${\mathcal T}_{X\times Y}$ be the collection of all unions of elements of 
${\mathcal B}_{X\times Y}$. Then ${\mathcal T}_{X\times Y}$ is a topology on $X\times Y$
called the product topology, and ${\mathcal B}_{X\times Y}$ is a 
basis for ${\mathcal T}_{X\times Y}$

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Do any 4 of the 6 problems. 
Each problem is worth 25 points. Solutions will be graded for correctness, clarity and style.

\begin{itemize}

\item[(1)] Consider the function $f:{\bf R}\to{\bf R}$ defined as
\begin{equation*}
f(x)= 
\begin{cases} x^2 & \text{if $x\neq1$,}
\\
0 &\text{if $x=1$.}
\end{cases}
\end{equation*}
Intuitively, we know that $f$ is not continuous. Prove it by
finding a closed subset $B\subseteq{\bf R}$ such that $f^{-1}(B)$ is not closed and
by finding an open subset $V\subseteq{\bf R}$ such that $f^{-1}(V)$ is not open
\vskip\baselineskip

\item[(2)] Let $X$ and $Y$ be sets, and let $A\subseteq X$ and $B\subseteq Y$ be subsets.
Prove that $(\pi_X)^{-1}(A)=A\times Y$ and $(\pi_Y)^{-1}(B)=X\times B$.
\vskip\baselineskip

\item[(3)] Let $X$ and $Y$ be topological spaces with topologies
${\mathcal T}_X$ and ${\mathcal T}_Y$, respectively.
Let $X\times Y$ have the product topology.
Prove that $\pi_X$ and $\pi_Y$ are continuous.
\vskip\baselineskip

\item[(4)] Let $f:X\to Y$ be a map of topological spaces. Let ${\mathcal B}_Y$ be a basis for the topology on $Y$.
Prove that $f$ is continuous if and only if $f^{-1}(V)$ is open in $X$ for every $V\in {\mathcal B}_Y$.
\vskip\baselineskip

\item[(5)] For any sets $A,B$ and $C$, prove that 
$(A\cup B)\cap C=(A\cap C)\cup(B\cap C)$
(i.e., intersection distributes over union).
\vskip\baselineskip

\item[(6)] For any sets $A,B$ and $C$, prove that 
$(A\cap B)\cup C=(A\cup C)\cap(B\cup C)$
(i.e., union distributes over intersection).
\vskip\baselineskip
\end{itemize}

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