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\begin{center}{\bf Homework 6, due Thursday, October 25, 2012}\end{center}
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Do any 5 of the 8 problems.
Each problem is worth 20 points. Solutions will be graded for correctness, clarity and style.
\begin{itemize}
\item[(1)] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Define $d:X\times Y\to {\bf R}$
by
$$d((x_1,y_1),(x_2,y_2))=\max\{d_X(x_1,x_2),d_Y(y_1,y_2)\}$$
whenever $(x_1,y_1),(x_2,y_2)\in X\times Y$. Show that $d$ is a metric.
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\item[(2)] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Define $d:X\times Y\to {\bf R}$
as in Problem 1, so $(X\times Y,d)$ is a metric space by Problem 1.
Let $x\in X$ and $y\in Y$,
and let $r>0$ be a real number. Show that
$D_{X\times Y}((x,y),r)=D_X(x,r)\times D_Y(y,r)$.
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\item[(3)] Let $X$ and $Y$ be sets. Let $U_i\subseteq X$ for all $i$ in some set $I$
and let $V_j\subseteq Y$ for all $j$ in some set $J$. Let $U=\cup_{i\in I}U_i$
and let $V=\cup_{j\in J}V_j$. Show that $U\times V=\cup_{(i,j)\in I\times J}U_i\times V_j$.
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\item[(4)] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Define $d:X\times Y\to {\bf R}$
as in Problem 1, so $(X\times Y,d)$ is a metric space by Problem 1.
Let $r>0$ and let $s>0$, and let $x\in X$ and let $y\in Y$. Show that
$D_X(x,r)\times D_Y(y,s)$ is open in the metric topology on $X\times Y$.
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\item[(5)] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Define $d:X\times Y\to {\bf R}$
as in Problem 1, so $(X\times Y,d)$ is a metric space by Problem 1.
Give $X$ and $Y$ their metric topologies,
let ${\mathcal T}_{prod}$ be the product topology on $X\times Y$ and
let ${\mathcal T}_{metric}$ be the metric topology on $X\times Y$ with respect to the metric $d$. Show that
${\mathcal T}_{prod}={\mathcal T}_{metric}$. (I.e., if $W$ is open in the metric topology on $X\times Y$,
show that $W$ is also open in the product topology, and vice versa. To do this think of $W$ as a union of basis
elements. Hint: Use Problem 2 to show that $W\in {\mathcal T}_{prod}$ if $W\in {\mathcal T}_{metric}$,
and use Problems 3 and 4 to show that $W\in {\mathcal T}_{metric}$ if $W\in {\mathcal T}_{prod}$.)
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\item[(6)] Let $({\bf R},d_E)$ be the usual Euclidean metric on the real numbers
(so $d_E(a,b)=|a-b|$) and let $d:{\bf R}^2\to {\bf R}$ be defined
by
$$d((x_1,y_1),(x_2,y_2))=\max\{d_E(x_1,x_2),d_E(y_1,y_2)\},$$
so $d$ is a metric on ${\bf R}^2$ by Problem 1.
Let $\delta_E$ be the Euclidean metric on ${\bf R}^2$, so
$$\delta_E((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}.$$
Show that the metric topology on ${\bf R}^2$ with respect to $d$ is the
same as the metric topology on ${\bf R}^2$ with respect to $\delta_E$.
[Hint: Denote the open disc centered at $p\in{\bf R}^2$ of radius $r$ with respect to $d$ by
$D_{({\bf R}^2,d)}(p,r)$ and denote the open disc centered at $p\in{\bf R}^2$
of radius $r$ with respect to $\delta_E$ by
$D_{({\bf R}^2,\delta_E)}(p,r)$. Show that $D_{({\bf R}^2,d)}(p,r)\subseteq
D_{({\bf R}^2,\delta_E)}(p,r\sqrt{2})$ and
that $D_{({\bf R}^2,\delta_E)}(p,r)\subseteq D_{({\bf R}^2,d)}(p,r)$.]
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\item[(7)] Let $X$, $Y$, $Z$ and $W$ be topological spaces.
Let $f:X\to Z$ and $g:Y\to W$ be continuous maps.
Define $H:X\times Y\to Z\times W$ by $H((x,y))=(f(x),g(y))$. Show that $H$ is continuous.
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\item[(8)] Let $X$ and $Y$ be topological spaces and give ${\bf R}$ and ${\bf R}^2$ the standard topologies.
Let $f:X\to {\bf R}$ and $g:Y\to {\bf R}$ be continuous maps. You may assume that
$p:{\bf R}^2\to{\bf R}$, defined by $p((a,b))=a+b$, is continuous.
Define $h:X\times Y\to{\bf R}$ by $h((x,y))=f(x)+g(y)$. Show that $h$ is continuous.
[Hint: Use the fact that the standard topology on ${\bf R}^2$ is the product topology to
apply Problem 7, using the fact that $h=p\circ H$, where
$H:X\times Y\to{\bf R}^2$ is defined by $H((x,y))=(f(x),g(y))$.]
\end{itemize}
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