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\noindent
\parbox{1.95in}{\bf Math 4710/5710}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{1.95in}{\bf \hfill Due: Wed., Mar. 19${}^\mathbf{th}$\!, 2025}

%\vspace{0.2in}
%
%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

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Some background: Let us define $\mathbb{R}^\omega = \mathbb{R} \times \mathbb{R} \times \cdots$ to be the set of sequences of real numbers: ${\bf x} = (x_1,x_2,x_3,\dots)$. If we give $\mathbb{R}$ its standard topology, then we can give $\mathbb{R}^\omega$ the corresponding product topology or box topology.

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We can also give $\mathbb{R}^\omega$ a {\it uniform topology} coming from the uniform metric on $\mathbb{R}^\omega$. Here we let:
$$\bar{\rho}({\bf x},{\bf y}) = \mathrm{sup} \{ \bar{d}(x_i,y_i) \;|\; i=1,2,\dots \}$$ 
where ${\bf x} = (x_1,x_2,\dots)$, ${\bf y} = (y_1,y_2,\dots)$, and $\bar{d}(x_i,y_i) = \min\{|x_i-y_i|,1\}$. {\it Note:} If we used $|x_i-y_i|$ instead of $\min\{|x_i-y_i|,1\}$, our supremum could be $\infty$ and then $\overline{\rho}$ would not be a metric.

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Let $\mathbb{R}^\infty = \{ {\bf x} \in \mathbb{R}^\omega \;|\; {\bf x} \mbox{ is eventually zero } \}$. So ${\bf x} =(x_1,x_2,\dots) \in \mathbb{R}^\infty$ implies that there exists some $N \in \mathbb{Z}_{>0}$ such that $x_n=0$ for all $n \geq N$. So in other words, the coordinates of ${\bf x}$ are non-zero only finitely many times.

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{\it Note:} If we think of $\mathbb{R}^n$ as a subset of $\mathbb{R}^{n+1}$ by identifying $(x_1,\dots,x_n)$ with $(x_1,\dots,x_n,0)$, then $\mathbb{R}^\infty$ is more-or-less the union of all $\mathbb{R}^n$'s.

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\begin{description}
\item[\#1 One is Enough:] We explore these interesting spaces.
\begin{enumerate}[(a)]
\item Give an example of a subset of $\mathbb{R}^\omega$ that is open in the box topology but not the product topology. 

You should carefully {\it prove} your set is {\it not} open in the product topology.

\item Assuming we already know $\bar{d}$ is a metric on $\mathbb{R}$, show that $\bar{\rho}$ is a metric on $\mathbb{R}^\omega$.
\item If $B_\epsilon({\bf x}) = \{ {\bf y} \in \mathbb{R}^\omega \;|\; \bar{\rho}({\bf x},{\bf y})<\epsilon \}$ denotes the (open) ball centered at ${\bf x}$ of radius $\epsilon$, then\\ \force\hfill describe $B_1({\bf 0})$ and $B_2({\bf 0})$ where ${\bf 0}=(0,0,\dots)$. 
\item Determine the closure of $\mathbb{R}^\infty$ in $\mathbb{R}^\omega$ in the (i) product, (ii) box, and {\bf [Grad. (iii)]} uniform topologies.
\end{enumerate}

\end{description}

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