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\def\N{\mathbb{N}} % Natural numbers
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\begin{document}
\begin{center}
\begin{large}
Math 70100: Functions of a Real Variable I \\
Homework 3, due Tuesday, September 23rd.
\end{large}
\end{center}
{\bf Name:} Insert your name here.
\begin{questions}
\question The Hilbert cube is the countable product $H=[0,1]^\N$
of all functions $\N \to [0,1]$ endowed with the product topology.
Give a direct proof that the Hilbert cube is sequentially compact.
That is, given a sequence $\{\alpha^n \in H\}_{n \in \N}$,
find a convergent subsequence.
({\em Hint:} You may want to use a version of the Cantor diagonalization argument.)
\begin{sol}
Insert answer here.
\end{sol}
\question {\em (Modified from Pugh Chapter 2 \#79)} A space $X$ is locally path-connected if given any $x \in X$
and any open set $U \subset X$ containing $x$, there is an open set $V \subset U$ containing $x$ which is path-connected.
Let $X$ be a topological space which is non-empty, compact, locally path-connected and connected.
Prove that $X$ is path-connected.
\begin{sol}
Insert answer here.
\end{sol}
\question Let $X$ be a compact metric space and let ${\mathcal U}$ be an open cover of $X$. Prove that there is an $\epsilon>0$
so that for every $x \in X$ there is a $U \in {\mathcal U}$ containing the open ball of radius $\epsilon$ about $x$. (Such an $\epsilon>0$ is called a {\em Lebesgue number} for the cover.)
\begin{sol}
Insert answer here.
\end{sol}
\question If $A$ and $B$ are subsets of $\R$, then we define
$$A+B=\{a+b~:~a \in A \text{ and } b\in B\} \subset \R.$$
Let $C$ be the standard middle third Cantor set. Prove that $C+C=[0,2]$. ({\em Hint}: Consider ternary expansions.)
\begin{sol}
Insert answer here.
\end{sol}
\end{questions}
\end{document}