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Math 70100: Functions of a Real Variable I \\
Homework 1, due Wednesday, September 10th.
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\noindent
{\bf Name: } Insert your name here.

\noindent
{\bf Date: } \today.

\begin{questions}
\question {\em (Based on Pugh 2.92)} Let $X$ be a topological space. 
Recall that a {\em neighborhood} $N$ of $x \in X$ is a subset $N \subset X$ so that there is an open set $U \subset X$ with $x \in U$ and $U \subset N$. A {\em boundary point} of a set $A \subset X$ is a point $x \in X$ so that every neighborhood of $x$ intersects both $A$ and $X \smallsetminus A$. The boundary of $A$, $\partial A$, is the set of all boundary points of $A$.
\begin{parts}
\part Show that $\partial A=X \smallsetminus \big(\mathrm{Int}(A) \cup \mathrm{Int}(X \smallsetminus A)\big)$. 
\begin{solution}
Insert your answer to problem 1a here.
\end{solution}

\part Explain why $\partial A$ is closed.
\begin{solution}
\end{solution}

\part Show that $\partial \partial A \subset \partial A$. 
\begin{solution}
\end{solution}

\part Show that $\partial \partial \partial A=\partial \partial A$. 
\begin{solution}
\end{solution}

\part Given an example which illustrates that
$\partial \partial A$ may not equal $\partial A$.
\begin{solution}
\end{solution}
\end{parts}

\question {\em (Pugh 2.37)} Let $C$ denote the vector space of continuous functions from $[0,1]$ to $\R$. This space can be endowed with the sup (or $L^\infty$) norm,
$$| f | = \sup~\{|f(x)|~:~x \in [0,1]\}$$
or the $L^1$ norm, 
$$\|f\|=\int_0^1 |f(x)|~dx.$$
Consider the identity map between $\mathit{id}$ from $(C,|\cdot|)$ to $(C,\|\cdot\|)$.
\begin{parts}
\part Show that $\mathit{id}$ is a continuous. (Thus it is a continuous linear bijection.)
\begin{solution}
\end{solution}
\part Show that the inverse $\mathit{id}^{-1}$ is not continuous.
\begin{solution}
\end{solution}
\end{parts}

\question {\em (Modified from Lang II.5.3a)} 
Let $\ell^1$ be the set of all sequences $\alpha=\{a_n\}_{n \in \N}$ of real numbers such that $\sum_{n \in \N} |a_n|$ converges. Define 
$$|\alpha|=\sum_{n \in \N} |a_n|.$$
\begin{parts}
\part Prove that $| \cdot |$ is a norm on $\ell^1$.
\begin{solution}
\end{solution}
\part Recall that a sequence $\{\alpha_n\}$ in a normed vector space is {\em Cauchy} if given any $\epsilon>0$,
there is an $N \in \N$ so that $|\alpha_m-\alpha_n|<\epsilon$ for $m,n \geq N$. A normed vector space is {\em complete}
if all Cauchy sequences converge. Show that $\ell^1$ is complete with the norm $|\cdot |$. 
\begin{solution}
\end{solution}
\end{parts}

\question {\em (Lang II.13)} The {\em diagonal} $\Delta$ is the set of all points $(x,x)$.
\begin{parts}
\part Show that a space $X$ is Hausdorff if and only if the diagonal is closed in $X \times X$. 
\begin{solution}
\end{solution}

\part Show that a product of Hausdorff spaces is Hausdorff.
\begin{solution}
\end{solution}
\end{parts}

\question {\em (Lang II.5.5c)} Let $X$ be a metric space. For each $x \in X$, define the function $f_x$ on $X$
by $f_x(y)=d(x,y)$. Let $\| \cdot \|$ be the sup norm.
\begin{parts}
\part Show that $d(x,y)=\|f_x-f_y\|$. 
\begin{solution}
\end{solution}

\part Let $a$ be a fixed element of $X$, and let $g_x=f_x-f_a$. Show that the map $x \mapsto g_x$ is a distance-preserving embedding of $X$ into the normed space of bounded functions on $X$. ({\em Remark:} This shows that every metric space is isometric to a subset of a normed vector space.)
\begin{solution}
\end{solution}
\end{parts}

\question {\em (Lang II.5.8ab)} Let $X$ be a topological space and $E$ a vector space with norm $|\cdot|$. Let $M(X,E)$
denote the set of all maps from $X$ to $E$. Let $B(X,E)$ denote the set of bounded maps from $X$ to $E$ endowed with the sup norm defined by $\|f\|=\sup \{|f(x)|~:~x \in X\}$. Let $BC(X,E) \subset B(X,E)$ be the set of bounded continuous maps. 
\begin{parts}
\part Show that $BC(X,E)$ is closed in $B(X,E)$. 
\begin{solution}
\end{solution}

\part A {\em Banach space} is a complete normed vector space. Show that if $E$ is a Banach space, then $B(X,E)$ 
is complete.
\begin{solution}
\end{solution}
\end{parts}

\question Let $X$ be a topological space. 
Then, $X$ is called {\em separable} if it has a countable base (or basis) for its topology.
A set $A \subset X$ is {\em dense (in $X$)} if its closure $\bar A=X$. 
\begin{parts}
\part {\em (Lang II.15)} Show that a separable space has a countable dense subset.
\begin{solution}
\end{solution}

\part {\em (Lang II.16a)} Show that if $X$ is a metric space and has a countable dense subset, then $X$ is separable.
\begin{solution}
\end{solution}
\end{parts}

\question (Lang II.5.17).
An {\em open covering} of a topological space $X$ is a collection ${\mathcal U}$ of open sets so that $X=\bigcup_{U \in {\mathcal U}} U$. A {\em subcover} is a subset ${\mathcal V} \subset {\mathcal U}$ which is still a cover (i.e., 
$X=\bigcup_{V \in {\mathcal V}} V$). 
\begin{parts}
\part Show that every open covering of a separable space has a countable subcovering. 
\begin{solution}
\end{solution}

\part Show that a disjoint collection of open sets in a separable space is countable.
\begin{solution}
\end{solution}

\part Show that a base (or basis) for the topology of a separable space contains a countable base.
\begin{solution}
\end{solution}

\end{parts}
\end{questions}
\end{document}
