\documentclass[12pt]{article}
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\def\Cx{\mathbb{C}}
\def\Nl{\mathbb{N}}
\def\Ra{\mathbb{Q}}
\def\Rl{\mathbb{R}}
\def\Ts{\mathbb{T}}
\def\Ir{\mathbb{Z}}


\begin{document}

\centerline{\textbf{Problem Set 2}}
\centerline{\textbf{Math 201A}: \textbf{Fall 2016}}

\bigskip\bigskip
\noindent
\textbf{Problem 1.}  (a) Prove that a closed subset of a complete metric space is complete.
(b) Prove that
a closed subset of a compact metric space is compact.
(c) Prove that a compact subset of a metric space is closed and bounded.

\bigskip
\noindent
\textbf{Problem 2.}  Let $A$ be a subset of a metric space $X$ with closure $\bar{A}$. Define the interior $A^\circ$ and boundary $\partial A$ of $A$ by
\[
A^\circ = \bigcup \left\{ G \subset A : \text{$G$ is open}\right\},\qquad
\partial A = \bar{A} \setminus A^\circ.
\]
(a) Why is $A^\circ$ open and $\partial A$ closed?

\smallskip\noindent
(b) Prove that $X\setminus \bar{A} = (X\setminus A)^\circ$.

\smallskip\noindent
(c) Prove that $A$ is closed if and only if $\partial A \subset A$, and $A$ is open
if and only if $\partial A \subset A^c$.

\smallskip\noindent
(d) If $A$ is open, does it follow that $(\bar{A})^\circ = A$?


\bigskip
\noindent
\textbf{Problem 3.} Let $X$ be a metric space with a dense subset $A\subset X$ such that every Cauchy
sequence in $A$ converges in $X$. Prove that $X$ is complete.



\bigskip
\noindent
\textbf{Problem 4.} Let $d: X\times X \to \Rl$ be the discrete metric on a set $X$,
\[
d(x,y) = \begin{cases} 1 &\text{if $x\ne y$},
\\
0 &\text{if $x = y$}. \end{cases}
\]
What are the compact subsets of the metric space $(X,d)$?


\bigskip
\noindent
\textbf{Problem 5.} Let $c_0$ be the Banach space of real sequences $(x_n)$ such that $x_n \to 0$ as $n\to \infty$ with the sup-norm
$\|(x_n)\| = \sup_{n\in \Nl} |x_n|$. Is the closed unit ball
\[
B = \left\{(x_n)\in c_0 : \|(x_n)\| \le 1\right\}
\]
compact?



\newpage
\noindent
\textbf{Problem 6.}  A metric (or topological) space $X$ is disconnected if there are non-empty open sets
$U,V\subset X$ such that $X= U\cup V$ and $U\cap V = \emptyset$. A space is connected if it is not disconnected.
A space $X$ is totally disconnected if its only non-empty connected subsets are the singleton sets $\{x\}$ with
$x\in X$.

\smallskip\noindent
(a) Show that the interval $[0,1]$ is connected (in its standard metric topology).
\textsc{Hint.} If $[0,1]$ is a disjoint union of open sets $U$, $V$ with $0\in U$, consider
$a = \sup\left\{x\in [0,1] : [0,x)\subset U\right\}$.

\smallskip\noindent
(b) Show that the set $\Ra$ of rational numbers is totally  disconnected.




\bigskip
\noindent
\textbf{Problem 7.} Let
\[
\mathcal{T} = \left\{G\subset \Rl : \text{$G^c$ is countable or $G^c=\Rl$}\right\}.
\]
(a) Show that $\mathcal{T}$ is a topology on $\Rl$.

\smallskip\noindent
(b) Let $I=(0,1)$ with closure
$\bar{I} = \bigcap\left\{ F\supset I : \text{$F$ is closed}\right\}$
in this topology. Show that $\bar{I} = \Rl$.

\smallskip\noindent
(c) Is there a sequence $(x_n)$ such that $x_n\in I$ and $x_n \to 2$ in this topology?

\smallskip\noindent
(d) What part of your proof in Problem 5 of Set 1 fails in this example?



\bigskip
\noindent
\textbf{Problem 8.} Let $S$ be the set of sequences whose terms are $0$ or $1$:
\[
S = \left\{ (s_k)_{k=1}^\infty : \text{$s_k = 0$ or $s_k=1$} \right\}.
\]
(a) Use a diagonal argument to show that $S$ is uncountable.

\smallskip\noindent
(b) Show that $S$ has the same cardinality as the power set $\mathcal{P}(\Nl)$ of the natural numbers.
\end{document}
