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\begin{document}

\centerline{\textbf{Problem Set 3}}
\centerline{\textbf{Math 201A}: \textbf{Fall 2016}}

\bigskip\bigskip
\noindent
\textbf{Problem 1.} (a) If $f : X\to Y$ and $\{ B_\alpha \subset Y : \alpha\in I\}$ is a family of subsets of $Y$, show that
\[
f^{-1}\left(\bigcap_{\alpha\in I} B_\alpha\right) = \bigcap_{\alpha\in I} f^{-1}\left( B_\alpha\right),
\qquad
f^{-1}\left(\bigcup_{\alpha\in I} B_\alpha\right) = \bigcup_{\alpha\in I} f^{-1}\left( B_\alpha\right).
\]
(b) If $\{ A_\alpha \subset X : \alpha\in I\}$ is a family of subsets of $X$, show that
\[
f\left(\bigcup_{\alpha\in I} A_\alpha\right) = \bigcup_{\alpha\in I} f\left( A_\alpha\right),
\]
but in general
\[
f\left(\bigcap_{\alpha\in I} A_\alpha\right) \ne \bigcap_{\alpha\in I} f\left( A_\alpha\right).
\]
What can you say in this case?


\bigskip
\noindent
\textbf{Problem 2.}  Define the distance $d(A,B)$ between two subsets $A, B\subset X$ of a metric space $(X,d)$
by
\[
d(A,B) = \inf\{d(x,y) : \text{$x \in  A$, $y \in  B$}\}.
\]
(a) Suppose that $F, K \subset X$ are nonempty subsets of $X$ such that $F$ is closed and $K$ is compact. Prove that $d(F, K) > 0$
if and only if $F \cap K = \emptyset$.

\smallskip\noindent(b) Give an example of two closed sets $E, F \subset \Rl$ such that $d(E,F) = 0$ but $E\cap F = \emptyset$.



\bigskip
\noindent
\textbf{Problem 3.}  Show that a metric space $X$ is sequentially compact if and only if every decreasing sequence of
nonempty closed sets has nonempty intersection. That is, if $F_n \subset X$ is closed, $F_n \ne \emptyset$, and
$F_n\supset F_{n+1}$ for all $n\in \Nl$, then
\[
\bigcap_{n=1}^\infty F_n \ne \emptyset.
\]
\textsc{Hint.} To prove that $X$ is sequentially compact, suppose that $(x_n)$ is a sequence in $X$ and consider $F_n = \overline{T_n}$
where $T_n = \{x_{n}, x_{n+1}, x_{n+2},\dots\}$.



\bigskip
\noindent
\textbf{Problem 4.}
Suppose that $(x_n)$ is a sequence in a compact metric space $X$
with the property that every convergent subsequence has the
same limit $x\in X$. Prove that $x_n \to x$ as $n\to \infty$.

\bigskip\noindent
\textbf{Problem 5.}
A real-valued function $f : X \to \Rl$ on a metric space $X$ is lower semi-continuous if
\[
f(x) \le \liminf_{n\to\infty} f(x_n)
\]
for every $x\in X$ and every sequence $(x_n)$ in $X$ such that $x_n\to x$ as $n\to \infty$. The epigraph $\text{epi} f$ of
$f$ is defined by
\[
\text{epi} f = \left\{(x,t) \in X\times \Rl : t \ge f(x)\right\}.
\]
Prove that $f$ is lower semi-continuous if and only if $\text{epi} f$ is closed in $X\times \Rl$.


\bigskip
\noindent
\textbf{Problem 6.} A metric space is perfect if every point in the space is an accumulation point,
meaning that it is a limit of a sequence of other points in the space.

\smallskip\noindent
Let $X$ be the space of all sequences consisting of zeros or ones:
\[
X = \left\{ (s_1,s_2,s_3,\dots) \mid s_n \in \{0,1\}\right\}.
\]
Define a metric $d: X\times X \to \Rl$ on $X$ by
\[
d(\mathbf{s},\mathbf{t}) = \sum_{n=1}^\infty \frac{\delta_n}{2^n}
\]
where $\mathbf{s} = (s_n)_{n=1}^\infty$, $\mathbf{t} = (t_n)_{n=1}^\infty$, and
\[
\delta_n = \left\{
\begin{array}{ll}
0 & \mbox{if $s_n=t_n$}, \\
1 & \mbox{if $s_n\ne t_n$}.
\end{array}\right.
\]
(a) Prove that $X$ is compact, totally disconnected, and perfect.

\smallskip\noindent
(b) Define the shift map $\sigma : X \to X$ by
\[
\sigma (s_1,s_2,s_3,\dots) = (s_2,s_3,s_4,\dots).
\]
Prove that $\sigma$ is continuous.

\smallskip\noindent
(c) Prove that there is a point $x\in X$ such that the orbit $\{\sigma^n (x) : n\in \Nl\}$
of $x$ under the shift map is dense in $X$.
Here, $\sigma^n = \sigma\circ\sigma\circ\dots\circ\sigma$
denotes the $n$th iterate of
$\sigma$.

\end{document}
