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\title{hw6-solutions}
\begin{document}
\centerline{\Large Math 171 Homework 6}
\centerline{\small (due May 13)}
\vskip .2in
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 42.3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{42.3}
Let $X_1, \ldots, X_n$ be a finite collection of compact subsets of a metric
space $M$. Prove that $X_1 \cup X_2 \cup \cdots \cup X_n$ is a compact metric
space. Show (by example) that this result does not generalize to infinite
unions.
\sol
Let $\mc U$ be an open cover of $X_1 \cup \ldots \cup X_n$. Since each
$X_i$ is compact, there exists a finite subcollection $\mc U_i$ of
$\mc U$ that covers $X_i$. Then $\bigcup_i \mc U_i$ is a finite subcollection
of $\mc U$ that covers $X_1 \cup \ldots \cup X_n$.
For an counterexample with infinite unions, consider
$\bigcup [-1 + 1/n, 1 - 1/n] = (0, 1)$. Since each interval
$[-1 + 1/n, 1 - 1/n]$ is closed and bounded, it is compact. However,
$(0, 1)$ is not closed, so it is not compact.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 42.8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{42.8}
Let $f$ be a continuous, real-valued function on a metric space $M$ which is
never zero. Prove that the collection of open sets $U$ for which either
$f(x) > 0$ for $x \in U$ or $f(x) < 0$ for $x \in U$ is an open cover of $M$.
\sol
It suffices to show that for every $x \in M$ there exists an element $U$ of
the collection containing $x$. Without loss of generality, assume that
$f(x) > 0$. Then, by continuity of $f$ there exists a ball $B_{\delta}(x)$
such that for every $y \in B_{\delta}(x)$, $f(y) > 0$. Then, $B_{\delta}(x)$
is the desired element of the collection of open sets.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 42.12
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{42.12}
A \emph{contractive mapping} on $M$ is a function $f$ from the metric space
$(M, d)$ into itself satisfying
\[
d(f(x), f(y)) < d(x, y)
\]
whenever $x, y \in M$ with $x \neq y$. Prove that if $f$ is a contractive
mapping on a compact metric space $M$, then there exists a unique point
$x \in M$ with $f(x) = x$. (Such a point is called a \emph{fixed point}.)
\sol
\emph{Existence.}
Consider the function $F: M \ra [0, \infty)$ given by
\[
F(x) = d(x, f(x)).
\]
We show that $F$ is continuous. Fix $x \in M$ and $\e > 0$. Since
$f$ is continuous, there exists $\delta > 0$ such that for every
$y$ in $B_{\delta}(x)$ we have that $d(f(x), f(y)) < \e/2$.
Let $\delta' = \min\{\e/2, \delta\}$. Then for every
$y$ in $B_{\delta'}(x)$ we have that
\begin{align*}
d(y, f(y))
& \leq d(x, y) + d(x, f(y)) \\
& \leq d(x, y) + d(x, f(x)) + d(f(x), f(y)) \\
& < \delta' + d(x, f(x)) + \frac \e 2 \\
& \leq \e + d(x, f(x)).
\end{align*}
and by the same argument with $x$ and $y$ switched we also have that
\[
d(x, f(x)) < \e + d(y, f(y)).
\]
Thus, for every $y$ in $B_{\delta'}(x)$ we have that
\[
|F(x) - F(y)| < \e,
\]
so $F$ is continuous.
Since $F$ is continuous,
by Theorem 42.6 $F(x)$ attains a minimum at a point $x_0 \in M$. If
$F(x_0) = 0$ then $f(x_0) = x_0$, so we are done. Assume $F(x_0) > 0$.
Then, $f(x_0) \neq x_0$, so
\[
F(f(x_0)) = d(f(f(x_0)), f(x_0)) < d(f(x_0), x_0) = F(x_0),
\]
contradicting the assumption that $x$ was a minimum of $F$. Thus,
$F(x_0) = 0$.
\emph{Uniquence.}
Assume that there are two distinct fixed point $x$ and $y$ of $f$. Then
on one hand
\[
d(f(x), f(y)) = d(x, y)
\]
because $f(x) = x$ and $f(y) = y$, but on the other hand
\[
d(f(x), f(y)) < d(x, y),
\]
so we get a contradiction. Thus, the fixed point is unique.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 43.2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{43.2}
Let $M_1$ and $M_2$ be compact metric spaces. Prove that the product metric
space $M_1 \times M_2$ is compact.
\sol
By Theorem 43.5, it suffices to show that any sequence
$\{(x_n, y_n\}$ in $M_1 \times M_2$ has a convergent subsequence.
Since, $M_1$ is compact, there exists a subsequence
$\{(x_{n_k}, y_{n_k}\}$ of $M_1 \times M_2$ whose first coordinates
\subsequence x converge to some $x \in M_1$. Since, $M_2$ is compact there
exists a further subsequence $\{(x_{n_{k_l}}, y_{n_{k_l}}\}$ of
$\{(x_{n_k}, y_{n_k}\}$ whose second coordinates
$\{ y_{n_{k_l}}\}$ converge to some $y \in M_2$. Since $\{x_{n_{k_l}}\}$
is a subsequence of \subsequence x,
$\{x_{n_{k_l}}\}$ converges to $x$. Therefore, by Problem 4 on the
midterm,
$\{(x_{n_{k_l}}, y_{n_{k_l}}\}$ converges to $(x, y) \in M_1 \times M_2$.
Thus, the sequence $\{(x_n, y_n\}$ has a convergent subsequence
$\{(x_{n_{k_l}}, y_{n_{k_l}}\}$, so $M_1 \times M_2$ is compact.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 43.7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{43.7}
Let $X$ be a compact subset of a metric space $M$. If $y \in X^c$, prove
that there exists a point $a \in X$ such that
\beq
\label{eq:min}
d(a, y) \leq d(x, y) \qquad \text{for all } x \in X.
\eeq
Give an example to show that the conclusion may fail if ``compact'' is
replaced by ``closed''.
\sol
Fix $y \in X^c$. Consider the function $f: X \ra [0, \infty)$ given by
\[
f(x) := d(x, y).
\]
We know that $f$ is continuous. Since $X$ is compact, $f$ attains a minimum
at some $a \in X$. The point $a$ satisfies (\ref{eq:min}).
\emph{Counterexample for closed.} Let $M = (0, 1] \cup (2, 3)$ with the
relative metric and let $X = (2, 3)$. Because $X$ is the intersection of
a closed subset $[2, 3]$ of $\R$ with $M$, $X$ is closed in $M$.
Let $y = 1$. For every $a \in X$ we have that $a > 2$. Choose some element
$x \in (2, a)$. Then $x \in X$ and
\[
d(x, y) = x - 1 < a - 1 = d(a, y).
\]
Thus, for $X$ only closed the conclusion of the problem fails.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 44.1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{44.1}
Give an example of metric spaces $M_1$ and $M_2$ and a continuous function $f$
from $M_1$ onto $M_2$ such that $M_2$ is compact, but $M_1$ is not compact.
\sol
Let $M_1 = \R$, let $M_2$ consist of a single point $p$ and let
$f: M_1 \ra M_2$ be given by $f(x) = p$ for all $x \in \R$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 44.6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{44.6}
Let $f$ be a one-to-one function from a metric space $M_1$ onto a metric
space $M_2$. If $f$ and $f^{-1}$ are continuous, we say that $f$ is a
\emph{homeomorphism} and that $M_1$ and $M_2$ are
\emph{homeomorphic} metric spaces.
\bi
\itema
Prove that any two closed intervals of $\R$ are homeomorphic.
\itemb
Prove (a) with ``closed'' replaced by ``open''; then by ``half-open.''
\itemc
Prove that a closed interval is not homeomorphic to either an open interval or
a half-open interval.
\itemd
Let $M$ be a metric space and let $G(M)$ denote the set of homeomorphisms of
$M$ onto $M$.
\bi
\itemi
Prove that $G(M)$ is a group under composition.
\itemii
Identify the group $G(M)$ in case $M$ is finite.
\itemiii
Prove that if $M_1$ and $M_2$ are homeomorphic metric spaces, then
$G(M_1)$ is isomorphic to $G(M_2)$.
\itemiv
Show, by example, that the converse of (iii) does not hold.
\ei
\iteme
Prove that any metric space $M$ is homeomorphic to a metric space
$(M^*, d)$ where $d$ is bounded by 1.
\itemf
Let $M$ be a separable metric space. Prove that there is a one-to-one
continuous function $f$ from $M$ to $H^\infty$.
\itemg
Prove that a metric space $M$ is compact if and only if $M$ is homeomorphic
to a closed subset of $H^\infty$.
\ei
\sol
\bi
\itema
Given two intervals $[a, b]$ and $[c, d]$, the function $f:[a, b] \ra [c, d]$
given by
\beq
\label{eq:fw}
f(x) = c + \frac {x - a}{b - a}(d - c)
\eeq
is continuous with a continuous inverse $g:[c, d] \ra [a, b]$
\beq
\label{eq:bw}
g(y) = a + \frac {y - c}{d - c}(b - a).
\eeq
\itemb
The same formulas (\ref{eq:fw}) and (\ref{eq:bw}) define pairs of continuous
functions which are inverses of each other: $f_{open}:(a, b) \ra (c, d)$ and
$g_{open}:(c, d) \ra (d, c)$, as well as
$f_{left}:[a,b) \ra [c, d)$ and $g_{left}:[c, d) \ra [a, b)$,
as well as
$f_{right}:(a,b] \ra [c, d)$ and $g_{right}:(c, d] \ra [a, b)$.
To show that $[a, b)$ is homeomorphic to $(c, d]$, we consider the continuous
function
$\tilde f: [a, b) \ra (c, d]$ given by
\[
\tilde f(x) = d - \frac {x - a}{b - a}(d - c)
\]
and its continuous inverse
\[
\tilde g(y) = a + \frac {d - y}{d - c}(b - a).
\]
\itemc
Because any closed interval is closed and bounded, it is compact.
Because any open interval and any half-open interval is not closed,
it is not compact.
\itemd
\bi
\itemi
We check the axioms of a group for $G(M)$:
\bi
\item \emph{Closure.} If $f, g: M \ra M$ are homeomorphisms then so is
$f \circ g$, because $f \circ g$ is continuous and has a continuous inverse
$g^{-1} \circ f^{-1}$.
\item \emph{Identity element.} The identity function $\id_M:M \ra M$ given by
$\id_M(x) = x$ satisfies the properties of the identity element of $G(M)$.
Namely, for every $f \in G(M)$ and $x \in M$, we have that
\[
(f \circ \id_M)(x) = f(\id_M(x)) = f(x)
\]
and
\[
(\id_M \circ f)(x) = \id_M(f(x)) = f(x)
\]
\item \emph{Associativity.} For any triple of elements $f, g, h \in G(M)$ and
$x \in M$ we have that
\[
(f \circ (g \circ h))(x) = f((g \circ h)(x)) = f(g(h(x))) =
(f \circ g)(h(x)) = ((f \circ g) \circ h)(x).
\]
\item \emph{Existence of inverses.} For any $f \in G(M)$, the inverse function
$f^{-1}: M \ra M$ is continuous and has a continuous inverse $f$, hence
$f^{-1}\in G(M)$. Moreover, $f^{-1}$ satisfies the properties of a group
inverse: $f \circ f^{-1} = \id_M$ and $f^{-1} \circ f = \id_M$.
\ei
\itemii
Let $M$ be a finite set $\{x_1, \ldots, x_n\}$. Any subset of $M$ is
open, hence by Theorem 40.5iii, any function $f: M \ra M$ is continuous.
Therefore, a function $f: M \ra M$ is a homeomorphism if and only if
it has an inverse function $f^{-1}$, i.e. if and only if $f$ is a bijection.
Such an $f$ is also known as a \emph{permutation}, so $G(M)$ is the
symmetric group on an $n$ letters $S_n$.
\itemiii
Given a homeomorphism $f: M_1 \ra M_2$, one can define an isomorphism
$F: G(M_1) \ra G(M_2)$ by
\[
F(g) = f \circ g \circ f^{-1}
\]
for $g \in G(M_1)$.
Because $f$, $g$, $f^{-1}$ is a homeomorphism, their composition $F(g)$ is
also a homeomorphism. The map $F$ has an inverse $F^{-1}: G(M_2) \ra G(M_1)$
given by
\[
F^{-1}(h) = f^{-1} \circ h \circ f
\]
for $h \in G(M_2)$,
so $F$ is a bijection between $G(M_1)$ and $G(M_2)$. Finally,
$F$ is a group homomorphism because
\[
F(g \circ h) = f \circ g \circ h \circ f^{-1} =
f \circ g \circ f^{-1} \circ f \circ h \circ f^{-1} = F(g) \circ F(h).
\]
\itemiv
Consider the sets $M_1 = (0, 1) \cup \{2\}$ and $M_2 = (0, 1)$.
Note that $M_1$ has an isolated point $2$ and $M_2$ has no isolated points,
so $M_1$ and $M_2$ are not homeomorphic.
We show that the groups $G(M_1)$ and $G(M_2)$ are isomorphic by proving
that every element of $G(M_1)$ restricts to an element of
$G(M_2)$ and every element of $G(M_2)$ uniquely extends to an
element of $G(M_1)$.
Given an element $f$ of $G(M_1)$, we know that
$\{f(2)\}$ is the preimage of a an open set
$\{2\}$ under the continuous map $f^{-1}$, so it has to be open in $M_1$.
Thus, $f(2)$ is an isolated point of $M_1$, so $f(2) = 2$.
Therefore the restriction $\tilde f$ of $f$ to $M_2$ defines an
a continuous function from $M_2$ to itself with a continuous
inverse $\tilde f^{-1}$ being the restriction of $f^{-1}$ to $M_2$.
Conversely, given any element $g \in G(M_2)$ we can uniquely extend it
to $\hat g: M_1 \ra M_1$ by setting $\hat g(2) := 2$. We have that
$\hat g$ is continuous and has a continuous inverse $\hat g^{-1}$
being the extension of $g^{-1}$ to $M_1$ via $\hat g^{-1}(2) = 2$.
\ei
\iteme
Consider the metric $d''$ from Problem 35.7:
\[
d''(x, y) = \min\{d(x, y), 1\}.
\]
By construction $d''$ is bounded by 1.
By Problem 37.10, the identity on $M$ viewed as a function from
$(M, d)$ to $(M, d'')$ is a homeomorphism.
\itemf
By part (e) we can assume that $d$ is bounded by 1. Let
$\{x_n: n\in \N\}$ be a dense subset of $M$. Define
$f: M \ra H^\infty$ by
\[
f(x) = (d(x, x_1), d(x, x_2), d(x, x_3), \ldots).
\]
We prove that $f$ is one-to-one and continuous.
To show that $f$ is one-to-one, it suffices to show that $f(x) \neq f(y)$
whenever $x \neq y$. Assume $x \neq y$ with $x, y \in M$. Then $d(x, y) > 0$.
Since $\{x_n: n\in \N\}$ is dense in $M$, we can choose $n$ such that
\[
d(x_n, x) < \frac {d(x, y)} {2}.
\]
Since by triangle inequality
\[
d(x, y) \leq d(x_n, x) + d(x_n, y),
\]
we have that
\[
d(x_n, y) \geq d(x, y) - d(x_n, x) > \frac {d(x, y)} 2.
\]
In particular, $d(x_n, x) \neq d(x_n, y)$, so the $n^{th}$ term of
$f(x)$ and $f(y)$ is different. Therefore, $f(x) \neq f(y)$.
Next we show that $f$ is continuous. Note that given $\e > 0$ and
$x, y \in M$ such that $d(x, y) < \e$, by triangle inequality we have that
\[
|d(x, x_n) - d(y, x_n)| \leq d(x, y) < \e
\]
for every $n$. Therefore,
\[
d_{H^\infty}(f(x), f(y)) = \sumint n \frac {|d(x, x_n) - d(y, x_n)|} {2^n}
< \sumint n \frac \e {2^n} = \e.
\]
Thus, $f$ is continuous (in fact we just
proved that $f$ is uniformly continuous).
\itemg
Assume that $M$ is compact. Then its image $f(M) \subset H^\infty$ is
also compact and hence closed in $H^\infty$. Since $f$ one-to-one,
by Theorem 44.3, $f$ is a homeomorphism from $M$ to $f(M)$.
Conversely, to show that every closed subset of $H^\infty$ is compact,
by Theorem 43.8 it suffices to show that $H^\infty$ is compact
(Problem 43.6).
Let $\{y^{(n)}\}$ be a sequence of elements of $H^\infty$.
Let $\mc S$ be the set of subsequences of $\{y^{(n)}\}$.
\begin{lemma}
\label{lem:subseq}
There exists $x \in H^\infty$, such that for every positive integer $k$
there exists a sequence $\{z^{(n)}\} \in \mc S$ such that
\[
\limn z^{(n)}_{l} = x_l
\]
for every $l$ from 1 to $k$.
\end{lemma}
Assuming the lemma, we construct a subsequence $\{y^{(n_k)}\}$ of
$\{y^{(n)}\}$ such that
\beq
\label{eq:ineq}
d(y^{(n_k)}, x) < \frac 1 {2^{k - 1}}.
\eeq
inductively as follows.
We can choose $y^{n_0}$ to be anything because $d(x, y) \leq 1$ for
every pair of points in $H^\infty$. Given $\{y^{(n_l)}\}$ with $l < k$
satisfying the condition (\ref{eq:ineq}), by Lemma \ref{lem:subseq} there
exists a subsequence
$\{y^{(m_j)}\}$ of $\{y^{(n)}\}$ such that
\[
\lim_{j \ra \infty} y^{(m_j)}_{l} = x_l
\]
for every $l$ from 1 to $k$. Then for every $l$ from between 1 and k
we can choose $N_l$ such that for every
$j \geq N_l$, we have that
\[
d(y^{(m_j)}_{l}, x_l) < \frac 1 {2^{k}}
\]
Let $n_k := \max(\{n_k + 1\} \cup \{N_l \| l = 1, \ldots, k\})$. Then
\begin{align*}
d(y^{(n_k)}, x)
& = \sum_{l = 1}^k \frac {|y^{(n_k)}_l - x_l|} {2^l}
+ \sum_{l = k + 1}^\infty \frac {|y^{(n_k)}_l - x_l|} {2^l} \\
& \leq \sum_{l = 1}^k \frac 1 {2^l} \cdot \frac 1 {2^k}
+ \sum_{l = k + 1}^\infty \frac 1 {2^l} \\
& = \frac 1 {2^k} \left(1 - \frac 1 {2^k}\right)
+ \frac 1 {2^k} \\
& < \frac 1 {2^{k - 1}},
\end{align*}
as desired. The constructed subsequence $\{y^{(n_k)}\}$ of $\{y^{(n)}\}$
converges to $x$ proving the compactness of $H^\infty$.
\begin{proof}[Proof of Lemma \ref{lem:subseq}]
We construct the desired $x$ term by term, using induction on $k$. For
$k = 0$ the statement is vacuous, so we may take $\{z^{(n)}\}$ to be
the whole sequence $\{y^{(n)}\}$.
Assume that for an integer $k \geq 1$
we have the first $k - 1$ terms $x_1, \ldots, x_{k - 1}$
of $x$ and a sequence $\{z^{(n)}\}$ that satisfies the premise of the
lemma for $k - 1$. Consider the sequence $\{z^{(n)}_k\}$ of $k^{th}$ terms
of $\{z^{(n)}\}$. Since $\{z^{(n)}_k\}$ is a sequence on a compact
metric space $[0, 1]$, it has a convergent subsequence
$\{z^{(n_j)}_k\}$. Let $x_k$ be the limit of $\{z^{(n_j)}_k\}$. Then
the sequence $\{z^{(n_j)}_k\} \in \mc S$ satisfies the premise of the
lemma for $k$. Thus, the inductive step is complete.
\end{proof}
\ei
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{1}
A point $x$ in a metric space is called \emph{isolated} if the set
$\{x\}$ is open. Prove that a complete (non-empty) metric space $M$ without
isolated points has an uncountable number of points.
\textbf{Possible Hint:} Since $M$ is complete, one can produce points in $M$
by producing Cauchy sequences in $M$. Using this reasoning, it suffices to
associate to each element $p$ of an uncountable set $P$ (such as the set of
binary sequences $S_{0, 1}$), a Cauchy sequence in $M$ such that if $p \neq q$,
the Cauchy sequence associated to $p$ must have a different limit from the
Cauchy sequence associated to $q$.
\sol
\begin{lemma}
\label{lem:disjoint-balls}
There exists functions from the set of \emph{finite} binary tuples
$f:S_{0, 1}^{finite} \ra M$ and $\e:S_{0, 1}^{finite} \ra (0, \infty)$
such that $\e(a) < 1 / (length(a) + 1)$
and for $C_a$ being the closed ball of radius $\e(a)$ around $f(a)$:
\[
C_{a} := \{x \in M \| d(f(a), x) \leq \e(a)\},
\]
the following two statements hold.
\bi
\item
Whenever a finite binary tuple $a$ is a prefix of a finite binary
tuple $b$ then then $C_a$ is contained in $C_b$.
\item
Given two finite binary tuple $a$ and $b$ such that neither is a prefix
of the other one, then $C_a$ and $C_b$ are disjoint.
\ei
\end{lemma}
Assuming Lemma \ref{lem:disjoint-balls}, we construct an injective
function $F: S_{0, 1} \ra M$ as follows.
Given any element $s \in S_{0, 1}$, let its
truncation $s^{(k)}$ of length $k$ be the binary $k$-tuple
consisting of the first $k$ elements of $s$:
\[
s^{(k)} = (s_1, \ldots, s_k).
\]
By Lemma \ref{lem:disjoint-balls}, for all $l \geq k$,
$f(s^{(l)})$ is contained in a closed ball of radius
$\e(s^{(k)}) < 1 / k$ around $s^{(k)}$. Therefore,
the sequence $\{s^{(k)}\}_{k\in \N}$ is Cauchy in $M$ and hence
converges to some point $F(s)$ in $M$. Moreover,
$F(s)$ is contained in $C_{s^(k)}$ for every $k$.
We show that
$F$ is injective. Given two distinct elements $s$ and $t$
of $S_{0, 1}$, let $k$ be the first index at such that $s_k \neq t_k$.
Then neither of $s^{(k)}$ and $t^{(k)}$ is a prefix of the other one,
hence, by Lemma \ref{lem:disjoint-balls}, the balls
$C_{s^{(k)}}$ and $C_{t^{(k)}}$ are disjoint. Since,
$F(s) \in C_{s^{(k)}}$ and $F(t) \in C_{t^{(k)}}$, it follows that
$F(s) \neq F(t)$. Thus, $M$ has an uncountable subset
$F(S_{0, 1})$ and, hence, is uncountable.
\begin{proof}[Proof of Lemma \ref{lem:disjoint-balls}]
We construct $f$ and $\e$ by induction on the length $l$ of the finite
binary tuples.
For $l = 0$ we have a single 0-tuple -- the empty tuple
$()$ to which we associate some element $x \in M$: $f(()) = x$.
Let $\e(()) = 1/2$.
Assume that we have constructed $f$ and $\e$ for all binary
$l$ tuples with $l\leq k$. Fix a $k$-tuple $a = (a_1, \ldots, a_k)$.
Next we define $f$ on $a0 = (a_1, \ldots, a_k, 0)$ and
$a1 = (a_1, \ldots, a_k, 1)$.
Let $f(a0) = f(a)$. Since $f(a)$ is not an isolated point, there
exists a point $y \in M$ such that $d(f(a), y) < \e(a)/2$.
Let $f(a1) = y$ and let
\[
\e(a0) = \e(a1) = \min\left\{\frac {d(f(a), y)} 3, \frac 1 {k + 3} \right\}.
\]
By construction $C_{a0}$ and $C_{a1}$ a disjoint and both contained in $C_a$
and $\e(a0)$ and $\e(a1)$ are both smaller than $1 / (k + 2)$.
This way we define $f$ and $\e$ on all $(k + 1)$-tuples. It remains to
check that $f$ and $\e$ still satisfy the two conditions of the lemma.
Given a prefix $a^{(l)}=(a_1, \ldots, a_l)$ of
$a^{(k + 1)}(a_1, \ldots, a_{k + 1})$,
by inductive assumption $C_{a^{(k)}}$ is contained $C_{a^{(l)}}$.
By the discussion in the previous paragraph $C_{a^{(k + 1)}}$
is contained in $C_{a^{(k)}}$, so $C_{a^{(k)}}$ is contained in
$C_{a^{(l)}}$.
Given two tuples $a$ and $b$ of length at most $k + 1$ that are not
prefixes of one another, we have three cases:
\bi
\item Case 1: the largest common prefix of $a$ and $b$ is of length $l < k$.
Then the tuples $a^{(l + 1)}$ and $b^{(l + 1)}$ of the first $(l + 1)$
elements of $a$ and $b$ are not prefixes of one another, so by inductive
assumption
$C_{a^{(l + 1)}}$ and $C_{b^{(l + 1)}}$ are disjoint. Since
$C_a$ and $C_b$ are contained in $C_{a^{(l + 1)}}$ and $C_{b^{(l + 1)}}$,
respectively, they are also disjoint.
\item Case 2: the largest common prefix $c$ of $a$ and $b$ is of length $k$.
Then then $\{a, b\} = \{c0, c1\}$ and we already mentioned that
$C_a$ and $C_b$ are disjoint right after the construction.
\ei
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{2}
If $f: M \ra N$ is a function, then recall that the \emph{graph of $f$} is
the following subset of $M \times N$:
\[
\Gamma_f := \{(m, f(m)) \| m \in M\}.
\]
On your midterm exam you were asked to prove that if $M$ and $N$ are metric
spaces and $f$ is continuous, then $\Gamma_f$ is a closed subset of
$M \times N$ (equipped with the product metric). This question explores the
converse assertion.
\bi
\itema
It is not always true that $\Gamma_f$ is closed implies $f$ is continuous.
Give an example (with justification) of a non-continuous function $f$ whose
graph is closed.
\itemb
Show that if the target $N$ is a \emph{compact} metric space, and
$\Gamma_f$ is closed, then $f$ is continuous.
\ei
\sol
\bi
\itema
Consider $f:[0, \infty) \ra [0, \infty)$ given by
\[
f(x) = \manylinedefinition{
0 & \text{if $x = 0$,} \\
\frac 1 x & \text{if $x > 0$.}
}
\]
We know that $g$ is not continuous, because $\lim_{x \ra 0} f(x)$
does not exist.
The graph of $f$ is s a union of $\{(0, 0)\}$ and the graph of
$g: (0, \infty) \ra [0, \infty)$ given by $g(x) = 1/x$. Hence,
if $\Gamma_g$ is closed in $[0, \infty) \times [0, \infty)$, then so is
$\Gamma_f$.
Let $(x, y)$ be a limit point of $\Gamma_g$ in $[0, \infty)$. Then there
exists a sequence $\{(x_i, y_i)\}$ in $\Gamma_g$ converging to $(x, y)$.
Then \sequence x and \sequence y converge to $x$ and $y$, respectively,
in $[0, \infty)$. Hence, on one hand, $x_iy_i$ converges to $xy$.
On the other hand, since $\{(x_i, y_i)\} \in \Gamma_g$, we have that
$x_iy_i = 1$, so $\{x_ny_n\}$ converges to 1. Thus, $xy = 1$, so $x > 0$
and $y = 1/x$, i.e. $(x, y) \in \Gamma_g$. Thus, $\Gamma_g$ contains all
of its limit points in $[0, \infty) \times [0, \infty)$.
\itemb
Assume $N$ is compact, $\Gamma_f$ is closed $f$ is not continuous to get a
contradiction.
Then there exists a sequence \sequence x in $M$ converging to some $x \in M$
such that $\{f(x_n)\}$ does not converge to $f(x)$. Therefore, there
exists $\epsilon > 0$ such that for infinitely many $n$,
\[d(f(x_n), f(x)) \geq \epsilon.
\]
Therefore, there exists a subsequence
\subsequence x of \sequence x such that
\beq
\label{eq:geq-e}
d(f(x_{n_k}), f(x))) \geq \epsilon, \> \forall k.
\eeq
Since $N$ is compact, there exists a subsequence
$\{f(x_{n_{k_l}})\}$ of $\{f(x_{n_k})\}$ that converges to some $n \in N$.
Then the sequence of points $\{(x_{n_{k_l}}, f(x_{n_{k_l}})\}$ converges
to $(x, n) \in M \times N$. Since each $\{(x_{n_{k_l}}, f(x_{n_{k_l}})\}$
is a point on the graph $\Gamma_f$ and $\Gamma_f$ is closed,
$(x, n) \in \Gamma_f$, i. e. $n = f(x)$.
However, by assumption (\ref{eq:geq-e}) $\{f(x_{n_{k_l}})\}$ cannot
converge to $f(x)$ leading to a contradiction.
\ei
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{3}
A metric space $M$ is \emph{locally compact} if every point $x$ has a
\emph{compact neighborhood} $K$ which is by definition a compact set in $M$
whose interior contains $x$.
\bi
\itema
Prove that $\R^n$ is locally compact.
\itemb
Prove that $\Q$ is not locally compact, and in fact that no point in $\Q$ has
a compact neighborhood.
\ei
\sol
\bi
\itema
Every point $x \in \R^n$ is contained in a compact set $[x - 1, x + 1]$ whose
interior $(x - 1, x + 1)$ contains $x$.
\itemb
Assume that $x \in \Q$ has a compact neighborhood $K$. Since the interior of
$K$ contains $x$, $B_{\e}^{\Q}(x) \subset K$ for some $\e > 0$. By denseness of
irrational numbers, we can pick an irrational number $y \in (x - \e, x + \e)$.
By denseness of rationals we can pick a sequence of rationals
\sequence q in $(x - \e, x + \e)$ converging to $y$. Then, by compactness
of $K$, \sequence q has a subsequence \subsequence q converging to some
$z \in K$ in the relative metric. Since the relative metric is the restriction
of the Euclidean metric, \subsequence q converges to $z$ in $\R$. However,
since we assumed that \sequence q converges to $y$ in $\R$,
$z = y$ which contradicts our assumptions that $y$ is irrational
and $z$ is rational. Thus, $x$ has not compact neighborhood.
\ei
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{4}
A collection $\mc F$ of subsets of a set $X$ is said to have the
\emph{finite intersection property} if
$F_1 \cap \cdots \cap F_n \neq \emptyset$
for any $n$ and any $F_1, \ldots, F_n \in \mc F$ (i.e. finite intersections
in $\mc F$ are non-empty). Prove that a metric space $M$ is compact if and
only if $\bigcap_{F \in \mc F} F \neq \emptyset$ for every family $\mc F$ of
closed subsets of $M$ with the finite intersection property.
\sol
We have the following chain of equivalences.
\bi
\item[] $M$ is compact.
\itemeq
Every family $\mc U$ of open sets in $M$
such that $\bigcup_{U \in \mc U} U = M$
has a finite subfamily $U_1, \ldots, U_n$ such that
$\bigcup_{i = 1}^n U_i = M$.
\itemeq
If $\mc U$ is a family of open sets in $M$ such that every
finite subfamily $U_1, \ldots, U_n$ of $\mc U$ satisfies
$\bigcup_{i = 1}^n U_i \neq M$ then
$\bigcup_{U \in \mc U}^n U \neq M$.
\itemeq
If $\mc U$ is a family of open sets in $M$ such that every
finite subfamily $U_1, \ldots, U_n$ of $\mc U$ satisfies
$\bigcap_{i = 1}^n U_i^c \neq \emptyset$ then
$\bigcap_{U \in \mc U}^n U^c \neq \emptyset$.
\itemeq
If $\mc F$ is a family of closed sets in $M$ such that every
finite subfamily $F_1, \ldots, F_n$ of $\mc F$ satisfies
$\bigcap_{i = 1}^n F_i \neq \emptyset$ then
$\bigcap_{F \in \mc F}^n F \neq \emptyset$.
\ei
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%problem 5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pb{5. Lebesgue's Covering Lemma}
An important fact that we have proved and used several times in class
(without stating the name) is call \textbf{Lebesgue's Covering Lemma}: If
$M$ is a compact metric space and $\mc U$ is any open cover of $M$, then there
exists a $\delta > 0$ (depending only on the cover), such that any
$\delta$-ball $B_{\delta}(x)$ (with $x \in M$) is contained in some element
$U$ of $\mc U$. (In your textbook, this appears as Lemma 43.3, as an
intermediate step in proving sequential compactness implies compactness. It
also is used to show that any continuous function from a compact metric space
is uniformly continous, see Theorem 44.5). Any such $\delta$ which satisfies
the condition above is called a \emph{Lebesgue number} for the cover $\mc U$.
\bi
\itema
Lemma 43.3 in the book uses the sequential compactness property of $M$ to prove
Lebesgue's covering lemma. Give another proof of Lebesgue's covering lemma
directly from the definition of compactness, in terms of every open cover
admitting a finite subcover.
\itemb
Show by example that Lebesgue's covering lemma is false when $M$ is
non-compact.
\ei
\sol
\bi
\itema
Because $\mc U$ is a cover of $M$, for every $x \in M$ we can choose an
element $U_x$ of $\mc U$ that contains $x$.
Because $U_x$ is open, we can choose $\delta_x > 0$ such that
$B_{\delta_x}(x)$ is contained in $U_x$. The collection
\[
\{B_{\delta_x / 2}(x) \| x \in M\}
\]
forms an open cover of $M$. By compactness of $M$, there exists
a finite subcollection $\{B_{\delta_{x_i} / 2}(x_i) \| i = 1, \ldots, n\}$
that covers $n$. We claim that
\[
\delta := \min\{\frac{\delta_{x_i}} 2 \| i = 1, \ldots, n \}.
\]
is a Lebesgue number of $\mc U$. Indeed, given any $x \in M$,
by assumptions, there exists $i$ such that
$x \in B_{\delta_{x_i} / 2}(x_i)$ and a $U \in \mc U$ such that
$B_{\delta_{x_i}}(x_i) \subset U$.
Then for any $y \in B_{\delta}(x)$ we have that
\[
d(y, x_i) \leq d(y, x) + d(x, x_i) < \delta + \frac {\delta_i} 2 \leq
\delta_i.
\]
Hence, $B_{\delta}(x) \subset B_{\delta_i}(x_i) \subset U$, as
desired.
\itemb
For example, take $M := (0, \infty)$ and
\[
\mc U := \{(r, 3r) \| r \in (0, \infty)\}.
\]
The collection $\mc U$ forms an open cover of $M$ because for every
$x \in M$, $x \in (r, 2r)$ for $r = 2x / 3$.
However, we show that for every $\delta > 0$, there exists $x \in M$ such that
the ball $B_{\delta}(x)$ is not contained in any of the elements of $M$.
Given $\delta > 0$, let $x = \delta / 3$. Then if an element
$(r, 2r)$ of $\mc U$ contains $x$, then $r < \delta / 3 $, so
$2r < 2\delta/3$. In particular, the element $x + 2\delta / 3 = \delta$
of $B_{\delta}(x)$ is not contained in $(r, 2r)$.
\ei
\end{document}