Baby Rudin Problems
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last update: 19 March 2014
Chapter 2

Prove that the empty set is a subset of every set.

A complex number $z$ is said to be algebraic if there are integers ${a}_{0},\dots ,{a}_{n},$
not all zero, such that
$${a}_{0}{z}^{n}+{a}_{1}{z}^{n1}+\cdots +{a}_{n1}z+{a}_{n}=0\text{.}$$
Prove that the set of all algebraic numbers is countable. Hint: For every positive integer $N$ there are only finitely many equations with
$$n+\left{a}_{0}\right+\left{a}_{1}\right+\cdots +\left{a}_{n}\right=N\text{.}$$

Prove that there exist real numbers which are not algebraic.

Is the set of all irrational real numbers countable?

Construct a bounded set of real numbers with exactly three limit points.

Let $E\prime $ be the set of all limit points of a set $E\text{.}$ Prove that
$E\prime $ is closed. Prove that $E$ and $\stackrel{\u203e}{E}$
have the same limit points. (Recall that $\stackrel{\u203e}{E}=E\cup E\prime \text{.)}$
Do $E$ and $E\prime $ always have the same limit points?

Let ${A}_{1},{A}_{2},{A}_{3},\dots $
be subsets of a metric space.

If ${B}_{n}=\bigcup _{i=1}^{n}{A}_{i},$
prove that ${\stackrel{\u203e}{B}}_{n}=\bigcup _{i=1}^{n}{\stackrel{\u203e}{A}}_{i},$
for $n=1,2,3,\dots \text{.}$

If $B=\bigcup _{i=1}^{\infty}{A}_{i},$
prove that $\stackrel{\u203e}{B}\supset \bigcup _{i=1}^{\infty}{\stackrel{\u203e}{A}}_{i}\text{.}$
Show, by an example, that this inclusion can be proper.

Is every point of every open set $E\subset {R}^{2}$ a limit point of $E\text{?}$
Answer the same question for closed sets in ${R}^{2}\text{.}$

Let ${E}^{\circ}$ denote the set of all interior points of a set $E\text{.}$
[See Definition 2.18(e); ${E}^{\circ}$ is called the interior of $E\text{.]}$

Prove that ${E}^{\circ}$ is always open.

Prove that $E$ is open if and only if ${E}^{\circ}=E\text{.}$

If $G\subseteq E$ and $G$ is open, prove that
$G\subset {E}^{\circ}\text{.}$

Prove that the complement of ${E}^{\circ}$ is the closure of the complement of $E\text{.}$

Do $E$ and $\stackrel{\u203e}{E}$ always have the same interiors?

Do $E$ and ${E}^{\circ}$ always have the same closures?

Let $X$ be an infinite set. For $p\in X$ and $q\in X,$ define
$$d(p,q)=\{\begin{array}{cc}1& \text{(if}\hspace{0.17em}p\ne q\text{)}\\ 0& \text{(if}\hspace{0.17em}p=q\text{).}\end{array}$$
Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?

For $x\in {R}^{1}$ and $y\in {R}^{1},$ define
$$\begin{array}{ccc}{d}_{1}(x,y)& =& {(xy)}^{2},\\ {d}_{2}(x,y)& =& \sqrt{xy},\\ {d}^{3}(x,y)& =& {x}^{2}{y}^{2},\\ {d}_{4}(x,y)& =& x2y,\\ {d}_{5}(x,y)& =& \frac{xy}{1+xy}\text{.}\end{array}$$
Determine, for each of these, whether it is a metric or not.

Let $K\subset {R}^{1}$ consist of $0$ and the numbers
$1/n,$ for $n=1,2,3,\dots \text{.}$
Prove that $K$ is compact directly from the definition (without using the HeineBorel theorem).

Construct a compact set of real numbers whose limit points form a countable set.

Give an example of an open cover of the segment $(0,1)$ which has no finite subcover.

Show that Theorem 2.36 and its Corollary become false (in ${R}^{1},$ for example) if the word
"compact" is replaced by "closed" or by "bounded."

Regard $Q,$ the set of all rational numbers, as a metric space, with
$d(p,q)=pq\text{.}$
Let $E$ be the set of all $p\in Q$ such that $2<{p}^{2}<3\text{.}$
Show that $E$ is closed and bounded in $Q,$ but that $E$ is not compact. Is
$E$ open in $Q\text{?}$

Let $E$ be the set of all $x\in [0,1]$ whose decimal
expansion contains only the digits $4$ and $7\text{.}$ Is $E$ countable?
Is $E$ dense in $[0,1]\text{?}$ Is
$E$ compact? Is $E$ perfect?

Is there a nonempty perfect set in ${R}^{1}$ which contains no rational number?


If $A$ and $B$ are disjoint closed sets in some metric space $X,$
prove that they are separated.

Prove the same for disjoint open sets.

Fix $p\in X,$ $\delta >0,$
define $A$ to be the set of all $q\in X$ for which
$d(p,q)<\delta ,$ define
$B$ similarly, with $>$ in place of $<\text{.}$ Prove that
$A$ and $B$ are separated.

Prove that every connected metric space with at least two points is uncountable. Hint: Use (c).

Are closures and interiors of connected sets always connected? (Look at subsets of ${R}^{2}\text{.)}$

Let $A$ and $B$ be separated subsets of ${R}^{k},$ suppose
$\text{a}\in A,$ $\text{b}\in B,$ and define
$$\text{p}\left(t\right)=(1t)\text{a}+t\text{b}$$
for $t\in {R}^{1}\text{.}$ Put
${A}_{0}={\text{p}}^{1}\left(A\right),$
${B}_{0}={\text{p}}^{1}\left(B\right)\text{.}$
[Thus $t\in {A}_{0}$ if and only if
$\text{p}\left(t\right)\in A\text{.]}$

Prove that ${A}_{0}$ and ${B}_{0}$ are separated subsets of
${R}^{1}\text{.}$

Prove that there exists ${t}_{0}\in (0,1)$ such that
$\text{p}\left({t}_{0}\right)\notin A\cup B\text{.}$

Prove that every convex subset of ${R}^{k}$ is connected.

A metric space is called separable if it contains a countable dense subset. Show that ${R}^{k}$ is separable.
Hint: Consider the set of points which have only rational coordinates.

A collection $\left\{{V}_{\alpha}\right\}$ of open subsets of $X$ is said to be a
base for $X$ if the following is true: For every $x\in X$ and every open set
$G\subset X$ such that $x\in G,$ we have
$x\in {V}_{\alpha}\subset G$ for some $\alpha \text{.}$
In other words, every open set in $X$ is the union of a subcollection of $\left\{{V}_{\alpha}\right\}\text{.}$
Prove that every separable metric space has a countable base. Hint: Take all neighborhoods with rational radius and center in some countable
dense subset of $X\text{.}$

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.
Hint: Fix $\delta >0,$ and pick ${x}_{1}\in X\text{.}$
Having chosen ${x}_{1},\dots ,{x}_{j}\in X,$
choose ${x}_{j+1}\in X,$ if possible, so that
$d({x}_{i},{x}_{j+1})\ge \delta $
for $i=1,\dots ,j\text{.}$ Show that this
process must stop after a finite number of steps, and that $X$ can therefore be covered by finitely many neighborhoods of radius
$\delta \text{.}$ Take $\delta =1/n$
$(n=1,2,3,\dots ),$
and consider the centers of the corresponding neighborhoods.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.
Hint: For every positive integer $n,$ there are finitely many neighborhoods of radius
$1/n$ whose union covers $K\text{.}$

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact.
Hint: By Exercises 23 and 24, $X$ has a countable base. It follows that every open cover of $X$ has a
countable subcover $\left\{{G}_{n}\right\},$
$n=1,2,3,\dots \text{.}$
If no finite subcollection of $\left\{{G}_{n}\right\}$ covers $X,$ then
the complement ${F}_{n}$ of ${G}_{1}\cup \cdots \cup {G}_{n}$
is nonempty for each $n,$ but $\bigcap {F}_{n}$ is empty. If
$E$ is a set which contains a point from each ${F}_{n},$ consider a limit point of
$E,$ and obtain a contradiction.

Define a point $p$ in a metric space $X$ to be a condensation point of a set
$E\subset X$ if every neighborhood of $p$ contains uncountable many points of $E\text{.}$
Suppose $E\subset {R}^{k},$ $E$ is uncountable, and let $P$
be the set of all condensation points of $E\text{.}$ Prove that $P$ is perfect and that at most countable
many points of $E$ are not in $P\text{.}$ In other words, show that
${P}^{c}\cap E$ is at most countable. Hint: Let
$\left\{{V}_{n}\right\}$ be a countable base of ${R}^{k},$
let $W$ be the union of those ${V}_{n}$ for which $E\cap {V}_{n}$
is at most countable, and show that $P={W}^{c}\text{.}$

Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.
(Corollary: Every countable closed set in ${R}^{k}$ has isolated points.) Hint: Use Exercise 27.

Prove that every open set in ${R}^{1}$ is the union of an at most countable collection of disjoint segments.
Hint: Use Exercise 22.

Imitate the proof of Theorem 2.43 to obtain the following result:
If ${R}^{k}=\bigcup _{1}^{\infty}{F}_{n},$
where each ${F}_{n}$ is a closed subset of ${R}^{k},$ then at least one
${F}_{n}$ has a nonempty interior.
Equivalent statement: If ${G}_{n}$ is a dense open subset of ${R}^{k},$
for $n=1,2,3,\dots ,$ then
$\bigcap _{1}^{\infty}{G}_{n}$ is not empty (in fact, it is dense in
${R}^{k}\text{).}$
(This is a special case of Baire's theorem; see Exercise 22, Chap. 3, for the general case.)
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