## Baby Rudin Problems

Last update: 19 March 2014

## Chapter 2

1. Prove that the empty set is a subset of every set.
2. A complex number $z$ is said to be algebraic if there are integers ${a}_{0},\dots ,{a}_{n},$ not all zero, such that $a0zn+a1zn-1 +⋯+an-1z+an=0.$ Prove that the set of all algebraic numbers is countable. Hint: For every positive integer $N$ there are only finitely many equations with $n+|a0|+|a1| +⋯+|an|=N.$
3. Prove that there exist real numbers which are not algebraic.
4. Is the set of all irrational real numbers countable?
5. Construct a bounded set of real numbers with exactly three limit points.
6. 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 $\stackrel{‾}{E}$ have the same limit points. (Recall that $\stackrel{‾}{E}=E\cup E\prime \text{.)}$ Do $E$ and $E\prime$ always have the same limit points?
7. Let ${A}_{1},{A}_{2},{A}_{3},\dots$ be subsets of a metric space.
1. If ${B}_{n}=\bigcup _{i=1}^{n}{A}_{i},$ prove that ${\stackrel{‾}{B}}_{n}=\bigcup _{i=1}^{n}{\stackrel{‾}{A}}_{i},$ for $n=1,2,3,\dots \text{.}$
2. If $B=\bigcup _{i=1}^{\infty }{A}_{i},$ prove that $\stackrel{‾}{B}\supset \bigcup _{i=1}^{\infty }{\stackrel{‾}{A}}_{i}\text{.}$
Show, by an example, that this inclusion can be proper.
8. 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 ${R}^{2}\text{.}$
9. 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 $E\text{.]}$
1. Prove that ${E}^{\circ }$ is always open.
2. Prove that $E$ is open if and only if ${E}^{\circ }=E\text{.}$
3. If $G\subseteq E$ and $G$ is open, prove that $G\subset {E}^{\circ }\text{.}$
4. Prove that the complement of ${E}^{\circ }$ is the closure of the complement of $E\text{.}$
5. Do $E$ and $\stackrel{‾}{E}$ always have the same interiors?
6. Do $E$ and ${E}^{\circ }$ always have the same closures?
10. Let $X$ be an infinite set. For $p\in X$ and $q\in X,$ define $d(p,q)= { 1 (if p≠q) 0 (if p=q).$ Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?
11. For $x\in {R}^{1}$ and $y\in {R}^{1},$ define $d1(x,y) = (x-y)2, d2(x,y) = |x-y|, d3(x,y) = |x2-y2|, d4(x,y) = |x-2y|, d5(x,y) = |x-y|1+|x-y|.$ Determine, for each of these, whether it is a metric or not.
12. 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 Heine-Borel theorem).
13. Construct a compact set of real numbers whose limit points form a countable set.
14. Give an example of an open cover of the segment $\left(0,1\right)$ which has no finite subcover.
15. 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."
16. Regard $Q,$ the set of all rational numbers, as a metric space, with $d\left(p,q\right)=|p-q|\text{.}$ 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 $Q\text{?}$
17. Let $E$ be the set of all $x\in \left[0,1\right]$ whose decimal expansion contains only the digits $4$ and $7\text{.}$ Is $E$ countable? Is $E$ dense in $\left[0,1\right]\text{?}$ Is $E$ compact? Is $E$ perfect?
18. Is there a nonempty perfect set in ${R}^{1}$ which contains no rational number?
1. If $A$ and $B$ are disjoint closed sets in some metric space $X,$ prove that they are separated.
2. Prove the same for disjoint open sets.
3. Fix $p\in X,$ $\delta >0,$ define $A$ to be the set of all $q\in X$ for which $d\left(p,q\right)<\delta ,$ define $B$ similarly, with $>$ in place of $<\text{.}$ Prove that $A$ and $B$ are separated.
4. Prove that every connected metric space with at least two points is uncountable. Hint: Use (c).
19. Are closures and interiors of connected sets always connected? (Look at subsets of ${R}^{2}\text{.)}$
20. Let $A$ and $B$ be separated subsets of ${R}^{k},$ suppose $\text{a}\in A,$ $\text{b}\in B,$ and define $p(t)= (1-t)a+ tb$ 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 $\text{p}\left(t\right)\in A\text{.]}$
1. Prove that ${A}_{0}$ and ${B}_{0}$ are separated subsets of ${R}^{1}\text{.}$
2. Prove that there exists ${t}_{0}\in \left(0,1\right)$ such that $\text{p}\left({t}_{0}\right)\notin A\cup B\text{.}$
3. Prove that every convex subset of ${R}^{k}$ is connected.
21. 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.
22. 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 $X\text{.}$
23. 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 $d\left({x}_{i},{x}_{j+1}\right)\ge \delta$ 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 $\delta =1/n$ $\left(n=1,2,3,\dots \right),$ and consider the centers of the corresponding neighborhoods.
24. 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 $K\text{.}$
25. 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.
26. 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 $P={W}^{c}\text{.}$
27. 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.
28. Prove that every open set in ${R}^{1}$ is the union of an at most countable collection of disjoint segments. Hint: Use Exercise 22.
29. 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.)