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

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 $$a_0{z}^n+a_1{z}^{n-1}+\cdots +a_{n-1}z+a_n=0\text{.}$$ Prove that the set of all algebraic numbers is countable. Hint: For every positive integer $N$ there are only finitely many equations with $$n+\left|a_0\right|+\left|a_1\right|+\cdots +\left|a_n\right|=N\text{.}$$
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 $\bar{E}$ have the same limit points. (Recall that $\bar{E}=E\cup E\prime \text{.)}$ 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 ${\bar{B}}_n=\bigcup _{i=1}^n{\bar{A}}_i,$ for $n=1,2,3,\dots \text{.}$
   2. If $B=\bigcup _{i=1}^{\infty }{A}_i,$ prove that $\bar{B}\supset \bigcup _{i=1}^{\infty }{\bar{A}}_i\text{.}$
   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 $\bar{E}$ 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 $$d(p,q)=\{\begin{array}{cc}1& \text{(if}\hspace{0.17em}p\ne q\text{)}\\ 0& \text{(if}\hspace{0.17em}p=q\text{).}\end{array}$$ Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?
11. For $x\in R^1$ and $y\in R^1,$ define $$\begin{array}{ccc}{d}_1(x,y)& =& {(x-y)}^2,\\ d_2(x,y)& =& \sqrt{|x-y|},\\ d^3(x,y)& =& |x^2-y^2|,\\ d_4(x,y)& =& |x-2y|,\\ d_5(x,y)& =& \frac{|x-y|}{1+|x-y|}\text{.}\end{array}$$ 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 $(0,1)$ 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 $d(p,q)=|p-q|\text{.}$ 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 $x\in [0,1]$ 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?
18. Is there a nonempty perfect set in $R^1$ which contains no rational number?
19. 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 $d(p,q)<\delta ,$ 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).
20. Are closures and interiors of connected sets always connected? (Look at subsets of $R^2\text{.)}$
21. Let $A$ and $B$ be separated subsets of $R^k,$ suppose $\text{a}\in A,$ $\text{b}\in B,$ and define $$\text{p}\left(t\right)=(1-t)\text{a}+t\text{b}$$ 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 $\text{p}\left(t\right)\in A\text{.]}$
    1. Prove that $A_0$ and $B_0$ are separated subsets of $R^1\text{.}$
    2. Prove that there exists $t_0\in (0,1)$ such that $\text{p}\left(t_0\right)\notin A\cup B\text{.}$
    3. Prove that every convex subset of $R^k$ is connected.
22. 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.
23. 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 $X\text{.}$
24. 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.
25. 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 $K\text{.}$
26. 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.
27. 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 $P=W^c\text{.}$
28. 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.
29. Prove that every open set in $R^1$ is the union of an at most countable collection of disjoint segments. Hint: Use Exercise 22.
30. 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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