Baby Rudin Problems

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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 a0,,an, 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 be the set of all limit points of a set E. Prove that E is closed. Prove that E and E have the same limit points. (Recall that E=EE.) Do E and E always have the same limit points?
  7. Let A1,A2,A3, be subsets of a metric space.
    1. If Bn=i=1nAi, prove that Bn=i=1nAi, for n=1,2,3,.
    2. If B=i=1Ai, prove that Bi=1Ai.
    Show, by an example, that this inclusion can be proper.
  8. Is every point of every open set ER2 a limit point of E? Answer the same question for closed sets in R2.
  9. Let E denote the set of all interior points of a set E. [See Definition 2.18(e); E is called the interior of E.]
    1. Prove that E is always open.
    2. Prove that E is open if and only if E=E.
    3. If GE and G is open, prove that GE.
    4. Prove that the complement of E is the closure of the complement of E.
    5. Do E and E always have the same interiors?
    6. Do E and E always have the same closures?
  10. Let X be an infinite set. For pX and qX, define d(p,q)= { 1 (ifpq) 0 (ifp=q). Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are compact?
  11. For xR1 and yR1, 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 KR1 consist of 0 and the numbers 1/n, for n=1,2,3,. 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 R1, 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|. Let E be the set of all pQ such that 2<p2<3. Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q?
  17. Let E be the set of all x[0,1] whose decimal expansion contains only the digits 4 and 7. Is E countable? Is E dense in [0,1]? Is E compact? Is E perfect?
  18. Is there a nonempty perfect set in R1 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 pX, δ>0, define A to be the set of all qX for which d(p,q)<δ, define B similarly, with > in place of <. 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 R2.)
  20. Let A and B be separated subsets of Rk, suppose aA, bB, and define p(t)= (1-t)a+ tb for tR1. Put A0=p-1(A), B0=p-1(B). [Thus tA0 if and only if p(t)A.]
    1. Prove that A0 and B0 are separated subsets of R1.
    2. Prove that there exists t0(0,1) such that p(t0)AB.
    3. Prove that every convex subset of Rk is connected.
  21. A metric space is called separable if it contains a countable dense subset. Show that Rk is separable. Hint: Consider the set of points which have only rational coordinates.
  22. A collection {Vα} of open subsets of X is said to be a base for X if the following is true: For every xX and every open set GX such that xG, we have xVαG for some α. In other words, every open set in X is the union of a subcollection of {Vα}. 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.
  23. Let X be a metric space in which every infinite subset has a limit point. Prove that X is separable. Hint: Fix δ>0, and pick x1X. Having chosen x1,,xjX, choose xj+1X, if possible, so that d(xi,xj+1)δ for i=1,,j. 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 δ. Take δ=1/n (n=1,2,3,), 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.
  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 {Gn}, n=1,2,3,. If no finite subcollection of {Gn} covers X, then the complement Fn of G1Gn is nonempty for each n, but Fn is empty. If E is a set which contains a point from each Fn, 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 EX if every neighborhood of p contains uncountable many points of E. Suppose ERk, E is uncountable, and let P be the set of all condensation points of E. Prove that P is perfect and that at most countable many points of E are not in P. In other words, show that PcE is at most countable. Hint: Let {Vn} be a countable base of Rk, let W be the union of those Vn for which EVn is at most countable, and show that P=Wc.
  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 Rk has isolated points.) Hint: Use Exercise 27.
  28. Prove that every open set in R1 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 Rk=1Fn, where each Fn is a closed subset of Rk, then at least one Fn has a nonempty interior.
    Equivalent statement: If Gn is a dense open subset of Rk, for n=1,2,3,, then 1Gn is not empty (in fact, it is dense in Rk). (This is a special case of Baire's theorem; see Exercise 22, Chap. 3, for the general case.)

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