A course on set theory

Corrections

• Page 77, Exercise 4.16, last sentence before the hint should instead say: y is maximal iff there is no z such that y ⊴ z but not z ⊴ y.
• Page 84, line 8: d(x,y) = |x-y|.
• Page 96: Exercise 5.12 is incorrectly stated; a correct version will be provided here later.
• Page 114, line 6: n₂ =
• Page 121, line 24: then pick k ∈ ω - ran(s_α)
• Page 129, line 22: dom(g) = dom(f_n) ∪ {a_n}
• Page 149, lines 15-17: "Instead of saying ultraproduct, we use the term ultrapower in this case because all the pairs (A_n,R_n) are the same."
• Pages 146-147: Exercise 7.4 should be ignored, it is wrong.

Additional exercises

1. (Section 7.2)
   1. Let < A_α | α < ω₁ > be a sequence such that
      • for every α < ω₁, A_α is a finite subset of ω₁, and
      • for all α < β < ω₁, A_α ≠ A_β.
      1. Prove that there is a stationary subset I of ω₁ and a set R such that for every α ∈ I, A_α ∩ α = R.
         Hint: Use Fodor's lemma.
      2. Prove that there exists an uncountable subset J of I such that for all α < β from J, max(A_α) < min(A_β - R).
      3. Conclude that for all α < β from J, A_α ∩ A_β = R.
   2. (Δ-system lemma) Let F be an uncountable family of finite sets. Prove that there is a set R and an uncountable family D ⊆ F such that for all distinct A, B ∈ D, A ∩ B = R.
2. (Section 7.2) Let R and S be isomorphic wellorderings of ω₁. Let I be the collection of α < ω₁ such that R ∩ (α x α) and S ∩ (α x α) are isomorphic.
   1. Prove that I contains a club.
   2. Give an example to show that I might not be closed.
      Hint: Try using 1 R 2 R 3 R ... 0 R ω R ω + 1 R ...
3. (Section 7.2) Let θ be a limit ordinal and T = { θ - C | C is a closed subset of θ }.
   1. Prove that T is a topology on θ
   2. Consider the intervals of ordinals of the form (α,β), [α,β), (α,β] and [α,β]. Which are open? Closed? Make a chart.
      Hint: It will depend on the sort of interval and on properties of the endpoints.
   3. Prove that every open set is a union of open intervals.
   4. Prove that the topology is not compact.
4. (Chapter 4) Assume the Continuum Hypothesis. That is, 2^ω = ω₁. Prove that there is a family F of subsets of ω₁ such that the cardinality of F is 2^ω₁ and, for all distinct members A and B of F, A ∩ B is countable.
   Hint: One possibility is to model your proof on Exercise 4.1.
5. (Chapter 6) Let (A,<_A) be a dense linear ordering without endpoints that has the least upper bound property. Give a direct proof that A has cardinality at least 2^ω.
   Hint: Use the assumption that (A,<_A) is a dense linear ordering without endpoints to find an injection from ^<ω2 to A such that is order perserving in a certain useful way. Then use the assumption that A has the least upper bound property to define an injection from ^ω2 to A.
6. (Section 7.2) Let λ be a regular uncountable cardinal and S be a stationary subset of λ. Let T = { α ∈ S | α = sup(α ∩ S) }. Prove that T is stationary in λ.
7. (Section 5.2) Let X and Y be topological spaces and f be a function from X to Y. We say that f is continuous iff for every open subset V of Y, f^-1[V] is an open subset of X. We say that f is a homeomorphism iff f is a bijection and both f and f^-1 are continuous.
   1. Let f be a function from the Baire Space to itself. Prove that the following are equivalent.
      1. f is continuous.
      2. f^-1[N_s] is open for every basic open set N_s.
      3. f^-1[C] is closed for every closed set C.
      4. If < x_i | i < ω > is a sequence that converges to y, then < f(x_i) | i < ω > is a sequence that converges to f(y).
   2. Prove that the identity function is a continuous injection from the Cantor Space to the Baire Space.
   3. Let A = { x ∈ ^ω2 | x(n) = 1 for infinitely many n < ω }. Find a homeomorphism between the Baire Space and the set A (with the topology that A inherits from the Cantor Space).
   4. (Wadge reduction) For subsets A and B of the Baire space, define A <_W B iff there is a continuous function f from the Baire space to itself such that A = f^-1[B].
      1. Prove that if B is not ^ωω, then ∅ <_W B.
      2. Prove that if B is not ∅, then ^ωω <_W B.
      3. Prove that ∅ and ^ωω are <_W- incomparable. That is, neither ∅ <_W ^ωω nor ^ωω <_W ∅.
      4. Prove that if A is clopen and B is neither ∅ nor ^ωω, then A <_W B.
      5. Prove that if A is open and B is not closed, then A <_W B.
      6. Prove that if A is closed and B is not open, then A <_W B.
      7. Prove that if A is closed but not open and B is open but not closed, then A and B are <_W-incomparable.