A course on set theory
Posted here is information about the book,
A course on set theory,
which was written by
Ernest Schimmerling and
published by Cambridge University Press in 2011.
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) = xy.
 Page 96: Exercise 5.12 is incorrectly stated;
a correct version will be provided here later.
 Page 114, line 6: n_{2} =
 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 1517:
"Instead of saying ultraproduct, we use the term ultrapower
in this case because all the pairs (A_{n},R_{n}) are the same."
 Pages 146147: Exercise 7.4 should be ignored, it is wrong.
Additional exercises

(Section 7.2)

Let < A_{α}  α < ω_{1} >
be a sequence such that
 for every α < ω_{1},
A_{α} is a finite subset of ω_{1}, and
 for all α < β < ω_{1},
A_{α} ≠ A_{β}.

Prove that there is a stationary subset I of ω_{1}
and a set R such that for every α ∈ I,
A_{α} ∩ α = R.
Hint: Use Fodor's lemma.

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).

Conclude that for all α < β from J,
A_{α} ∩ A_{β} = R.

(Δsystem 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.
 (Section 7.2)
Let R and S be isomorphic wellorderings of ω_{1}.
Let I be the collection of α < ω_{1} such that
R ∩ (α x α) and S ∩ (α x α) are isomorphic.
 Prove that I contains a club.
 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 ...
 (Section 7.2)
Let θ be a limit ordinal and T = { θ  C  C is a closed subset of
θ }.
 Prove that T is a topology on θ
 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.
 Prove that every open set is a union of open intervals.
 Prove that the topology is not compact.
 (Chapter 4) Assume the Continuum Hypothesis.
That is, 2^{ω} = ω_{1}.
Prove that there is a family F of subsets of ω_{1} 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.
 (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 ^{<ω}2 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 ^{ω}2 to A.
 (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.
 (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.
 Let f be a function from the Baire Space to itself.
Prove that the following are equivalent.
 f is continuous.
 f^{1}[N_{s}] is open for every basic open set N_{s}.
 f^{1}[C] is closed for every closed set C.
 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).
 Prove that the identity function is a continuous injection from the
Cantor Space to the Baire Space.
 Let A = { x ∈ ^{ω}2  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).
 (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].
 Prove that if B is not ^{ω}ω,
then ∅ <_{W} B.
 Prove that if B is not ∅,
then ^{ω}ω <_{W} B.
 Prove that ∅ and ^{ω}ω are <_{W} incomparable.
That is, neither ∅ <_{W} ^{ω}ω nor ^{ω}ω <_{W} ∅.
 Prove that if A is clopen and B is neither ∅
nor ^{ω}ω,
then A <_{W} B.
 Prove that if A is open and B is not closed, then A <_{W} B.
 Prove that if A is closed and B is not open, then A <_{W} B.
 Prove that if A is closed but not open and
B is open but not closed, then A and B are <_{W}incomparable.