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 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."
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.