Set theory, Spring 2015
Homework 7 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Schimmerling 4.6). Prove that, for every ordinal , there is a
cardinal > such that cf() = and = .
Lets rst forget about the conality requireme

Set theory, Spring 2015
Homework 8 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (*). Suppose that (X, d) is a separable metric space. Show that
|X| 20 .
Unsurprisingly, I will follow my own hint. Fix a countable dense D X,

Set theory, Spring 2015
Homework 12 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1. Suppose that I has a winning strategy for the game G . Show
A
that 2 injects into A.
As suggested by the hint, we x a winning strategy for I

Set theory, Spring 2015
Homework 11 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1. (ZFC) Show that there exists A such that the game GA
is determined while the game G( \A) is not determined.
In class we constructed a set B s

Set theory, Spring 2015
Homework 10 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (*). This problem connects topological properties of sequence
spaces with the cardinality of the underlying set. Given a set X, we equip the
s

Set theory, Spring 2015
Homework 9 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1. This problem concerns the middle thirds Cantor set, which is
a special subset of R equipped with the usual metric d(x, y) = |x y|. Dene
recurs

Set theory, Spring 2015
Homework 13 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1. Suppose that (Q, ) is any partially ordered set. Show that
there is a set X and a set P P(X) such that there is a bijection : Q P
with q0
q1

Set theory, Spring 2015
Homework 2 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Schimmerling 2.8). In general, dene
A
B = cfw_f | f is a function from A to B.
Consider a function of the form (a, b) S(a,b) with domain A B.

Set theory, Spring 2015
Homework 5 Solutions
Clinton Conley
(Please contact me if you nd any errors!)
Problem 1 (Schimmerling 4.1). Let
1. Prove that
<
<
2=
n
n<
2.
2 is countable.
2. Let F = cfw_x n | n < | x 2. Prove that |F| = 20 .
3. Prove that there