MA 3205 – Set Theory – Homework due Week 13

MA 3205 – Set Theory – Homework due Week 13
Frank Stephan, fstephan@comp.nus.edu.sg, 6516-2759, Room S14#04-13.
Homework. The homework follows the lecture notes. You have to hand in at least
three starred homeworks throughout the semester. Further homework can be checked
on request. Homework to be marked should be handed in after the lecture on Tuesday
of the week when the homework is due.
Exercise 17.12. Consider the following partial ordering given on the set NN of all
functions from N to N:
f @ g ⇔ ∃n ∀m > n (f (m) < g(m)).
This partial ordering only shares some but not all of the properties of the ordering
<lin considered in the lecture. In order to see this, show the following two properties:
• For countably many functions f0 , f1 , . . . there is a function g such that ∀n ∈
N (fn @ g);
• There are uncountably many f below the exponential
function n 7→ 2n . Namely
P
n−m−1
for every A ⊆ N the function cA : n 7→
· A(m) is below the
m∈n 2
exponential function.
Note that cA @ cB ⇔ A <lex B. Thus there is an uncountable linearly ordered set of
functions below the exponential function.
Exercise 17.13∗ . Use the Axiom of S
Choice to prove the following: If |A| = ℵ1 and
every B ∈ A satisfies |B| ≤ ℵ1 then | A| ≤ ℵ1 .
Exercise 18.9. Verify that Hausdorff’s axioms are true for the set R. That is, verify
that (R, {A ⊆ R | A is open}) is a Hausdorff space.
Exercise 18.10. Let α be any ordinal. Define a topology on α by saying that β is
open iff β is an ordinal and β ⊆ α. Verify that the first three axioms of Hausdorff are
satisfied, but not the last fourth one.
Exercise 18.11. Find a topology on the set of ordinals up to a given ordinal α which
satisfies the Axioms of Hausdorff and in which an ordinal β ∈ α is isolated iff it is
either a successor ordinal or 0.
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